Linear Operators in Spaces with an Indefinite Metric (Pure and Applied Mathematics: A Wiley Series of Texts, Monographs and Tracts)

✍ Scribed by T. Ya. Azizov, I. S. Iokhvidov

• Publisher: Wiley
• Year: 1989
• Tongue: English
• Leaves: 315
• Edition: 1
• Category: Library

✦ Synopsis

A monograph on the fundamentals of the theory of linear operators in spaces with an indefinite metric. Introduces the geometry of spaces with indefinite metric, and the central topics of operator theory, and explores variations on the theme. The theory is set out in Krein and Pontryagin spaces, with passing reference made to generalizations to Banach spaces and simply normed spaces. Also considers the application of the geometry and operator theory to actual problems. Includes examples, exercises and problems.

✦ Table of Contents

Title Page
Copyright Page
Contents
Preface
1 The geometry of spaces with an indefinite metric
§1 Linear spaces with an Hermitian form
§2 Krein spaces (axiomatics)
§3 Canonical projectors P§and canonical symmetry J
§4 Semi-definite and definite lineals and subspaces in a Krein space
§5 Uniformly definite (regular) lineals and subspaces. Subspaces of the classes h§
§6 Decomposability of lineals and subspaces of a Krein space. The Gram operator of a subspace. W-spaces and G-spaces
§7 J-orthogonal complements and projections. Projectional completeness
§8 The method of angular operators
§9 Pontryagin spaces II. W-spaces and G()- spaces
§10 Dual pairs. J-orthonormalized systems and bases
Remarks and bibliographical indications on Chapter 1
2 Fundamental classes of operators in spaces with an indefinite metric
§1 The adjoint operator T`
§2 Dissipative operators
§3 Hermitian, symmetric, and self-adjoint operators
§4 Plus-operators, J-non-contractive and J-bi-non-contractive operators
§5 Isometric, semi-unitary, and unitary operators
§6 The Cayley-Neyman transformation
Remarks and bibliographical indications on Chapter 2
3 Invariant semi-definite subspaces
§1 Statement of the problems
§2 Invariant subspaces of a J-non-contractive operator
§3 Fixed points of linear-fractional transformations and invariant subspaces
§4 Invariant subspaces of a family of operators
§5 Operators of the classes H and K(H)
Remarks and bibliographical indications on Chapter 3
4 Spectral topics and some applications
§1 The spectral function
§2 Completeness and basicity of a system of root vectors of J- dissipative operators
§3 Examples and applications
Remarks and bibliographical indications on Chapter 4
5 Theory of extensions of isometric and symmetric operators in spaces with an indefinite metric
§1 Potapov-Ginzburg linear-fractional transformations and extensions of operators
§2 Extensions of standard isometric and symmetric operators
§3 Generalized resolvents of symmetric operators
Remarks and bibliographical indications on Chapter 5
References
Index