# Indefinite Inner Product Spaces, Schur Analysis, and Differential Equations

This is volume 263 of the Birkhäuser series on *Operator Theory Advances and Applications*. It is devoted to Heinz Langer on the occasion of his eightieth birthday. Two other volumes in this series were celebrating Langer: *Contributions to operator theory in spaces with an indefinite metric* (OT106, 1995) on the occasion of his sixtieth birthday and *Operator theory and indefinite inner product spaces* (OT163, 2004) on the occasion of his retirement at the University of Vienna.

The titles of these three volumes already illustrate that operator theory in indefinite inner product spaces form the focus of Langer's research. Langer was born in Dresden in 1935. After his his PhD and his habilitation at the TH Dresden he became a professor leading the institute of probability and mathematical statistics. His stay in Odessa in 1968-69 where he met M.G. Krein strongly influenced his career and his research interests. He also spent research stays at several Western universities as well, which was not obvious in the time of the DDR. In 1969 he left East Germany permanently to become a professor in Dortmund, later in Regensburg, and finally, in 1991, he accepted a position in Vienna where he stayed until his retirement.

This is just a very brief summary, but Bernd Kirstein has a much longer, and richly illustrated contribution in this book. It is the ceremonial address on the occasion of the honorary doctorate awarded to Heinz Langer by the TU Dresden in 2016. He received many other prizes among which another Dr. h.c. from Stockholm University in 2015. Kirstein sketches in detail the people that were influential on Langer's career. Many of them became colleagues and friends. Among them are the most important names in the domain: Krein, Nudelman, Iokvidov, Potapov, Sakhnovich, Gohberg (who founded the OT series in 1979), Adamyan, Arov, Potapov, and many more. Kirstein describes this from his own perspective, hence the paper describes also the history of the Schur analysis group in Leipzig that he is leading together with his mathematical twin brother Bernd Fritzsche. Kirstein also illustrates the difficulties in maintaining relationships among mathematicians in an East block country and their colleagues who had left for Israel or another Western country before the fall of the Iron Curtain in 1989.

A list of the publications of Heinz Langer (op to January 2017) is also included in the biographical part I of this book. A similar list in OT163 in 2006 had 171 entries, while the current one has 203 (the last one from 2017) which illustrates that Heinz Langer at his age is still an active researcher and collaborator. And the latter is what the main content of this book really is: an illustration of the influence that Langer had on other people who worked on topics related to the subjects that are close to the heart of his own research, always prepared to listen and collaborate. These topics include nonlinear eigenvalue problems, indefinite inner product spaces such as Krein and Pontryagin spaces and applications in mathematical physics.

A collection of sixteen research papers, (some are longer surveys, others are short communications, all together over 420 pages) form the main part II of this volume. The titles of the papers and their authors are available on the publisher's website (see this book's meta-data elsewhere on this page) so that I do not need to repeat them here. The papers are listed in alphabetical order of the first author, but in their introduction, the editors subdivide them into five (overlapping) classes. The largest group falls under the broad title *Schur analysis, linear systems and related topics*. These papers are about Carthéodory and Weyl functions, Nevanlinna-Pick interpolation, scattering theory, L-systems and an inverse monodromy problem. In the group about *Differential operators, inverse problems and related topics* which is broad as well, we find papers related to the pantograph delay equation, and spectral and other properties for a selection of other operators. Two papers are explicitly dealing with *Pontryagin spaces* and one paper is about probability and is classified as *Non-commutative analysis*. *Positivity* is a keyword that can be assigned to almost all the papers in the volume, but it groups the remaining three texts where positivity has a key role.

This volume will of course be of interest to anyone who knows or collaborated with Heinz Langer, but more generally for anyone working in one of the topics that he was, and still is, interested in, and this is a broad field as illustrated by the papers in this volume. So it may be that not all the papers are interesting for a particular reader, but in that case there is of course also the possibility to download an electronic version of a particular paper from the publisher's website, like one would do for a journal paper.

**Submitted by Adhemar Bultheel |

**13 / Mar / 2018