European Mathematical Society - 47 Operator theory
https://euro-math-soc.eu/msc/47-operator-theory
enThe moment problem
https://euro-math-soc.eu/review/moment-problem
<div class="field field-name-field-review-review field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><div class="tex2jax"><p>
The moment problem, or one should say moment problems (plural) because there are several different classical moment problems. Some ideas can be found in work of Chebyshev and Markov, but Stieltjes at the end of the nineteenth century was one of the first to formally consider the moment problem named after him. Given a sequence of numbers ($m_k$), is there a positive measure $\mu$ such that $m_k=\int x^k \mu(dx), k=0,1,2,\ldots$? In the case of Stieltjes, the measure was supposed to have a support on the positive real line. First of all one wants to find out under what conditions such a measure exists, then when the solution is unique, and when it is not unique to characterize all possible solutions. Soon (around 1920) other versions were formulated by Hamburger (when the support of the measure is the whole real line) and Hausdorff (when the support is a finite interval) and some ten years later the trigonometric moment problem was tackled by Verblunsky, Akhiezer and Krein where the support is the complex unit circle. There is a basic difference between the trigonometric moment problem and the other classical moment problems on (parts of) the real line. In the latter situation, the existence of a solution is guaranteed by requiring the positivity of Hankel matrices whose entries are the moments. In the trigonometric case, the Hankel matrices are replaced by Toeplitz matrices. The latter involve also the moments $m_{-k}=\overline{m}_k, k=1,2,\ldots$ which are automatically matched as well. Not so for the other moment problems. When also imposing moments with a negative index in those cases, this is called a <em>strong</em> moment problem. When only a finite number of moments are prescribed, this is called a <em>truncated</em> moment problem.</p>
<p>
The importance of the moment problem is a consequence of the fact that it is at the crossroads of several branches and applications of mathematics. It relates to linear algebra, functional analysis and operator theory, stochastic processes, approximation theory, optimization, orthogonal polynomials, systems theory, scattering theory, signal processing, probability, and many more. No wonder that the greatest names in mathematics have contributed to the problem with papers and monographs. Because of the many connections to different fields also many approaches and many generalizations have been considered. The previously described moments are called power moments because of the $x^k$, but one could also prescribe moments based on a set of other functions $M_k(x)$. Traditionally, the Hausdorff moment problem is formulated for the interval [0,1], but one may consider any finite interval $[a,b]$ just like the Stieltjes moment problem could be formulated for any half line $[\alpha,\infty)$. Other generalizations lifts these problems to a block version, by assuming that the moments are matrices and the measure is matrix-valued, or the variable $x$ can have several components, resulting in a multivariate moment problem.</p>
<p>
The fact that today, 100 years after Hamburger and Hausdorff, this is still an active research field is another proof of the importance of moment problems. Many books did appear already that were devoted to moment problems or where moment problems played an essential role. Some classics are Shohat and Tamarkin <em>The Problem of Moments</em> (1943), Akhiezer <em>The classical moment problem and some related questions in analysis</em> (1965), Krein and Nudelman <em>The Markov moment problem and extremal problems</em> (1977). The present book is a modern update of the situation. It gives an operator theoretic approach to moment problems, leaving aside the applications. The univariate classical problems of Hamburger ($\mathbb{R}$), Stieltjes ($[0,\infty)$) and Hausdorff ($[a,b]$), appear both in their full and their truncated version. Also the trigonometric moment problem is represented but by only one chapter.<br />
The introduction to these problems is quite general. It is showing how integral representations for linear functionals can be obtained, and in particular how this works for finite dimensional spaces, and for truncated moment problems. Another essential tool is giving some examples of how moment problems can be defined on a commutative *-semigroup. Indeed, all what is needed is a structure with an involution (which could be the identity) and it should allow the definition of a positive definite linear functional so that it can give rise to an inner product on the space of polynomials (and its completion). With gross oversimplification one could say that a sequence is a moment sequence if the associated linear functional is positive and the solution corresponds to the measure that appears in an integral representation of the functional. For real problems, the involution is the identity: $x^*=x$, for complex problems, the involution $x^*=1/\overline{x}$ allows to treat the trigonometric moment problem at the same level as the real moment problems.<br />
This general approach is not really needed for the classical one dimensional moment problems that are treated in part I and the truncated version in part II, but the generality of the introduction allows more easy generalizations to the multivariate case and its truncated version that are discussed in parts III and IV respectively. What is treated in the first two parts are the classical results: the representation of positive polynomials, conditions for the existence of a solution of the moment problem, Hankel matrices, orthogonal polynomials and the Jacobi operator, determinacy (i.e. uniqueness) of the solution, the characterization of all solutions in the indeterminate case, and the relation with complex interpolation problems for Pick functions. For truncated moment problems one may look for some special, so called N-extremal, solutions which lie on the boundary of the solution set, or a canonical solution or solutions that maximize the mass in a particular point of an atomic solution.</p>
<p>
For the multivariate case, it takes some more work and we do not have the classical cases where the measure should be supported and generalizations can go in many different directions. Nevertheless, the corresponding chapters in parts III and IV go through the same steps as in the univariate case as much as possible. What are representing measures and when are polynomials positive? By defining the moment problem for a finitely generated abelian unital algebra, and using a fiber theorem that characterizes moment functionals, some generalizations of the one-dimensional case can be obtained (like for example a rational moment problem) or moment problems on some cubics. Determinacy of the multivariate moment problem is given in the form of a generalized Carleman condition, moments for the Gaussian measure on the unit sphere, and complex one- and two-sided moment problems are all discussed. Characterizing a canonical or extreme solution(s) is not as simple as in the one-dimensional case. Only for the truncated multivariate problems Hankel matrices are introduced and atomic solutions with maximization of a point mass can be characterized.</p>
<p>
The book appears in the series <em>Graduate Texts in Mathematics</em> which means that it is conceived as a as a text that could be used for lecturing with proofs fully included and extra exercises after every chapter as well as notes the refer to the history and the related literature. It is however marvellously capturing the present state of the art of the topic. So it will be also a reference work for researchers. It captures a survey of the univariate case and indicates research directions for the multivariate problem. The list of references at the end of the book has both historical as recent publications, but it is restricted to what has been discussed in the present book. Schmüdgen has published two books before on operator theory, so he knows how to write a book on a difficult subject and still keep it accessible for the audience that he is addressing (graduate students and researchers). Lists of symbols are really helpful to remember notation. The fact that on page 4 Chebyshev and Markov are situated in 1974 and 1984 respectively is just a glitch in an otherwise carefully edited text.</p>
</div></div></div></div><div class="field field-name-field-review-reviewer field-type-text field-label-inline clearfix"><div class="field-label">Reviewer: </div><div class="field-items"><div class="field-item even">Adhemar Bultheel</div></div></div><div class="field field-name-field-review-desc field-type-text-long field-label-hidden"><div class="field-items"><div class="field-item even"><div class="tex2jax"><p>
This is a modern operator approach surveying classical one-dimensional moment problems, but the setting is general by formulating the problem on an abelian *-semigroups. This allows to also capture an introduction to multivariate moment problems which is much more recent and a subject that is still in evolution. The characterization of moment sequences, associated linear moment functionals, and determinate as well as indeterminate problems for the full or the truncated problems are discussed. Particular canonical and N-extremal solutions or solutions with a maximal mass point are discussed.</p>
</div></div></div></div><span class="vocabulary field field-name-field-review-author field-type-taxonomy-term-reference field-label-inline clearfix"><h2 class="field-label">Author: </h2><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/author/konrad-schm%C3%BCdgen" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Konrad Schmüdgen</a></li></ul></span><span class="vocabulary field field-name-field-review-publisher field-type-taxonomy-term-reference field-label-inline clearfix"><h2 class="field-label">Publisher: </h2><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/publisher/springer-internationa" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Springer Internationa</a></li></ul></span><div class="field field-name-field-review-pub field-type-number-integer field-label-inline clearfix"><div class="field-label">Published: </div><div class="field-items"><div class="field-item even">2017</div></div></div><div class="field field-name-field-review-isbn field-type-text field-label-inline clearfix"><div class="field-label">ISBN: </div><div class="field-items"><div class="field-item even">978-3-319-64545-2 (hbk); 978-3-319-64546-9 (ebk)</div></div></div><div class="field field-name-field-review-price field-type-text field-label-inline clearfix"><div class="field-label">Price: </div><div class="field-items"><div class="field-item even">84,79 € (hbk); 67,82 € (ebk)</div></div></div><div class="field field-name-field-review-pages field-type-number-integer field-label-inline clearfix"><div class="field-label">Pages: </div><div class="field-items"><div class="field-item even">535</div></div></div><span class="vocabulary field field-name-field-review-class field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/imu/analysis-and-its-applications" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Analysis and its Applications</a></li></ul></span><div class="field field-name-field-review-website field-type-text field-label-hidden"><div class="field-items"><div class="field-item even"><a href="http://www.springer.com/gp/book/9783319645452" title="Link to web page">http://www.springer.com/gp/book/9783319645452</a></div></div></div><span class="vocabulary field field-name-field-review-msc field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/msc/47-operator-theory" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">47 Operator theory</a></li></ul></span><span class="vocabulary field field-name-field-review-msc-full field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/msc-full/47a57" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">47A57</a></li></ul></span><span class="vocabulary field field-name-field-review-msc-other field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/msc-full/42a70" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">42A70</a></li><li class="vocabulary-links field-item odd"><a href="/msc-full/30e05" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">30e05</a></li><li class="vocabulary-links field-item even"><a href="/msc-full/44a60" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">44A60</a></li></ul></span>Tue, 13 Mar 2018 07:38:36 +0000Adhemar Bultheel48323 at https://euro-math-soc.euhttps://euro-math-soc.eu/review/moment-problem#commentsLarge Truncated Toeplitz Matrices, Toeplitz Operators, and Related Topics
https://euro-math-soc.eu/review/large-truncated-toeplitz-matrices-toeplitz-operators-and-related-topics
<div class="field field-name-field-review-review field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><div class="tex2jax"><p>
This book is a volume 259 in the Bikhäuser OT Series <em>Operator Theory Advances and Applications</em>. It contains 30 contributions celebrating Albert Böttcher's 60th birthday.</p>
<p>
Albert Böttcher is a professor of mathematics at the TU Chemnitz in Germany. His main research topic is functional analysis. At his 18th he won the silver medal at the International Math Olympiad in Moscow. He studied mathematics at the TU Karl-Marx-Stadt (now TU Chemnitz) and finished a PhD in 1984 entitled <em>The finite section method for the Wiener-Hopf integral operator</em> under supervision of V.B. Dybin at Rostov-on-Don State University in Russia (this was all while the Berlin wall was still up). Since then he stayed at the TU Chemnitz. At the time of writing he has (co)authored 9 books and over 220 papers. The complete list is in the beginning of the book but one may also consult his <a href="https://www-user.tu-chemnitz.de/~aboettch/" target="_blank">website</a> where he keeps his list of publications up to date.</p>
<p>
The contributions start with reminiscences and best wishes by friends, colleagues and students of Albrecht Böttcher. Besides personal recollections, there is some discussion of his work, some photographs and reproductions of slides he used in presentations to illustrate that he is not only an excellent mathematician but also a passionate teacher and lecturer.</p>
<p>
That leaves about 700 pages of original research papers all of which relate from far or near to subjects that Böttcher has worked on. The Toeplitz operators and Toeplitz matrices of the title are indeed well represented, but there are all the other "Related Topics" which are close to his work too. About fifty renowned authors are involved.</p>
<p>
The Toeplitz operator (and hence also its spectrum) is characterized by a function, which is called its symbol. It features in a multiplication or convolution in the definition of the operator. With respect to a standard monomial basis, Toeplitz operators are represented by (infinite) Toeplitz matrices that have constant entries along diagonals. Of course the spectral and other properties of truncations of the infinite matrices to large finite ones relate to corresponding properties of Toeplitz operators, and similarly it can be related to other operators such as convolution and Wiener-Hopf operators. These matrices and operators have applications in differential and integral equations, systems and control, signal processing, and many more. Depending on the application the symbol may get an interpretation of transfer function of a system, power spectrum or autocorrelation of a signal, the kernel of an integral equation, or just a weight function in a Hilbert space. So, Toeplitz matrices and operators are also related to numerical methods for solving functional equations after discretization. Or to orthogonal polynomials (on the unit circle), which then in turn links to (trigonometric) moment problems, quadrature, and approximation theory (on the unit circle, but in a similar way also to analogs on the real line).</p>
<p>
Obviously this is not the place to discuss every paper in detail. The table of contents is available on the publisher's website and for convenience the research papers are also listed below. From the titles you will recognize the papers on determinants and eigenvalues for Toeplitz matrices, in particular their asymptotic behaviour as their size goes to infinity. Of course circulant and Hankel operators and combinations of these as operators or matrices are not far off the central theme and they are thus also treated in some of the chapters. The majority of the papers present new results. Note that most of them are (functional) analysis. Only a few exceptions are more linear algebra or make a link to physics or explicitly discuss numerical aspects (see [14, 16, 18, 23, 25, 27] below).</p>
<p>
Some of the papers are quite long (more than 30 pages and some even up to 50 pages). They are basically true research papers, sometimes a bit more expository, but they are not of the introductory broad survey type. So this is not the book you should read to be introduced to the subject, but is is more a sketch of the state-of-the-art for who is already famiiar. The style of course depends on the authors, but the book is homogeneous because of the subjects that all somehow relate to Böttcher's work. These topics discussed here are also close to the core idea of this book series <em>Operators Theory Advances and Applications</em>, founded by Israel Gohberg as a complement to the journal <em>Integral Equations and Operator Theory</em>. Only one of Böttcher's books appeared in this series though (<em>Convolution Operators and Factorization of Almost Periodic Matrix Functions </em> (2002) authored with Yu. I. Karlovich, and I. M. Spitkovsky appeared as volume 131) but several of his books are with Springer / Birkhäuser. That these topics are still a main focus of research is illustrated by the successful annual IWOTA conferences (<em>International Workshop on Operator Theory and its Applications</em>), the proceedings of which are also published in this OT series. The IWOTA 2017 is organized by A. Böttcher, D. Potts and P. Stollmann at the TU Chemnitz.</p>
<p>
Thus for anyone interested in the general topics of this book series, this collection will be a worthy addition. For those who are more selective, there is of course still the possibility to get some separate chapters, which is the advantage of having it also available as an ebook.</p>
<p>
Here are the titles and authors of the research papers in this volume:</p>
<p>
<br />
7. <em>Asymptotics of Eigenvalues for Pentadiagonal Symmetric Toeplitz Matrices, </em> Barrera, M. (et al.), Pages 51-77<br />
8. <em>Echelon Type Canonical Forms in Upper Triangular Matrix Algebras, </em> Bart, H. (et al.), Pages 79-124<br />
9. <em>Asymptotic Formulas for Determinants of a Special Class of Toeplitz + Hankel Matrices, </em> Basor, E. (et al.), Pages 125-154<br />
10. <em>Generalization of the Brauer Theorem to Matrix Polynomials and Matrix Laurent Series, </em> Bini, D.A. (et al.), Pages 155-178<br />
11. <em>Eigenvalues of Hermitian Toeplitz Matrices Generated by Simple-loop Symbols with Relaxed Smoothness, </em> Bogoya, J.M. (et al.), Pages 179-212<br />
12. <em>On the Asymptotic Behavior of a Log Gas in the Bulk Scaling Limit in the Presence of a Varying External Potential II, </em> Bothner, T. (et al.), Pages 213-234<br />
13. <em>Useful Bounds on the Extreme Eigenvalues and Vectors of Matrices for Harper's Operators, </em> Bump, D. (et al.), Pages 235-265<br />
14. <em>Fast Inversion of Centrosymmetric Toeplitz-plus-Hankel Bezoutians, </em> Ehrhardt, T. (et al.), Pages 267-300<br />
15. <em>On Matrix-valued Stieltjes Functions with an Emphasis on Particular Subclasses, </em> Fritzsche, B. (et al.), Pages 301-352<br />
16. <em>The Theory of Generalized Locally Toeplitz Sequences: a Review, an Extension, and a Few Representative Applications, </em> Garoni, C. (et al.), Pages 353-394<br />
17. <em>The Bézout Equation on the Right Half-plane in a Wiener Space Setting, </em> Groenewald, G.J. (et al.), Pages 395-411<br />
18. <em>On a Collocation-quadrature Method for the Singular Integral Equation of the Notched Half-plane Problem, </em> Junghanns, P. (et al.), Pages 413-462<br />
19. <em>The Haseman Boundary Value Problem with Slowly Oscillating Coefficients and Shifts, </em> Karlovich, Yu.I., Pages 463-500<br />
20. <em>On the Norm of Linear Combinations of Projections and Some Characterizations of Hilbert Spaces, </em> Krupnik, N. (et al.), Pages 501-510<br />
21. <em>Pseudodifferential Operators in Weighted Hölder-Zygmund Spaces of Variable Smoothness, </em> Kryakvin, V. (et al.), Pages 511-531<br />
22. <em>Commutator Estimates Comprising the Frobenius Norm - Looking Back and Forth, </em> Lu, Zhiqin (et al.), Pages 533-559<br />
23. <em>Numerical Ranges of 4-by-4 Nilpotent Matrices: Flat Portions on the Boundary, </em> Militzer, E. (et al.), Pages 561-591<br />
24. <em>Traces on Operator Ideals and Related Linear Forms on Sequence Ideals (Part IV), </em> Pietsch, A., Pages 593-619<br />
25. <em>Error Estimates for the ESPRIT Algorithm, </em> Potts, D. (et al.), Pages 621-648<br />
26. <em>The Universal Algebra Generated by a Power Partial Isometry, </em> Roch, S., Pages 649-662<br />
27. <em>Norms, Condition Numbers and Pseudospectra of Convolution Type Operators on Intervals, </em> Seidel, M., Pages 663-680<br />
28. <em>Paired Operators in Asymmetric Space Setting, </em> Speck, F.-O., Pages 681-702<br />
29. <em>Natural Boundary for a Sum Involving Toeplitz Determinants, </em> Tracy, C.A. (et al.), Pages 703-718<br />
30. <em>A Riemann-Hilbert Approach to Filter Design, </em> Wegert, E., Pages 719-740</p>
</div></div></div></div><div class="field field-name-field-review-reviewer field-type-text field-label-inline clearfix"><div class="field-label">Reviewer: </div><div class="field-items"><div class="field-item even">Adhemar Bultheel</div></div></div><div class="field field-name-field-review-desc field-type-text-long field-label-hidden"><div class="field-items"><div class="field-item even"><div class="tex2jax"><p>
This is a collection of papers dedicated to Albrecht Böttcher's 60th birthday. The contributions are by friends, colleagues and students. After the impressive list of his publications, many of which dealing with asymptotics of Toeplitz and related operators, the book has some birthday addresses sketching Böttcher as a person and some of his work. The major part however consists of research papers written on invitation by specialists on topics related by far or near to the work of Böttcher. <br />
</p>
</div></div></div></div><span class="vocabulary field field-name-field-review-author field-type-taxonomy-term-reference field-label-inline clearfix"><h2 class="field-label">Author: </h2><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/author/dario-bini" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Dario A. Bini</a></li><li class="vocabulary-links field-item odd"><a href="/author/torsten-ehrhardt" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Torsten Ehrhardt</a></li><li class="vocabulary-links field-item even"><a href="/author/alexei-yu-karlovich" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Alexei Yu. Karlovich</a></li><li class="vocabulary-links field-item odd"><a href="/author/ilya-matvey-spitkovsky" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Ilya Matvey Spitkovsky</a></li><li class="vocabulary-links field-item even"><a href="/author/eds-1" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">(eds.)</a></li></ul></span><span class="vocabulary field field-name-field-review-publisher field-type-taxonomy-term-reference field-label-inline clearfix"><h2 class="field-label">Publisher: </h2><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/publisher/birkh%C3%A4user-basel" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">birkhäuser basel</a></li></ul></span><div class="field field-name-field-review-pub field-type-number-integer field-label-inline clearfix"><div class="field-label">Published: </div><div class="field-items"><div class="field-item even">2017</div></div></div><div class="field field-name-field-review-isbn field-type-text field-label-inline clearfix"><div class="field-label">ISBN: </div><div class="field-items"><div class="field-item even">978-3-319-49180-6 (hbk)</div></div></div><div class="field field-name-field-review-price field-type-text field-label-inline clearfix"><div class="field-label">Price: </div><div class="field-items"><div class="field-item even">174,89 € (hbk)</div></div></div><div class="field field-name-field-review-pages field-type-number-integer field-label-inline clearfix"><div class="field-label">Pages: </div><div class="field-items"><div class="field-item even">766</div></div></div><span class="vocabulary field field-name-field-review-class field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/imu/analysis-and-its-applications" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Analysis and its Applications</a></li></ul></span><div class="field field-name-field-review-website field-type-text field-label-hidden"><div class="field-items"><div class="field-item even"><a href="http://www.springer.com/gp/book/9783319491806" title="Link to web page">http://www.springer.com/gp/book/9783319491806</a></div></div></div><span class="vocabulary field field-name-field-review-msc field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/msc/47-operator-theory" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">47 Operator theory</a></li></ul></span><span class="vocabulary field field-name-field-review-msc-full field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/msc-full/47b35" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">47B35</a></li></ul></span><span class="vocabulary field field-name-field-review-msc-other field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/msc-full/30e05" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">30e05</a></li><li class="vocabulary-links field-item odd"><a href="/msc-full/45e10" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">45E10</a></li><li class="vocabulary-links field-item even"><a href="/msc-full/47a57" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">47A57</a></li><li class="vocabulary-links field-item odd"><a href="/msc-full/15b05" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">15B05</a></li><li class="vocabulary-links field-item even"><a href="/msc-full/65d15" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">65D15</a></li><li class="vocabulary-links field-item odd"><a href="/msc-full/65g50" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">65G50</a></li><li class="vocabulary-links field-item even"><a href="/msc-full/65j10" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">65J10</a></li></ul></span>Sat, 20 May 2017 12:07:31 +0000Adhemar Bultheel47678 at https://euro-math-soc.euhttps://euro-math-soc.eu/review/large-truncated-toeplitz-matrices-toeplitz-operators-and-related-topics#commentsOperator Theory
https://euro-math-soc.eu/review/operator-theory
<div class="field field-name-field-review-review field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><div class="tex2jax"><p>
Operator theory is a relatively young mathematical subject area that grew out of increasing abstraction in (linear) algebra and analysis. Still it has grown to become a very broad subject with a wide range of applications in different branches of mathematics but also in engineering, physics, etc. Many aspects and advances are covered in journals and in the book series <em>Operator Theory, Advances and Applications</em> by Birkhäuser. This book attempts to give a modern introduction to the subject by collecting a set of survey papers. Even by restricting the topics of this book to the mathematical aspects, and leaving out all the proofs, the nearly 2000 pages of this book barely suffice to cover everything.</p>
<p>
The contributions are grouped into 8 parts, each part with its own editors. The first four relate to complex function spaces with many ideas coming from systems theory and signal processing. They form the first volume in the printed version. The next two parts deal with multivariate and infinite dimensional analysis and the last two with the case where the complex variables are replaced by quaternions or by a Clifford algebra. The different parts and their editors are:</p>
<ol>
<li>
Reproducing Kernel Hilbert Spaces (F.H. Szafraniec)</li>
<li>
Indefinite Inner Product Spaces (M. Langer, H. Woracek)</li>
<li>
de Branges Spaces (A. Baranov, H. Woracek)</li>
<li>
Linear Systems Theory (D. Alpay, M. Mboup)</li>
<li>
Multivariable Operator Theory (J.A. Ball)</li>
<li>
Infinite Dimensional Analysis (P.E.T. Jorgensen)</li>
<li>
General Aspects of Quaternionic and Clifford Analysis (F. Colombo, I. Sabadini, M. Shapiro)</li>
<li>
Further Developments of Quaternionic and Clifford Analysis (F. Colombo.I. Sabadini, M. Shapiro)</li>
</ol>
<p>
</p>
<p>
It is impossible to strictly separate the parts and there is always some overlap in the fuzzy boundary region. Whether or not you consider it an advantage or a disadvantage of the electronic version of the book that the links can bring you to the referenced sections with a mouse click or a tap of the finger is a matter of personal preference. Being survey papers, they do contain theorems, but the proofs are not included and there is usually an extensive list of references. Each part starts with a brief survey by the editor in principle followed by an introductory survey.</p>
<p>
Discussing the 64 papers separately would lead us too far. We give only a telegraphic survey with some namedropping so that one gets an idea of the topics that were discussed. It should illustrate that this is quite different from a classical textbook on operator theory and functional analysis.</p>
<p>
1. The reproducing Hilbert spaces (RKHS) start with an introduction that is a translation of the editor's Polish book. There are also the applications for Nevanlinna-Pick interpolation, Bergman kernels. sampling theory, wavelets and coherent states. The RKHS are fundamental and they also show up in other parts. That is in fact a general observation as we noted above.</p>
<p>
2. In indefinite inner product spaces (in particular Kreĭn and Pontryagin spaces) it needs some adaptation for the classical definitions (e.g. symmetric, isometric, selfadjoint). Furthermore one encounters contractions and commutant lifting, definitizable operators and their spectrum, the Nevanlinna-Pick problem and Schur analysis returns with generalizations and also differential equations and indefinite Hamiltonians, and in the finite dimensional case applications in numerical analysis occur with e.g. Riccati equations.</p>
<p>
3. The de Branges spaces are used in studying entire functions and are surveyed here with some of the applications and generalizations (e.g. canonical systems, moment problems,...) but this part also discusses de Branges-Rovnyak spaces where contraction operators in Hilbert spaces are the main study objects.</p>
<p>
4. Linear systems developed a strong symbiosis with operator theory at an early stage and they have mutually influenced each other. So this part is rightfully included in this volume. There is the realization theory of operators, but also time-frequency analysis, coding theory, optimal control, and semi- and quasi-separable systems, a topic recently of focussed interest in (numerical) linear algebra.</p>
<p>
5. Multivariable operator theory is still being developed. Generalizations for the dilation theory of Sz.-Nagy are discussed, also Hilbert modules.</p>
<p>
6. In the part on infinite dimensional systems we experience again the intimate relation of operator theory with systems theory and signal processing. we encounter here multiresolution analysis, harmonic analysis, Lie algebras, Von Neumann algebras, and even number theory.</p>
<p>
7-8. Finally in the last two parts, quaternionic and Clifford analysis come to full expansion. This is a very abstract subject but with surprisingly nice practical applications (e.g. in boundary value problems, orthogonal polynomials or wavelets).</p>
<p>
Most papers are extensive survey papers, only few are relatively short. The overall impression is one of uniformity both in style of writing and the way the theory is presented. This is quite an achievement, knowing that there were 71 contributing authors all writing on related but different topics. For a volume of this size, the index at the end is surprisingly short (only 7 pages). There are some typos but never affecting the overall message. A few examples: page 19, a sentence ending abruptly with `is related to Szaf...'; on page 23 there is a `i' between two formulas where it should be `and'; page 87 `RHKS' instead of `RKHS'; page 1218 there is twice \$H\$ which is probably not intended.</p>
<p>
I do not think this is a book that one would like to read as a whole. In fact there is no necessary order in which one should read the different parts. It is like bundling eight books into these two volumes. There is some introductory chapter in every part, but even there one could just pick the chapter that one is interested in, independently of the others. Each of them is high standard and obviously written by experts. Although I had no access to a printed version, I believe that the electronic one is the most easy to use and to be preferred over the printed one. It is also the most flexible in modifying and adapting as new findings come along since it should be clear from this book that operator theory is still in full expansion and updates may be needed in a not too distant future.</p>
</div></div></div></div><div class="field field-name-field-review-reviewer field-type-text field-label-inline clearfix"><div class="field-label">Reviewer: </div><div class="field-items"><div class="field-item even">Adhemar Bultheel</div></div></div><div class="field field-name-field-review-desc field-type-text-long field-label-hidden"><div class="field-items"><div class="field-item even"><div class="tex2jax"><p>
This is a massive volume that collects 64 survey papers on operator theory and its applications. Each one can in principle be read independently. The papers are grouped in eight parts ranging from reproducing kernel Hilbert spaces over linear systems to Clifford analysis.</p>
</div></div></div></div><span class="vocabulary field field-name-field-review-author field-type-taxonomy-term-reference field-label-inline clearfix"><h2 class="field-label">Author: </h2><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/author/daniel-alpay" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">daniel alpay</a></li><li class="vocabulary-links field-item odd"><a href="/author/ed-1" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">(ed.)</a></li></ul></span><span class="vocabulary field field-name-field-review-publisher field-type-taxonomy-term-reference field-label-inline clearfix"><h2 class="field-label">Publisher: </h2><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/publisher/springer-verlag-0" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">springer verlag</a></li></ul></span><div class="field field-name-field-review-pub field-type-number-integer field-label-inline clearfix"><div class="field-label">Published: </div><div class="field-items"><div class="field-item even">2015</div></div></div><div class="field field-name-field-review-isbn field-type-text field-label-inline clearfix"><div class="field-label">ISBN: </div><div class="field-items"><div class="field-item even">978-3-0348-0666-4 (hbk) 978-3-0348-0667-1 (ebk)</div></div></div><div class="field field-name-field-review-price field-type-text field-label-inline clearfix"><div class="field-label">Price: </div><div class="field-items"><div class="field-item even">793,94 € (hbk) 906,29 € (ebk)</div></div></div><div class="field field-name-field-review-pages field-type-number-integer field-label-inline clearfix"><div class="field-label">Pages: </div><div class="field-items"><div class="field-item even">1851</div></div></div><span class="vocabulary field field-name-field-review-class field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/imu/analysis-and-its-applications" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Analysis and its Applications</a></li></ul></span><div class="field field-name-field-review-website field-type-text field-label-hidden"><div class="field-items"><div class="field-item even"><a href="http://www.springer.com/gp/book/9783034806664" title="Link to web page">http://www.springer.com/gp/book/9783034806664</a></div></div></div><span class="vocabulary field field-name-field-review-msc field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/msc/47-operator-theory" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">47 Operator theory</a></li></ul></span><span class="vocabulary field field-name-field-review-msc-full field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/msc-full/47-02" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">47-02</a></li></ul></span><span class="vocabulary field field-name-field-review-msc-other field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/msc-full/93a99" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">93A99</a></li></ul></span>Mon, 16 Nov 2015 09:24:22 +0000Adhemar Bultheel46527 at https://euro-math-soc.euhttps://euro-math-soc.eu/review/operator-theory#commentsA Sequence of Problems on Semigroups
https://euro-math-soc.eu/review/sequence-problems-semigroups
<div class="field field-name-field-review-review field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><div class="tex2jax"><p>This is an interesting problem book in the area of one-parameter semi-groups of operators written by an expert in the field. The book contains more that 400 questions and problems where neither the proofs nor the solutions are presented. They are distributed along 24 sections. Starting from elementary sections it develops many different topics in the area like linear and non-linear operator semigroups, local semigroups, Lie generators...<br />
The book is strongly recommendable for a motivation and wide immersion in this important area.</p>
</div></div></div></div><div class="field field-name-field-review-reviewer field-type-text field-label-inline clearfix"><div class="field-label">Reviewer: </div><div class="field-items"><div class="field-item even">Francisco L. Hernandez</div></div></div><div class="field field-name-field-review-legacy-affiliation field-type-text field-label-inline clearfix"><div class="field-label">Affiliation: </div><div class="field-items"><div class="field-item even">Universidad Complutense Madrid</div></div></div><div class="field field-name-field-review-appendix field-type-file field-label-hidden"><div class="field-items"><div class="field-item even"><span class="file"><img class="file-icon" alt="PDF icon" title="application/pdf" src="/modules/file/icons/application-pdf.png" /> <a href="https://euro-math-soc.eu/sites/default/files/book-review/Neubergerbook.pdf" type="application/pdf; length=27904">Neubergerbook.pdf</a></span></div></div></div><span class="vocabulary field field-name-field-review-author field-type-taxonomy-term-reference field-label-inline clearfix"><h2 class="field-label">Author: </h2><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/author/jb-neuberger" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">J.B. Neuberger</a></li></ul></span><span class="vocabulary field field-name-field-review-publisher field-type-taxonomy-term-reference field-label-inline clearfix"><h2 class="field-label">Publisher: </h2><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/publisher/springer" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">springer</a></li></ul></span><div class="field field-name-field-review-pub field-type-number-integer field-label-inline clearfix"><div class="field-label">Published: </div><div class="field-items"><div class="field-item even">2011</div></div></div><div class="field field-name-field-review-isbn field-type-text field-label-inline clearfix"><div class="field-label">ISBN: </div><div class="field-items"><div class="field-item even">978-1-4614-0430-9</div></div></div><div class="field field-name-field-review-pages field-type-number-integer field-label-inline clearfix"><div class="field-label">Pages: </div><div class="field-items"><div class="field-item even">141</div></div></div><span class="vocabulary field field-name-field-review-class field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/imu/analysis-and-its-applications" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Analysis and its Applications</a></li></ul></span><span class="vocabulary field field-name-field-review-msc field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/msc/47-operator-theory" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">47 Operator theory</a></li></ul></span><span class="vocabulary field field-name-field-review-msc-full field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/msc-full/47-02" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">47-02</a></li></ul></span>Fri, 07 Aug 2015 08:11:18 +0000Francisco L. Hernandez46346 at https://euro-math-soc.euhttps://euro-math-soc.eu/review/sequence-problems-semigroups#commentsMorse Theoretic Aspects of p-Laplacian Type Operators
https://euro-math-soc.eu/review/morse-theoretic-aspects-p-laplacian-type-operators
<div class="field field-name-field-review-review field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><div class="tex2jax"><p>Standard Morse theory is a useful tool to compute homology groups of finite dimensional manifolds from the critical points of a sufficiently generic function. The tools of Morse theory can be developed in an infinite dimensional setting to some extent. This point of view can be applied to solve differential equations, by means of writing a Morse functional from a Banach vector space whose critical points are the solutions to the original equation. Each critical point has a degree and some cohomological information. This together with some standard tools of algebraic topology may allow finding restrictions to the number and properties of critical points.</p>
<p>The current book is devoted to apply infinite dimensional Morse theory to study the p-Laplacian and other quasilinear operators where the Euler functional is not defined on a Hilbert space or is not $C^2$ or where there are no eigenspaces to work with. The p-Laplacian operator arises in many applications such as non-Newtonian fluid flows and turbulent filtration in porous media. The p-Laplacian problem consists on finding the eigenfunctions of $\Delta_p u= \lambda |u|^{p-2} u$, where $\Delta_p u= div( |\nabla u|^{p-2} \nabla u)$. </p>
<p>The book is very technical, and it is only accessible to experts in the field. Even the introduction and overview is written in a technical language, assuming some knowledge with the notation. Chapters 2, 3 and 4 are more readable, containing background material. A large part of the rest of the book is of research level. Researchers in the area will find the book of interest, containing many results, and with a large bibliography.</p>
</div></div></div></div><div class="field field-name-field-review-reviewer field-type-text field-label-inline clearfix"><div class="field-label">Reviewer: </div><div class="field-items"><div class="field-item even">Vicente Muñoz</div></div></div><div class="field field-name-field-review-legacy-affiliation field-type-text field-label-inline clearfix"><div class="field-label">Affiliation: </div><div class="field-items"><div class="field-item even">UCM</div></div></div><div class="field field-name-field-review-desc field-type-text-long field-label-hidden"><div class="field-items"><div class="field-item even"><div class="tex2jax"><p>The purpose of this book is to present a Morse theoretic study of a very general class of homogeneous operators that includes the p-Laplacian as a special case. The p-Laplacian operator is a quasilinear differential operator that arises in many applications such as non-Newtonian fluid flows and turbulent filtration in porous media.</p>
</div></div></div></div><span class="vocabulary field field-name-field-review-author field-type-taxonomy-term-reference field-label-inline clearfix"><h2 class="field-label">Author: </h2><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/author/k-perera" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">k. perera</a></li><li class="vocabulary-links field-item odd"><a href="/author/rp-agarwal" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">r.p. agarwal</a></li><li class="vocabulary-links field-item even"><a href="/author/d-oregan" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">d. o'regan</a></li></ul></span><span class="vocabulary field field-name-field-review-publisher field-type-taxonomy-term-reference field-label-inline clearfix"><h2 class="field-label">Publisher: </h2><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/publisher/ams" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">ams</a></li></ul></span><div class="field field-name-field-review-pub field-type-number-integer field-label-inline clearfix"><div class="field-label">Published: </div><div class="field-items"><div class="field-item even">2011</div></div></div><div class="field field-name-field-review-isbn field-type-text field-label-inline clearfix"><div class="field-label">ISBN: </div><div class="field-items"><div class="field-item even">978-082184968</div></div></div><div class="field field-name-field-review-price field-type-text field-label-inline clearfix"><div class="field-label">Price: </div><div class="field-items"><div class="field-item even">$69</div></div></div><div class="field field-name-field-review-website field-type-text field-label-hidden"><div class="field-items"><div class="field-item even"><a href="http://www.ams.org/bookstore-getitem/item=SURV-161" title="Link to web page">http://www.ams.org/bookstore-getitem/item=SURV-161</a></div></div></div><span class="vocabulary field field-name-field-review-msc field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/msc/47-operator-theory" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">47 Operator theory</a></li></ul></span><span class="vocabulary field field-name-field-review-msc-full field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/msc-full/47j05" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">47j05</a></li></ul></span>Sat, 18 Feb 2012 13:15:58 +0000Anonymous45443 at https://euro-math-soc.euhttps://euro-math-soc.eu/review/morse-theoretic-aspects-p-laplacian-type-operators#commentsNonlinear Analysis
https://euro-math-soc.eu/review/nonlinear-analysis
<div class="field field-name-field-review-review field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><div class="tex2jax"><p>This book deals with several important topics in nonlinear analysis, presented both on their own and as a basic tool for solving a broad class of nonlinear problems. It includes problems arising in the theory of partial differential equations, in particular the theories of boundary value problems, control theory, and calculus of variations. The book is written as a self-contained textbook. The reader will be pleased to find, in a rather large appendix, all the basic facts on topology, measure theory and functional analysis. But even when reading the book from the beginning, the reader will find that the book can serve as a well-written textbook, providing the basic knowledge and containing material of a deeper level. </p>
<p>Successively, one learns about Hausdorff measures and capacity, covering theorems, Dini derivatives, area formulas, Lebesgue-Bochner and Sobolev spaces, vector valued integration, evolution triples needed for PDE theory, together with the standard inequalities and embedding theorems that are the core of the theory. The modern concepts of nonlinear operators and Young measures, also in the context of the Nemytskii operators, and the theory of superposed convergences are dealt with. The book will be valuable both for postgraduate students beginning their professional career in the field and for experts who are looking for a well-written and comprehensive handbook on the field.</p>
</div></div></div></div><div class="field field-name-field-review-reviewer field-type-text field-label-inline clearfix"><div class="field-label">Reviewer: </div><div class="field-items"><div class="field-item even">mrok</div></div></div><span class="vocabulary field field-name-field-review-author field-type-taxonomy-term-reference field-label-inline clearfix"><h2 class="field-label">Author: </h2><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/author/l-gasi%C5%84ski" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">l. gasiński</a></li><li class="vocabulary-links field-item odd"><a href="/author/ns-papageorgiou" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">n.s. papageorgiou</a></li></ul></span><span class="vocabulary field field-name-field-review-publisher field-type-taxonomy-term-reference field-label-inline clearfix"><h2 class="field-label">Publisher: </h2><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/publisher/chapman-hallcrc-boca-raton-series-mathematical-analysis-and-applications-vol-9" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">chapman & hall/crc, boca raton: series in mathematical analysis and applications, vol. 9</a></li></ul></span><div class="field field-name-field-review-pub field-type-number-integer field-label-inline clearfix"><div class="field-label">Published: </div><div class="field-items"><div class="field-item even">2005</div></div></div><div class="field field-name-field-review-isbn field-type-text field-label-inline clearfix"><div class="field-label">ISBN: </div><div class="field-items"><div class="field-item even">1-58488-484-3 </div></div></div><div class="field field-name-field-review-price field-type-text field-label-inline clearfix"><div class="field-label">Price: </div><div class="field-items"><div class="field-item even">USD 99,95</div></div></div><span class="vocabulary field field-name-field-review-msc field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/msc/47-operator-theory" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">47 Operator theory</a></li></ul></span>Sun, 23 Oct 2011 12:03:21 +0000Anonymous40031 at https://euro-math-soc.euTopological Degree Theory and Applications
https://euro-math-soc.eu/review/topological-degree-theory-and-applications
<div class="field field-name-field-review-review field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><div class="tex2jax"><p>After introducing the Brouwer degree theory in Rn, the authors consider the Leray-Schauder degree for compact mappings in normed spaces. A description of degree theory for condensing mappings (chapter 3) is followed by a chapter dedicated to studies of degree theory for A-proper mappings. The focus then turns to the construction of the Mawhin coincidence degree for L-compact mappings, degree theory for mappings of class (S+) and their perturbation with other monotone-type mappings. The last chapter is dedicated to fixed point index theory in a cone of a Banach space and presents a new fixed point index for countably condensing maps. Each chapter is accompanied by important applications illustrating the reason that it was necessary to change the previous concept of topological degree to a more general one. Many examples and exercises conclude each chapter. The book forms a good text for a self-study course or special topic courses and it is an important reference for anybody working in differential equations, analysis or topology. In summary, we can say that the book is an up-to-date exposition of the theory and applications of an important part of mathematics.</p>
</div></div></div></div><div class="field field-name-field-review-reviewer field-type-text field-label-inline clearfix"><div class="field-label">Reviewer: </div><div class="field-items"><div class="field-item even">oj</div></div></div><span class="vocabulary field field-name-field-review-author field-type-taxonomy-term-reference field-label-inline clearfix"><h2 class="field-label">Author: </h2><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/author/d-oregan" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">d. o'regan</a></li><li class="vocabulary-links field-item odd"><a href="/author/y-j-cho" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">y. j. cho</a></li><li class="vocabulary-links field-item even"><a href="/author/y-q-chen" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">y.-q. chen:</a></li></ul></span><span class="vocabulary field field-name-field-review-publisher field-type-taxonomy-term-reference field-label-inline clearfix"><h2 class="field-label">Publisher: </h2><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/publisher/chapman-hallcrc-boca-raton-series-mathematical-analysis-and-applications-vol-10" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">chapman & hall/crc, boca raton: series in mathematical analysis and applications, vol. 10</a></li></ul></span><div class="field field-name-field-review-pub field-type-number-integer field-label-inline clearfix"><div class="field-label">Published: </div><div class="field-items"><div class="field-item even">2006</div></div></div><div class="field field-name-field-review-isbn field-type-text field-label-inline clearfix"><div class="field-label">ISBN: </div><div class="field-items"><div class="field-item even">1-58488-648-X </div></div></div><div class="field field-name-field-review-price field-type-text field-label-inline clearfix"><div class="field-label">Price: </div><div class="field-items"><div class="field-item even">USD 79,95</div></div></div><span class="vocabulary field field-name-field-review-msc field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/msc/47-operator-theory" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">47 Operator theory</a></li></ul></span>Sat, 22 Oct 2011 18:47:16 +0000Anonymous40008 at https://euro-math-soc.euSmooth Homogeneous Structures in Operator Theory
https://euro-math-soc.eu/review/smooth-homogeneous-structures-operator-theory
<div class="field field-name-field-review-review field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><div class="tex2jax"><p>This book summarizes (and acquaints the reader with) some methods of differential geometry that are applied to functional analysis, mainly operator theory. Homogeneous spaces are the focus of interest, i.e. the orbits of group actions endowed with the structure of smooth manifolds. These spaces appear in many settings in the theory of operators and operator algebras. Only a few books have been written on this topic and the present one is very valuable, in part because it presents in terse and clear form the recent results that have previously only been available as journal articles. The author also raises new ideas, e.g. the investigation of operator ideals from the point of view of Lie theory.<br />
Part 1 of the book is an introduction to Lie theory in infinite dimensions. In part 2, geometry of homogeneous spaces is studied. In part 3, the orbits are presented as manifolds, where differential geometric structure carries a lot of operator theoretic information. Many questions concerning further development of this field are set forth. The author also suggests tools that can be used to approach the problems studied. The book is very well arranged. It brings new and fresh ideas and is therefore a challenge and encouragement to those interested in the field.</p>
</div></div></div></div><div class="field field-name-field-review-reviewer field-type-text field-label-inline clearfix"><div class="field-label">Reviewer: </div><div class="field-items"><div class="field-item even">jdr</div></div></div><span class="vocabulary field field-name-field-review-author field-type-taxonomy-term-reference field-label-inline clearfix"><h2 class="field-label">Author: </h2><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/author/d-beltita" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">d. beltita</a></li></ul></span><span class="vocabulary field field-name-field-review-publisher field-type-taxonomy-term-reference field-label-inline clearfix"><h2 class="field-label">Publisher: </h2><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/publisher/chapman-hallcrc-boca-raton-monographs-and-surveys-pure-and-applied-mathematics-vol-137" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">chapman & hall/crc, boca raton: monographs and surveys in pure and applied mathematics, vol. 137</a></li></ul></span><div class="field field-name-field-review-pub field-type-number-integer field-label-inline clearfix"><div class="field-label">Published: </div><div class="field-items"><div class="field-item even">2005</div></div></div><div class="field field-name-field-review-isbn field-type-text field-label-inline clearfix"><div class="field-label">ISBN: </div><div class="field-items"><div class="field-item even">1-58488-617-X</div></div></div><div class="field field-name-field-review-price field-type-text field-label-inline clearfix"><div class="field-label">Price: </div><div class="field-items"><div class="field-item even">USD 89,95</div></div></div><span class="vocabulary field field-name-field-review-msc field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/msc/47-operator-theory" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">47 Operator theory</a></li></ul></span>Fri, 21 Oct 2011 11:23:56 +0000Anonymous39901 at https://euro-math-soc.euNonlinear Evolution Equations
https://euro-math-soc.eu/review/nonlinear-evolution-equations
<div class="field field-name-field-review-review field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><div class="tex2jax"><p>This book explains several methods for investigation of the existence of solutions of nonlinear evolution equations and asymptotic properties of global solutions. The first chapter has introductory outlines and contains also a list of important evolution equations. Linear contraction semigroups are briefly introduced in chapter 2 and their properties are used in proving the existence results for semilinear equations. The compactness method and the method of monotone operators are described in chapter 3. It is shown in chapter 4 how the comparison principle can be used for the convergence of monotone iterations. Construction of invariant regions is also explained in this section. The methods given in chapters 2 -- 4 generally yield solutions that can blow up in finite time. The problem of the existence of small global solutions is examined in chapter 5. In particular, a priori estimates in Lp-norms are presented here. Chapter 6 is devoted to the convergence of global solutions to stationary ones and the existence of global attractors. Since the book concentrates on the main features of describing methods and leaves out various technical generalizations it is readable and is recommended mainly to graduate students in various fields of nonlinear science with a good background in mathematics.</p>
</div></div></div></div><div class="field field-name-field-review-reviewer field-type-text field-label-inline clearfix"><div class="field-label">Reviewer: </div><div class="field-items"><div class="field-item even">jmil</div></div></div><span class="vocabulary field field-name-field-review-author field-type-taxonomy-term-reference field-label-inline clearfix"><h2 class="field-label">Author: </h2><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/author/s-zheng" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">s. zheng</a></li></ul></span><span class="vocabulary field field-name-field-review-publisher field-type-taxonomy-term-reference field-label-inline clearfix"><h2 class="field-label">Publisher: </h2><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/publisher/chapman-hallcrc-boca-raton-monographs-and-surveys-pure-and-applied-mathematics-vol-133" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">chapman hall/crc, boca raton: monographs and surveys in pure and applied mathematics, vol. 133</a></li></ul></span><div class="field field-name-field-review-pub field-type-number-integer field-label-inline clearfix"><div class="field-label">Published: </div><div class="field-items"><div class="field-item even">2004</div></div></div><div class="field field-name-field-review-isbn field-type-text field-label-inline clearfix"><div class="field-label">ISBN: </div><div class="field-items"><div class="field-item even">1-58488-452-5</div></div></div><div class="field field-name-field-review-price field-type-text field-label-inline clearfix"><div class="field-label">Price: </div><div class="field-items"><div class="field-item even">USD 99,95</div></div></div><span class="vocabulary field field-name-field-review-msc field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/msc/47-operator-theory" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">47 Operator theory</a></li></ul></span>Mon, 03 Oct 2011 14:26:38 +0000Anonymous39894 at https://euro-math-soc.euEquivariant Degree Theory
https://euro-math-soc.eu/review/equivariant-degree-theory
<div class="field field-name-field-review-review field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><div class="tex2jax"><p>The aim of the book is the development and applications of the degree theory in the context of equivariant maps. (Equivariant simply means that the mapping has certain symmetries, e.g., being even/odd, periodic, rotational invariant, etc.). The theory is developed both in finite and infinite dimension. The first chapter gives necessary preliminaries. The second chapter brings the definition of the degree and studies its basic properties. As the definition is somewhat abstract (the degree is defined as an element of the group of equivariant homotopy classes of maps between two spheres), it is useful to compute the degree in various particular cases. This is accomplished in Chapter 3. The last and also the longest chapter, deals with applications to particular ODE’s and to bifurcation theory. The aim of the authors was to write a book that would be easily accessible even to non-specialists, thus the exposition is accompanied by a number of examples and the use of abstract special tools is limited. It is also worth noting that each chapter is accompanied by detailed bibliographical remarks.</p>
</div></div></div></div><div class="field field-name-field-review-reviewer field-type-text field-label-inline clearfix"><div class="field-label">Reviewer: </div><div class="field-items"><div class="field-item even">dpr</div></div></div><span class="vocabulary field field-name-field-review-author field-type-taxonomy-term-reference field-label-inline clearfix"><h2 class="field-label">Author: </h2><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/author/j-ize" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">j. ize</a></li><li class="vocabulary-links field-item odd"><a href="/author/vignoli" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">a. vignoli</a></li></ul></span><span class="vocabulary field field-name-field-review-publisher field-type-taxonomy-term-reference field-label-inline clearfix"><h2 class="field-label">Publisher: </h2><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/publisher/walter-de-gruyter" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">walter de gruyter</a></li></ul></span><div class="field field-name-field-review-pub field-type-number-integer field-label-inline clearfix"><div class="field-label">Published: </div><div class="field-items"><div class="field-item even">2003</div></div></div><div class="field field-name-field-review-isbn field-type-text field-label-inline clearfix"><div class="field-label">ISBN: </div><div class="field-items"><div class="field-item even">3-11-017550-9</div></div></div><div class="field field-name-field-review-price field-type-text field-label-inline clearfix"><div class="field-label">Price: </div><div class="field-items"><div class="field-item even">€98</div></div></div><span class="vocabulary field field-name-field-review-msc field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/msc/47-operator-theory" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">47 Operator theory</a></li></ul></span>Mon, 12 Sep 2011 13:49:22 +0000Anonymous39675 at https://euro-math-soc.eu