Introduction to the Theory of Toeplitz Operators with Infinite Index

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.