Factorization, Singular Operators and Related Problems / Edition 1

by Stefan Samko, Amarino Lebre, António F. dos Santos

Overview

These proceedings comprise a large part of the papers presented at the International Conference Factorization, Singular Operators and related problems, which was held from January 28 to February 1, 2002, at the University of th Madeira, Funchal, Portugal, to mark Professor Georgii Litvinchuk's 70 birth­ day. Experts in a variety of fields came to this conference to pay tribute to the great achievements of Professor Georgii Litvinchuk in the development of various areas of operator theory. The main themes of the conference were focussed around the theory of singular type operators and factorization problems, but other topics such as potential theory and fractional calculus, to name but a couple, were also presented. The goal of the conference was to bring together mathematicians from var­ ious fields within operator theory and function theory in order to highlight recent advances in problems many of which were originally studied by Profes­ sor Litvinchuk and his scientific school. A second aim was to stimulate in­ ternational collaboration even further and promote the interaction of different approaches in current research in these areas. The Proceedings will be of great interest to researchers in Operator The­ ory, Real and Complex Analysis, Functional and Harmonic Analysis, Potential Theory, Fractional Calculus and other areas, as well as to graduate students looking for the latest results.

Product Details

ISBN-13:	9781402014079
Publisher:	Springer Netherlands
Publication date:	07/31/2003
Edition description:	2003
Pages:	334
Product dimensions:	8.27(w) x 11.69(h) x 0.03(d)

Table of Contents

	Preface	ix
	Foreword	xiii
	A few words about Georgii Litvinchuk	xvii
	Contributed Papers
	Singular Integrals Along Flat Curves with Kernels in the Hardy Space H[superscript 1] (S[superscript n-1])	1
1	Introduction and statement of results	1
2	The Hardy space H[superscript 1] (S[superscript n-1])	3
3	Preparation	4
4	Proof of main results	6
	On Functional Equations with Operator Coefficients	13
1	Introduction	13
2	C*-algebras, generated by dynamical systems. The isomorphism theorem	14
3	The symbolic calculus for FPDO	16
4	The invertibility conditions for functional operators. The hyperbolic approach	17
5	The case of a finite group. Formulas for the index	19
6	The essential spectra of weighted shift operators	20
	Elliptic Systems with Almost Regular Coefficients: Singular Weight Integral Operators	25
1	Introduction	25
2	Definitions. Embedding theorems	26
3	Integral equations. Weight integral operators	31
4	Boundary value problems for generalized analytic functions	38
5	Smoothness of quasi-conformal mappings	39
	Toeplitz Matrices with Slowly Growing Pseudospectra	43
1	Introduction and Main Results	43
2	Toeplitz determinants	45
3	Slow growth of the resolvent norm	51
	A Numerical Procedure for the Inverse Sturm-Liouville Operator	55
1	Introduction	55
2	Formulation of the method	56
3	Some examples	59
4	Using different gradients	60
5	Conclusion	63
	A Geometrical Proof of a Theorem of Crum	65
1	Introduction	65
2	The strongly continous unitary group associated to f	66
3	Construction of the function f[superscript c]	69
4	Positive definiteness of f[superscript o]	71
	Localization and Minimal Normalization of some Basic Mixed Boundary Value Problems	73
1	Introduction to mixed boundary value problems and normalization	74
2	Associated operators	77
3	Localization	79
4	Reduction to semi-homogeneous problems	83
5	The Fredholm property	86
6	Normalization - the basic idea	89
7	Minimal normalization in the scalar case	94
8	Concluding remarks	97
	Factorization of some Classes of Matrix Functions and the Resolvent of a Hankel Operator	101
1	Introduction	101
2	Factorization of a class of hermitian matrix functions	103
3	The resolvent of a Hankel operator	106
	Compactness of Commutators Arising in the Fredholm Theory of Singular Integral Operators with Shifts	111
1	Introduction	111
2	Preliminaries	113
3	Compactness of commutators on L[superscript 2]	118
4	Compactness of commutators on rearrangement-invariant spaces	125
5	Some corollaries	125
	Some Problems in the Theory of Integral and Differential Equations of Fractional Order	131
1	Introduction	131
2	Laplace transform method	132
3	Operational calculus method	137
4	Compositional method	140
5	Problems and new trends of research	144
	Fractional Differential Equations: A Emergent Field in Applied and Mathematical Sciences	151
1	Introduction	152
2	The Complexity Systems	154
3	Fractional Integral and Fractional Derivative Operators	158
4	A New Model for the Super-Diffusion Processes	161
	Boundary Value Problems for Analytic and Harmonic Functions of Smirnov Classes in Domains with Non-Smooth Boundaries	175
1	The Riemann Problem with Boundary Values from the Zygmund Class	177
2	The Dirichlet Problem for Harmonic Functions from the Smirnov Classes ep (D) and ep (D, [rho])	182
3	The Dirichlet Problem in the Class e 1 (D) when the Boundary Function Belongs to the Zygmund Class	189
	An Estimate for the Dimension of the Kernel of a Singular Operator with a non-Carleman Shift	197
1	Introduction	197
2	An estimate for dim ker T	198
	On the Solution of Integral Equations on the Circular Disk by Use of Orthogonal Polynomials	205
1	Introduction	205
2	Basic results	207
3	The fully discretised Galerkin method	212
4	Error estimates	215
	Singular and Fredholm Integral Equations for Dirichlet Boundary Problems for Axial-Symmetric Potential Fields	219
1	Introduction	219
2	Preliminary notes and notation	221
3	Dirichlet boundary problem for the axial-symmetric potential	222
4	Dirichlet boundary problem for the Stokes flow function	228
	On the Analyticity of the Schwarz Operator with Respect to a Curve	237
1	Introduction	237
2	Preliminaries and notation	240
3	Real analyticity of the modified Schwarz operator	245
4	Regularity of another variants of the Schwarz operator	251
	Integral Operators with Shifts on Homogeneous Groups	255
1	Introduction	255
2	The limit operators method	256
3	Operators on homogeneous groups	259
4	Fredholmness of convolution operators with shifts	264
	On the Algebra Generated by a Poly-Bergman Projection and a Composition Operator	273
1	Introduction	273
2	Symbol algebra of R n,A = R(CI; B n, B n,A)	275
3	Symbol algebra of R n = R(C(G)I; B n, W B n W)	281
4	Symbol algebra of R n,W = R(C(G)I; B n, W)	284
5	Proof of Theorem 1.1	286
	How to Compute the Partial Indices of a Regular and Smooth Matrix-Valued Function?	291
1	Introduction	291
2	Toeplitz operators and their finite sections	292
3	Modified finite sections	295
4	Speed of convergence	296
5	Collocation-based approximations	298
	The Multiplicative and Spectral Structure of Analytic Operator-Valued Functions	301
1	Introduction	301
2	Limiting values of multiplicative integrals	302
3	The case of the derivative in the Hardy class	312
	Toeplitz Operators on the Bergman Space	315
1	Introduction	315
2	Commutative algebras of Toeplitz operators	316
3	Bergman space structure and spectral form of special classes of Toeplitz operators	318
4	Unbounded symbols	322
5	Commutator properties and representations of C*-algebras	324
6	Dynamics of properties of Toeplitz operators	326