Factorization, Singular Operators and Related Problems

by Litvinchuk, G. S.; Samko, Stefan G.; Lebre, Amarino; Dos Santos, Antonio Ferreira

• ISBN13: 9781402014079
• ISBN10: 1402014074
• Format: Hardcover
• Copyright: 2003-08-01
• Publisher: Kluwer Academic Pub

Summary

The International Conference on Factorization, Singular Operators and Related Problems held from January 28 to February 1, 2002, at the University of Madeira, Funchal, Portugal, was dedicated to mark Professor Georgii Litvinchuk's 70th birthday. Experts in various fields came to this conference to pay tribute to the great achievements of Professor Georgii Litvinchuk in the development of various topics of operator theory. The main topics of the conference were in the theory of singular type operators and in factorization problems, but other topics such as potential theory, fractional calculus and others were also presented. The 21 articles published in this book will be of great interest to researchers in Operator Theory, Real and Complex Analysis, Functional and Harmonic Analysis, Potential Theory, Fractional Calculus and others, as well as to graduate students requiring the most up-to-date results.

Table of Contents

Preface	ix
Foreword	xiii
A few words about Georgii Litvinchuk	xvii
Francisco Teixeira
Contributed Papers
Singular Integrals Along Flat Curves with Kernels in the Hardy Space H1(Sn-1)	1	(12)
Ahmad Al-Salman and Hussain Al-Qassem
1 Introduction and statement of results	1	(2)
2 The Hardy space H1(Sn-1)	3	(1)
3 Preparation	4	(2)
4 Proof of main results	6	(7)
On Functional Equations with Operator Coefficients	13	(12)
Anatolij Antonevich
1 Introduction	13	(1)
2 C*-algebras, generated by dynamical systems. The isomorphism theorem	14	(2)
3 The symbolic calculus for FPDO	16	(1)
4 The invertibility conditions for functional operators. The hyperbolic approach	17	(2)
5 The case of a finite group. Formulas for the index	19	(1)
6 The essential spectra of weighted shift operators	20	(5)
Elliptic Systems with Almost Regular Coefficients: Singular Weight Integral Operators	25	(18)
Stanislav Antontsev
1 Introduction	25	(1)
2 Definitions. Embedding theorems	26	(5)
3 Integral equations. Weight integral operators	31	(7)
4 Boundary value problems for generalized analytic functions	38	(1)
5 Smoothness of quasi-conformal mappings	39	(4)
Toeplitz Matrices with Slowly Growing Pseudospectra	43	(12)
Albrecht Böttcher and Sergei Gradsky
1 Introduction and Main Results	43	(2)
2 Toeplitz determinants	45	(6)
3 Slow growth of the resolvent norm	51	(4)
A Numerical Procedure for the Inverse Sturm-Liouville Operator	55	(10)
B. Malcolm Brown, Valentin Samko, Ian Knowles and Marco Marietta
1 Introduction	55	(1)
2 Formulation of the method	56	(3)
3 Some examples	59	(1)
4 Using different gradients	60	(3)
5 Conclusion	63	(2)
A Geometrical Proof of a Theorem of Grum	65	(8)
Ramón Bruzual and Marisela Dominguez
1 Introduction	65	(1)
2 The strongly continuous unitary- group associated to f	66	(3)
3 Construction of the function fc	69	(2)
4 Positive definiteness of fo	71	(2)
Localization and Minimal Normalization of some Basic Mixed Boundary Value Problems	73	(28)
Luis P. Castro, Roland Duduchava and Frank-Olme Speck
1 Introduction to mixed boundary value problems and normalization	74	(3)
2 Associated operators	77	(2)
3 Localization	79	(4)
4 Reduction to semi-homogeneous problems	83	(3)
5 The Fredholm property	86	(3)
6 Normalization - the basic idea	89	(5)
7 Minimal normalization in the scalar case	94	(3)
8 Concluding remarks	97	(4)
Factorization of some Classes of Matrix Functions and the Resolvent of a Hankel Operator	101	(10)
Ana C. Conceição, Viktor G. Kravchenko and Francisco S. Teixeira
1 Introduction	101	(2)
2 Factorization of a class of hermitian matrix functions	103	(3)
3 The resolvent of a Hankel operator	106	(5)
Compactness of Commutators Arising in the Fredholm Theory of Singular Integral Operators with Shifts	111	(20)
Alexei Karlovich and Yuri Karlovich
1 Introduction	111	(2)
2 Preliminaries	113	(5)
3 Compactness of commutators on L2	118	(7)
4 Compactness of commutators on rearrangement-invariant spaces	125	(1)
5 Some corollaries	125	(6)
Some Problems in the Theory of Integral and Differential Equations of Fractional Order	131	(20)
Anatoly A. Kubas
1 Introduction	131	(1)
2 Laplace transform method	132	(5)
3 Operational calculus method	137	(3)
4 Compositional method	140	(4)
5 Problems and new trends of research	144	(7)
Fractional Differential Equations: A Emergent Field in Applied and Mathematical Sciences	151	(24)
Anatoly A. Kubas, Hari M. Srivasta.va and Juan J. Trujillo
1 Introduction	152	(2)
2 The Complexity Systems	154	(4)
3 Fractional Integral and Fractional Derivative Operators	158	(3)
4 A New Model for the Super-Diffusion Processes	161	(14)
Boundary Value Problems for Analytic and Harmonic Functions of Smirnov Classes in Domains with Non-Smooth Boundaries	175	(22)
Vakhtang Kokilashvili, Zurab Meshveliani and Vakhtang Paatashvili
1 The Riemann Problem with Boundary Values from the Zygmund Class	177	(5)
2 The Dirichiet Problem for Harmonic Functions from the Smirnov Classes eP(D) and eP(D,p)	182	(7)
3 The Dirichlet Problem in the Class e1(D) when the Boundary Function Belongs to the Zygmund Class	189	(8)
An Estimate for the Dimension of the Kernel of a Singular Operator with a non-Carleman Shift	197	(8)
Viktor G. Kravchenko and Rul C. Marreiros
1 Introduction	197	(1)
2 An estimate for dim ker T	198	(7)
On the Solution of Integral Equations on the Circular Disk by Use of Orthogonal Polynomials	205	(14)
Frank Penzel
1 Introduction	205	(2)
2 Basic results	207	(5)
3 The fully discretised Galerkin method	212	(3)
4 Error estimates	215	(4)
Singular and Fredholm Integral Equations for Dirichlet Boundary Problems for Axial-Symmetric Potential Fields	219	(18)
Sergiy Plaksa
1 introduction	219	(2)
2 Preliminary notes and notation	221	(1)
3 Dirichlet boundary problem for the axial-symmetric potential	222	(6)
4 Dirichlet boundary problem for the Stokes flow function	228	(9)
On the Analyticity of the Schwarz Operator with Respect to a Curve	237	(18)
Luca Preciso and Sergei Rognsin
1 Introduction	237	(3)
2 Preliminaries and notation	240	(5)
3 Real analyticity of the modified Schwarz operator	245	(6)
4 Regularity of another variants of the Schwarz operator	251	(4)
Integral Operators with Shifts on Homogeneous Groups	255	(18)
Vladimir S. Rabinovich and Steffen Roch
1 Introduction	255	(1)
2 The limit operators method	256	(3)
3 Operators on homogeneous groups	259	(5)
4 Fredholmness of convolution operators with shifts	264	(9)
On the Algebra Generated by a Poly-Bergman Projection and a ComPosition Operator	273	(18)
Josue Ramirez and Ilya M. Spitkovsky
1 Introduction	273	(2)
2 Symbol algebra of Rn,A = R(CI; Bn, Bn,A)	275	(6)
3 Symbol algebra of 1Zn = R(C(G)I; Bn, WBW)	281	(3)
4 Symbol algebra of Rn,W = R(C(G)I; Bn, W)	284	(2)
5 Proof of Theorem 1.1	286	(5)
How to Compute the Partial Indices of a Regular and Smooth Matrix-Valued Function?	291	(10)
Bernd Silbermann
1 Introduction	291	(1)
2 Toeplitz operators and their finite sections	292	(3)
3 Modified finite sections	295	(1)
4 Speed of convergence	296	(2)
5 Collocation-based approximations	298	(3)
The Multiplicative and Spectral Structure of Analytic Operator-Valued Functions	301	(14)
Anna Tarasenko
1 Introduction	301	(1)
2 Limiting values of multiplicative integrals	302	(10)
3 The case of the derivative in the Hardy class	312	(3)
Toeplitz Operators on the Bergman Space	315
Nikolai Vasilevski
1 Introduction	315	(1)
2 Commutative algebras of Toeplitz operators	316	(2)
3 Berman space structure and spectral form of special classes of Bergman operators	318	(4)
4 Unbounded symbols	322	(2)
5 Commutator properties and representations of C*-algebras	324	(2)
6 Dynamics of properties of Toeplitz operators	326