Introduction to the Theory of Toeplitz Operators with Infinite Index

Inhaltsverzeichnis1 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.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.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.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.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.