# Operator Theory

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 *Operator Theory, Advances and Applications* 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.

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:

- Reproducing Kernel Hilbert Spaces (F.H. Szafraniec)
- Indefinite Inner Product Spaces (M. Langer, H. Woracek)
- de Branges Spaces (A. Baranov, H. Woracek)
- Linear Systems Theory (D. Alpay, M. Mboup)
- Multivariable Operator Theory (J.A. Ball)
- Infinite Dimensional Analysis (P.E.T. Jorgensen)
- General Aspects of Quaternionic and Clifford Analysis (F. Colombo, I. Sabadini, M. Shapiro)
- Further Developments of Quaternionic and Clifford Analysis (F. Colombo.I. Sabadini, M. Shapiro)

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.

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.

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.

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.

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.

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.

5. Multivariable operator theory is still being developed. Generalizations for the dilation theory of Sz.-Nagy are discussed, also Hilbert modules.

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.

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).

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.

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.

**Submitted by Adhemar Bultheel |

**16 / Nov / 2015