Introduction to the Theory of Toeplitz Operators with Infinite Index

ISBN-13: 9783764367282 / Angielski / Twarda / 2002 / 300 str.

Berthold Wolfgang Schaaf; S. M. Grudsky; V. B. Dybin

Opis

This book is devoted to Toeplitz and singular integral operators with symbols that have discontinuities of the oscillating type. Criteria for the normal solvability of such operators are established and several methods for describing the kernel and image spaces of the operators are presented. The approach is based on the idea of modelling discontinuities with an "infinite index" by appropriate inner functions, especially by infinite Blaschke products. The corresponding techniques have been elaborated by the authors during the last two decades, and they are applicable to both symbols with slowly and rapidly increasing arguments. Moreover, the book reveals exciting connections between invariant subspaces of the shift operator, bases in Banach spaces, and various classes of entire and meromorphic functions. The book aims at making advanced topics accessible to a broad readership. It is addressed to graduate and postgraduate students and to mathematicians interested in functional analysis, the theory of functions of a complex variable, or mathematical physics.

Spis treści

1 Examples of Toeplitz Operators with Infinite Index Auxiliary material.- 1.1 The space Lp(?, ?) and the operator S?.- 1.2 The classes Lp± (?, ?).- 1.3 Normally solvable operators.- 1.4 Toeplitz operators.- Examples of operators with infinite index.- 1.5 Blaschke products.- 1.6 An elementary singular function.- 1.7 Boundary degeneracy.- References and comments.- 2 Factorization and Invertibility.- (p, ?)-factorization and (?-theory.- 2.1 The space Lp(?, ?) and the operator S?.- 2.2 Classes of bounded and continuous functions.- 2.3 The classes Lp± (?, ?).- 2.4 The class fact(p, ?).- 2.5 A sufficient condition for (p, ?)-factorizability.- Factorization and Toeplitz operators with infinite index.- 2.6 Inner-outer factorization.- 2.7 The class fact(?, p, ?) and one-sided invertibility.- 2.8 Examples of functions in fact(?, p, ?).- 2.9 The argument of a Blaschke product.- 2.10 The argument of an outer function.- References and comments.- 3 Model Subspaces Model operator and model subspaces.- 3.1 Model subspaces.- 3.2 Deformation of the contour.- 3.3 Model subspaces on ?.- 3.4 Boundary behavior.- Bases and interpolation in model subspaces.- 3.5 Bases.- 3.6 The Carleson condition and interpolation in Hp, ? (?±).- 3.7 Sine-type functions.- 3.8 Bases of ent?e functions.- 3.9 Bases of meromorphic functions.- 3.10 Boundary interpolation.- References and comments.- 4 Toeplitz Operators with Oscillating Symbols Almost periodic discontinuities.- 4.1 Uniformly almost periodic functions.- 4.2 Model subspaces on bounded smooth curves.- 4.3 Standard almost periodic discontinuities.- 4.4 Well-posed problems for the Toeplitz equation.- 4.5 General discontinuities of almost periodic type.- Semi-almost periodic discontinuities.- 4.6 The class SAP.- 4.7 Model function.- 4.8 Generalized factorization of SAP functions.- 4.9 Model subspaces.- Wh?l points of power type.- 4.10 Two-sided wh?ls.- 4.11 One-sided wh?ls.- References and comments.- 5 Generalized Factorization of u-periodic Functions and Matrix Functions.- 5.1 Block Toeplitz operators.- 5.2 Generalized factorization of matrix functions.- 5.3 u-periodic matrix functions.- 5.4 Infinite index of logarithmic type.- 5.5 Infinite index of arbitrary order.- 5.6 Sufficient conditions for the theorem on.- general oscillations. Examples.- 5.7 Slow oscillations.- 5.8 Modelling of oscillations.- 5.9 Generalized almost periodic discontinuities.- 5.10 Generalized matrix periodic discontinuities.- References and comments.- 6 Toeplitz Operators Whose Symbols Have Zeros.- The normalization principle.- 6.1 Normally solvable operators.- 6.2 Normalization of linear operators.- Normalization of Toeplitz operators.- 6.3 Symbols with polynomial degeneracy.- 6.4 Symbols with locally-polynomial degeneracy.- 6.5 Basic examples.- References and comments.- References.