Introduction to the Theory of Toeplitz Operators with Infinite Index

We offer the reader of this book some specimens of "infinity" that we seized from the "mathematical jungle" and trapped within the solid cage of analysis The creation of the theory of singular integral equations in the mid 20th century is associated with the names of N.1. Muskhelishvili, F.D. Gakhov, N.P. Vekua and their numerous students and followers and is marked by the fact that it relied principally on methods of complex analysis. In the early 1960s, the development of this theory received a powerful impulse from the ideas and methods of functional analysis that were then brought into the picture. Its modern architecture is due to a constellation of brilliant mathemati- cians and the scientific collectives that they produced (S.G. Mikhlin, M.G. Krein, B.V. Khvedelidze, 1. Gohberg, LB. Simonenko, A. Devinatz, H. Widom, R.G. Dou- glas, D. Sarason, A.P. Calderon, S. Prossdorf, B. Silbermann, and others). In the ensuing period, the Fredholm theory of singular integral operators with a finite index was completed in its main aspects in wide classes of Banach and Frechet spaces.

Table of Contents:
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 Modelfunction.- 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.