Skip to main content

2002 | Buch

Introduction to the Theory of Toeplitz Operators with Infinite Index

verfasst von: Vladimir Dybin, Sergei M. Grudsky

Verlag: Birkhäuser Basel

Buchreihe : Operator Theory: Advances and Applications

insite
SUCHEN

Inhaltsverzeichnis

Frontmatter
Introduction
Abstract
The Toeplitz operator T(a) on the real line ℝ induced by a function a given on ℝ is defined by
$$T(a)f = {P^ + }(af),$$
(1)
where P+ is the analytic projector of the space L p (ℝ) , 1 < p < ∞, onto the Hardy subspace H p (∏+) for the upper complex half-plane ∏+. The function a is called the symbol of the operator T(a). To study such operators in the Hardy space H p (∏+) we will assume from now on that aL∞(ℝ). In the simplest case the symbol a admits a representation
$$a(x) = {a^ + }(x){\left( {\frac{{x - i}}{{x + i}}} \right)^\kappa }{a^ - }(x),$$
(2)
where
$${({a^ + })^{ \pm 1}} \in {H^\infty }({\prod _ + }), {({a^ - })^{ \pm 1}} \in {H^\infty }({\prod _ - }), \kappa \in \mathbb{Z},$$
(3)
which is known as a canonical factorization. The rational function
$$r(x) = {\left( {\frac{{x - i}}{{x + i}}} \right)^\kappa }$$
(4)
plays a determining role in the description of the main qualitative characteristics of the operator T(a). Specifically, for κ > 0 the operator T(a) is left-invertible and dim coker T(a) = κ for κ < 0 it is right-invertible and dim ker T(a) = −κ and for κ = 0 it is invertible. As a consequence, the index of the operator T(a) is equal to −κ, and is also equal to minus the topological index of the function r, which is defined by
$$in{d_\mathbb{R}}r = \frac{1}{{2\pi }}\arg r(x)|_{ - \infty }^\infty .$$
Vladimir Dybin, Sergei M. Grudsky
Chapter 1. Examples of Toeplitz Operators with Infinite Index
Abstract
In this chapter, after introducing the necessary definitions and studying the simplest properties of Toeplitz operators, we will show on a number of examples how a Toeplitz operator may turn out to have an infinite index.
Vladimir Dybin, Sergei M. Grudsky
Chapter 2. Factorization and Invertibility
Abstract
In this chapter we introduce the notion of the (p, ϱ)-factorization of a bounded measurable function and establish a connection between this notion and the Fredholmness of Toeplitz operators. After that we will propose a generalization of the theory of (p, ϱ)-factorization to the case of Toeplitz operators with infinite index.
Vladimir Dybin, Sergei M. Grudsky
Chapter 3. Model Subspaces
Abstract
In this chapter we study Toeplitz operators of the form T(h±1), where hL + (г) ∩ GL(г). Special attention will be devoted to the subspaces ker T(h−1) in terms of which the kernels of the operators T(a) with symbols a ∈ fact (∈, p, ϱ) were described in Theorem 2.6. It is in this chapter that we establish a connection between model subspaces, which are standard objects of nonclassical spectral theory (see Remark 2.1), and Toeplitz operators with infinite index.
Vladimir Dybin, Sergei M. Grudsky
Chapter 4. Toeplitz Operators with Oscillating Symbols
Abstract
In this chapter we consider Toeplitz operators whose symbols possess the best known kinds of discontinuities of oscillatory type: almost periodic (a.p.) discontinuities, semi-almost periodic discontinuities, and whirl points of power type. For each of these cases the central problem is the construction of a generalized factorization (2.30) of the symbol, which enables us to apply Theorem 2.6.
Vladimir Dybin, Sergei M. Grudsky
Chapter 5. Generalized Factorization of u-periodic Functions and Matrix Functions
Abstract
In this chapter we confine ourselves to the setting of the space L p 0), 1 <p <∞. First, we will give the needed definitions and facts from the theory of matrix Toeplitz operators. Then we will introduce the concept of the generalized factorization of a matrix valued function (matrix function for short) and formulate the corresponding invertibility theory for the operatorT(a)with matrix symbol a. Conceptually, the main section of this chapter is Section 5.3, devoted to the theory of generalized factorization of u-periodic matrix functions. After that we will consider new classes of Toeplitz operators with infinite index, first in the scalar and then in the matrix case.
Vladimir Dybin, Sergei M. Grudsky
Chapter 6. Toeplitz Operators Whose Symbols Have Zeros
Abstract
Let X be a topological linear space and let the operator AL(X) = L (X,X) have a nonclosed image (im A ≠ clos im A). Since a linear operator is in fact a triplet (correspondence rule, domain of definition, domain of values), its properties depend in an essential manner on each of the components of the triplet and there are reasons to hope that by appropriately perturbing these components one can achieve that the image of the “new” (perturbed) operator will be closed. This problem is intimately connected with problems that arise when solving the equation Ax = y.If we are interested in describing all its right-hand sides for which the solution lies in a given space, then we fix the domain of definition of the operator and try to restrict its domain of values, simultaneously modifying the topology in the latter. If, however, we are interested in describing all “reasonable” solutions for a given class of right-hand sides, then the domain of values of the operators is kept fixed, while the operator itself and its domain of definition are subject to a certain extension. The “new” operator with closed image, as well and the procedure for constructing it, are called a normalization of the operator A.
Vladimir Dybin, Sergei M. Grudsky
Backmatter
Metadaten
Titel
Introduction to the Theory of Toeplitz Operators with Infinite Index
verfasst von
Vladimir Dybin
Sergei M. Grudsky
Copyright-Jahr
2002
Verlag
Birkhäuser Basel
Electronic ISBN
978-3-0348-8213-2
Print ISBN
978-3-0348-9476-0
DOI
https://doi.org/10.1007/978-3-0348-8213-2