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**Invitation to linear operators. From matrices to bounded linear operators on a Hilbert space.**
*(English)*
Zbl 1029.47001

London: Taylor and Francis. x, 255 p. (2001).

The book under review deals with bounded linear operators on Hilbert space. Emphasis is put on properties of operators that are already present on matrices, and most of the proofs are given using elementary tools.

There is a basic introduction to Hilbert spaces and their bounded linear operators, which occupies the first half of the book. The second half is devoted to “further developments on linear operators” which surveys (in detail) a big part of the author’s work on the subject, together with related facts. The exact topics are mentioned below, but they could be summarized as “operator inequalities and norm inequalities”. In these areas the book is quite comprehensive.

Chapter I, “Hilbert Spaces”, provides some of the usual elementary facts from Hilbert spaces, with some emphasis (at the end) on finite-dimensional Hilbert spaces: inner product, parallelogram identity, orthogonal complements, Gram-Schmidt orthonormalization.

Chapter II, “Fundamental Properties of Bounded Linear Operators”, covers boundedness and continuity of linear operators, partial isometries, polar decomposition, elementary spectral theory, numerical range, and it ends with a more detailed coverage of several classes of non-normal operators (quasinormal, subnormal, hyponormal, paranormal, transaloid, normaloid, convexoid, spectraloid).

Chapter III, “Further development of bounded linear operators”, deals mainly with operator inequalities (Young, Hölder-McCarthy, Löwner-Heinz, Furuta, powers of \(p\)-hyponormal operators, Kantorovich, weighted mixed Schwarz, generalized Schwarz, Selberg, Heinz-Kato). There are also subsections on chaotic order and relative operator entropy, Aluthge transformation on \(p\)-hyponormal operators, and variations on paranormality.

There is a basic introduction to Hilbert spaces and their bounded linear operators, which occupies the first half of the book. The second half is devoted to “further developments on linear operators” which surveys (in detail) a big part of the author’s work on the subject, together with related facts. The exact topics are mentioned below, but they could be summarized as “operator inequalities and norm inequalities”. In these areas the book is quite comprehensive.

Chapter I, “Hilbert Spaces”, provides some of the usual elementary facts from Hilbert spaces, with some emphasis (at the end) on finite-dimensional Hilbert spaces: inner product, parallelogram identity, orthogonal complements, Gram-Schmidt orthonormalization.

Chapter II, “Fundamental Properties of Bounded Linear Operators”, covers boundedness and continuity of linear operators, partial isometries, polar decomposition, elementary spectral theory, numerical range, and it ends with a more detailed coverage of several classes of non-normal operators (quasinormal, subnormal, hyponormal, paranormal, transaloid, normaloid, convexoid, spectraloid).

Chapter III, “Further development of bounded linear operators”, deals mainly with operator inequalities (Young, Hölder-McCarthy, Löwner-Heinz, Furuta, powers of \(p\)-hyponormal operators, Kantorovich, weighted mixed Schwarz, generalized Schwarz, Selberg, Heinz-Kato). There are also subsections on chaotic order and relative operator entropy, Aluthge transformation on \(p\)-hyponormal operators, and variations on paranormality.

Reviewer: Martin Argerami (Regina)