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Dual algebras with applications to invariant subspaces and dilation theory. (English) Zbl 0569.47007

Regional Conference Series in Mathematics 56. Providence, RI: American Mathematical Society (AMS) (ISBN 0-8218-0706-4). xi, 108 p. (1985).
The theory of dual algebras has been developing since the appearance of Scott Brown’s poineering paper in 1978, which showed the utility of this concept for studying the structure theory of bounded linear operators on Hilbert spaces. This book is an expanded and revised version of the lecture notes concerning dual algebra from the NSF/CBMS regional conference held in Tempe, Arizona in May, 1984 which C. Pearcy, H. Bercovici and C. Foiaş lectured. The main purpose of this book is to present an approach to the study of nonselfadjoint dual algebras that allow one to obtain results on invariant subspaces and dilation theory.
Chapter I is devoted to present general natures concerning dual algebras. Most of the chapters II, III, IV and X are taken from the paper ”Invariant subspaces, dilation theory and the structure of the predual of a dual algebra” by C. Apostol and the authors which will be published in J. Funct. Anal.
Some part of the Chapters III and VI are new materials; for example, \(\cap^{\infty}_{n=1}A_ n({\mathcal H})=A_{\aleph_ 0}\). Chapter IV and V are taken from the paper ”Dilation theory and systems of simultaneous equations in the predual of an operator algebra”, written by the authors which was published in Mich. Math. J. 30, 335-354 (1983; Zbl 0541.47007)]. Chapter VIII comes from the papers ”A reflexivity theorem for weakly closed subspaces of operators” by H. Bercovici, published in Trans. Am. Math. Soc. 288, 139-146 (1985; review below), and ”Invariant subspaces, dilation theory, and the structure of the predual of a dual algebra” written by C. Apostol, and the authors, which will appear in Indiana Univ. Math. J. Chapter IX contains the materials in the paper ”On the reflexivity of algebras and linear spaces of operators” by the authors which are submitted to Mich. Math. J. Chapter X is devoted to the description of application for shifts and subnormal operators by using the results obtained from Chapter I through Chapter IX. At the end of this chapter the authors present the following conjecture:
If \(T\in A_{\aleph_ 0}({\mathcal H})\) and \(K\in {\mathcal C}_ 1({\mathcal H})\) is such that \(T+K\in A({\mathcal H})\), is it always true that \(T+K\in A_{\aleph_ 0}({\mathcal H})?\)
Reviewer: J.C.Rho

MSC:

47A15 Invariant subspaces of linear operators
47A20 Dilations, extensions, compressions of linear operators
47A53 (Semi-) Fredholm operators; index theories
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