Includes Part 1, Number 2: Books and Pamphlets, Including Serials and Contributions to Periodicals July - December)

One of two volumes, this text combines distinct topics of modern analysis and its applications: Hardy classes of holomorphic functions; spectral theory of Hankel and Toeplitz operators. Each topic has important implications for complex analysis.

In the present paper the author studies the analogue of the class [italic capital]C0 within a class of operators having a functional calculus based on the algebra of bounded holomorphic functions in a finitely connected domain with an analytic boundary. The latter class consists of the operators having the closure of the domain as a spectral set and having no normal direct summands with spectra contained in the boundary of the domain. (If the domain is the disk the preceding class reduces to the class of completely nonunitary contractions.) The basic properties known for the case of the disk, including the model theory, are established. The extension, even the mere construction of the functional calculus, is not routine, in part because it is unknown whether the analogue of Sz.-Nagy's dilation theorem is true in the author's multiply connected setting.

Boris Pavlov (1936-2016), to whom this volume is dedicated, was a prominent specialist in analysis, operator theory, and mathematical physics. As one of the most influential members of the St. Petersburg Mathematical School, he was one of the founders of the Leningrad School of Non-self-adjoint Operators. This volume collects research papers originating from two conferences that were organized in memory of Boris Pavlov: “Spectral Theory and Applications”, held in Stockholm, Sweden, in March 2016, and “Operator Theory, Analysis and Mathematical Physics – OTAMP2016” held at the Euler Institute in St. Petersburg, Russia, in August 2016. The volume also includes water-color paintings by Boris Pavlov, some personal photographs, as well as tributes from friends and colleagues.

"In a certain sense, subnormal operators were introduced too soon because the theory of function algebras and rational approximation was also in its infancy and could not be properly used to examine the class of operators. The progress in the last several years grew out of applying the results of rational approximation." from the Preface. This book is the successor to the author's 1981 book on the same subject. In addition to reflecting the great strides in the development of subnormal operator theory since the first book, the present work is oriented towards rational functions rather than polynomials. Although the book is a research monograph, it has many of the traits of a textbook including exercises. The book requires background in function theory and functional analysis, but is otherwise fairly self-contained. The first few chapters cover the basics about subnormal operator theory and present a study of analytic functions on the unit disk. Other topics included are: some results on hypernormal operators, an exposition of rational approximation interspersed with applications to operator theory, a study of weak-star rational approximation, a set of results that can be termed structure theorems for subnormal operators, and a proof that analytic bounded point evaluations exist.

Overall, this work combines together - in two volumes - four formally distinct topics of modern analysis and their applications: Hardy classes of holomorphic functions; spectral theory of Hankel and Toeplitz operators; function models for linear operators and free interpolations; and infinite-dimensional system theory and signal processing. This, the second volume, contains parts C and D of the whole.