# Elements of Hilbert Spaces and Operator Theory

Books on functional analysis and operator theory appear regularly, but they are often dealing with a special topic such as differential operators, integral equations, spectral theory, or particular classes of operators. Not so remarkable since the topic is relatively new and mainly developed during the previous century by some of the best: Hilbert, Banach, Riesz, von Neumann, etc. The subject has grown very rapidly. Many of the general introductions therefore date back to the twentieth century as well. Only few are published after 2000. Given that the subject is still growing and has become a standard tool in theory and applications from analytic number theory to dynamical systems, a new modern introduction such as this one is very welcome.

In its most elementary form engineers only need finite dimensional vector spaces, and if we restrict to linear operators, there is linear algebra to solve all the problems. However these problems are in many cases approximations for a phenomenon taking place in an infinite dimensional space, and when infinity is involved the mathematics become tricky and one needs good foundations to develop the proper theorems. Of course the subject is immense as is illustrated by the classic 3 volume set *Linear Operators* (1958-1971) by N. Dunford and J. Schwartz that contains over 2500 pages. So, it is a difficult balance to be kept between an encyclopedic work and an introduction that is both complete in the theoretical foundations and yet has attention for the applicability. This requires a lot of craftsmanship that is achieved remarkably well in this book.

The author has made a selection in the massive amount of material available to provide a very readable introduction to the subject. To study the operators, one first needs to understand the spaces on which they operate. These are in practice mostly Hilbert spaces and Banach spaces. Then the linear operators can be defined and their spectral analysis can be studied. This defines the structure of the book shaped by the 5 chapters: a short general introduction recalling the necessary preliminaries, then inner product spaces, the linear operators, their spectral theory, and finally the Banach spaces. There is an extra chapter with hints and solutions to the many exercises that are amply sprinkled throughout the text.

The emphasis is definitely on linear operators on Hilbert spaces and the spectral analysis of special classes of operators. So a first target is to introduce inner product spaces, that is the spaces of $L^2$ type, so that one can talk about orthogonality of a basis, define projections and discuss approximations. The most important operators are the normal, unitary, and isometric operators. The special classes for which the spectral analysis is studied in more detail include compact operators, trace class, self adjoint, and Hilbert-Schmidt operators. In the Banach space chapter, topics include the Hahn-Banach theorem, the Baire category theorem, and the open mapping and closed graph theorems. There is also a section on unbounded operators and at some point also invariant subspaces are discussed.

The style is the typical mathematical approach of definition-theorem-proof kind of sequencing. However this is made lightly digestible by including many examples, remarks, and illustrations of what these formal definitions or theorems mean in practice and what the applications can be. Thus, although this is a quite mathematical subject, I think also engineers of a more theoretical kind will certainly appreciate this book very much. Among the applications we can mention Fourier analysis, orthogonal polynomials, approximation and convergence, Müntz theorem, Browder fixed point theorem, the mean ergodic theorem, numerical range, and much more. All the proofs are fully worked out. Only few theorems are mentioned without a proof if it is really too long and complicated and thus beyond the scope of these notes. Of course such a book cannot be read without an appropriate preparation which should include analysis and linear algebra. Most sections are followed by a set of exercises. These include often applications and examples, or ask to prove some extra properties. The level of difficulty nicely matches the level of the text. To assimilate the material, one should solve at least some of these exercises. As mentioned above, some 100 pages with hints and solutions are summarized in the last chapter.

To conclude, I think this is a marvelous introduction to the topic. Certainly applied mathematicians and engineers who need a stronger mathematical background, or for mathematicians with an interest in applications, will appreciate this most. Obviously it can be used, or at least parts of it, as a perfect set of lecture notes for a course on the subject. Note however, that it is a general introduction. Let me give two examples of what it is *not*. It does *not* discuss in any detail the solution of differential or integral equations (Sobolev spaces are too specialized and out of the picture). Neither is there a direct link to the extensive literature on systems theory. The books of the Birkhäuser series on *Operator Theory Advances and Applications* for example are much more specialized. However this book introduces the preliminaries to engage in all these topics, just because it is paying special attention to the operators that are most common in applications such as these.

**Submitted by Adhemar Bultheel |

**20 / Apr / 2017