These lecture notes present an extensive review of the factorization theory of matrix functions relative to a curve with emphasis on the developments of the last 20–25 years. The classes of functions considered range from rational and continuous matrix functions to matrix functions with almost periodic or even semi almost periodic entries. Also included are recent results about explicit factorization based on the state space method from systems theory, with examples from linear transport theory. Related applications to Riemann-Hilbert boundary value problems and the Fredholm theory of various classes of singular integral operators are described too. The applications also concern inversion of singular integral operators of different types, including Wiener-Hopf and Toeplitz operators.

An Overview of Matrix Factorization Theory and Operator Applications

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Constructive methods for factorization of matrix-functions

Pseudo‐Differential Operators on Manifolds with Singularities

This is an overview of the Erlangen Program at Large. Study of objects and properties, which are invariant under a group action, is very fruitful far beyond traditional geometry. In this paper we demonstrate this on the example of the group SL2(ℝ). Starting from the conformal geometry we develop analytic functions and apply these to functional calculus. Finally we link this to quantum mechanics and conclude by a list of open problems.

Erlangen Program at Large: An Overview

Sufficient conditions for the projective freeness of Banach algebras

1 Convolution Operators and Their Symbols.- 2 Introduction to Scalar Wiener-Hopf Operators.- 3 Scalar Wiener-Hopf Operators with SAP Symbols.- 4 Some Phenomena Caused by SAP Symbols.- 5 Introduction to Matrix Wiener-Hopf Operators.- 6 Factorization of Matrix Functions.- 7 Bohr Compactification.- 8 Existence and Uniqueness ofAPFactorization.- 9 Matrix Wiener-Hopf Operators withAPWSymbols.- 10 Matrix Wiener-Hopf Operators withSAPWSymbols.- 11 Left Versus Right Wiener-Hopf Factorization.- 12 Corona Theorems.- 13 The Portuguese Transformation.- 14 Some Concrete Factorizations.- 15 Scalar Trinomials.- 16 Toeplitz Operators.- 17 Zero-Order Pseudodifferential Operators.- 18 Toeplitz Operators with SAP Symbols on Hardy Spaces.- 19 Wiener-Hopf Operators with SAP Symbols on Lebesgue Spaces.- 20 Hankel Operators on Besicovitch Spaces.- 21 Generalized AP Factorization.- 22 Canonical Wiener-Hopf Factorization via Corona Problems.- 23 Canonical APW Factorization via Corona Problems.

Convolution Operators and Factorization of Almost Periodic Matrix Functions

We treat the Dirichlet problem for the 1D heat equation in a bounded domain G ⊂ R. The boundary of G is assumed to be smooth and noncharacteristic except for two points where it has contact of degree less than 2 with lines orthogonal to the t -axis. At these points the boundary has cuspidal singularities which have to be treated with particular care. We prove that this problem fits into the framework of analysis on manifolds with singular points elaborated by V. Rabinovich et al. (2000). The results extend to general parabolic equations. Mathematics Subject Classification (2010). Primary 35K35; Secondary 35G15, 58J35.

Ilya M. Spitkovskiĭ

Papers

Convolution Operators and Factorization of Almost Periodic Matrix Functions

Operator theory, pseudo-differential equations, and mathematical physics : the Vladimir Rabinovich anniversary volume