Singular Integral Operators, Factorization and Applications

by Kaashoek, M. A.; Lebre, Amarino Brites; Dos Santos, Antonio Ferreira; Speck, Frank-Olme; Bottcher, Albrecht

• ISBN13: 9783764369477
• ISBN10: 3764369477
• Format: Hardcover
• Copyright: 2003-06-01
• Publisher: Birkhauser

Summary

This book contains the proceedings of the International Workshop on Operator Theory and Applications held in Faro, Portugal, September 12 to 15, 2000. It includes 20 selected articles centered on the analysis of various classes of singular operators, the factorization of operator and matrix functions, algebraic methods in approximation theory, and applications in diffraction theory. Some papers are related to topics from fractional calculus, complex analysis, operator algebras, and partial differential equations.

Table of Contents

Editorial Introduction	xi
Note on the Memory of Professor Erhard Meister	xiii
Program of IWOTA-Portugal 2000	xiv
List of Participants	xx
Covariant Symbolic Calculi on Real Symmetric Domains
by JONATHAN ARAZY AND HARALD UPMEIER
1. Introduction	1	(1)
2. Real symmetric domains	2	(3)
3. Quantization Hilbert spaces	5	(4)
4. Covariant symbolic calculi on real symmetric domains	9	(5)
5. The Toeplitz-Berezin calculus	14	(12)
References	26	(3)
Singular Integral operators with Piecewise Almost Periodic Coefficients and Carleman Shift
by M. AMI LIA BASTOS, ANTÓNIO BRAVO AND YURI KARLOVICH
1. Introduction	29	(2)
2. Preliminaries	31	(2)
3. The algebra of singular integral operators with piecewise continuous coefficients	33	(4)
4. The Fredholm study of operators with semi-almost periodic coefficients	37	(8)
5. Singular integral operators with piecewise almost periodic coefficients	45	(6)
6. Fredholmness of singular integral operators with Carleman shift and piecewise almost periodic coefficients	51	(5)
References	56	(3)
On the Essential Spectrum of Toeplitz Operators with Semi-Almost Periodic Symbols
by ALBRECHT BÖTTCHER, SERGEI GRUDSKY AND ILYA SPITKOVSKY
1. Introduction and main results	59	(4)
2. Sarason's criterion	63	(1)
3. The local shape of the essential spectrum	64	(1)
4. Essential spectra of prescribed local shapes	65	(5)
5. Essential spectra of prescribed global shapes	70	(4)
6. Separated almost periodic representatives	74	(2)
References	76	(3)
Well-Posedness of Diffraction Problems Involving n Coplanar Strips
by LUÍS PINHEIRO DE CASTRO AND FRANK-OLME SPECK
1. Introduction	79	(1)
2. Formulation of the problem	79	(2)
3. Relations with Wiener-Hopf operators	81	(3)
4. Sufficient conditions for the invertibility	84	(3)
5. Example: The Rawlins problem for n strips	87	(2)
References	89	(2)
Factorization of Matrix Functions and the Resolvents of Certain Operators
by ANA CONCEIÇAO, VIKTOR KRAVCHENKO AND FRANCISCO TEIXEIRA
1. Introduction	91	(1)
2. Preliminaries	92	(1)
3. Resolvents of N_ and N+ through the factorization of A7	93	(1)
4. Factorization of A7 through the resolvent of N_ and N+	94	(6)
References	100	(1)
Finite Difference Cauchy-Riemann Operators and Their Fundamental Solutions in the Complex Case
by KLAUS GURLEBECK AND ANGELA HOMMEL
1. Introduction	101	(1)
2. Cauchy-Riemann operators and fundamental solutions in the complex case	102	(1)
3. Discrete theory	103	(11)
References	114	(3)
Factorization Theorem for the Transfer Function Associated with an Unbounded Non-Self-Adjoint 2 x 2 Operator Matrix
by VOLKER HARDT, REINHARD MENNICKEN AND ALEXANDER K. MOTOVILOV
1. Introduction	117	(2)
2. Analytic continuation	119	(2)
3. The transformation equations	121	(3)
4. Factorization	124	(5)
5. An example	129	(2)
References	131	(2)
Boundedness of Some Convolution and Twisted Convolution Operators
by ALEXEY N. KARAPETYANTS AND ENRIQUE RAMÍREZ DE ARELLANO
1. Introduction	133	(2)
2. Integral representations for the operators Ao, Bo and some properties of the kernel Oo(x)	135	(3)
3. Boundedness of convolution operators	138	(1)
4. Boundedness of twisted convolution operators	139	(3)
5. Concluding remarks	142	(1)
References	143	(4)
Invertibility of Functional Operators with Slowly Oscillating Non-Carleman Shifts
by ALEXEI KARLOVICH, YURI KARLOVICH AND AMARINO LEBRE
1. Introduction	147	(3)
2. Sufficient conditions for one-sided invertibility	150	(5)
3. Slowly oscillating functions and shifts	155	(4)
4. Limit operators for a functional operator	159	(9)
5. Necessary conditions for invertibility	168	(5)
References	173	(2)
Hadamard-Type Integral Equations and Fractional Calculus Operators
by ANATOLY A. KILBAS
1. Introduction	175	(3)
2. Hadamard-type fractional integrals and derivatives in Xu(a,b)	178	(3)
3. Hadamard-type integral equations in Xu(a, b)	181	(5)
4. Hadamard-type integral equations in Xu(a, b) in the case 0 less than o less than 1 183
5. The existence of Hadamard-type fractional derivatives and Hadamard-type differentiation and integration as reciprocal operations	186	(2)
References	188	(1)
Factorization of Singular Integral operators with a Carleman Shift via Factorization of Matrix Functions
by VIKTOR KRAVCHENKO, AMARINO LEBRE AND JUAN RODRÍGUEZ
1. Introduction	189	(3)
2. Preliminaries	192	(3)
3. Analysis of the case of a forward shift	195	(15)
References	210	(3)
Single Elements of Operator Algebras
by WILLIAM E. LONGSTAFF
1. Introduction	213	(1)
2. Results	214	(6)
References	220	(3)
Skew Incidence Half Plane Diffraction Problems: Particular Aspects of the Factorization Problem
by ERNST LUNEBURG AND HAMIT A. SERBEST
1. Introduction	223	(3)
2. Preliminaries	226	(4)
3. The two-face impedance half plane problem	230	(12)
4. Other examples	242	(5)
5. Concluding remarks	247	(1)
References	248	(5)
Aperiodic Wave Diffraction by a Half-Plane
by ERHARD MEISTER
1. Introduction	253	(1)
2. The field equations in elasto- and viscoelastodynamics	254	(1)
3. Initial boundary value problems	255	(5)
4. Conclusions	260	(1)
References	260	(3)
An Axiomatic Approach to the Limit Operators Method
by VLADIMIR S. RABINOVICH AND STEFFEN ROCH
1. Introduction	263	(1)
2. Abstract scheme for the method of limit operators	264	(5)
3. Fredholmness of convolution operators on homogeneous groups	269	(9)
4. Pseudodifferential operators	278	(6)
References	284	(3)
Algebras of Approximation Sequences: Structure of Fractal Algebras
by STEFFEN ROCH
1. Introduction	287	(4)
2. Fractality and Fredholmness of approximation sequences	291	(3)
3. Structure of certain fractal homomorphisms	294	(8)
4. The first half of the standard model	302	(2)
5. The second half of the standard model	304	(5)
References	309	(2)
Noncanonical Factorizations of Almost Periodic Multivariable Matrix Functions
by LEIBA RODMAN, ILYA M. SPITKOVSKY AND HUGO J. WOERDEMAN
1. Introduction	311	(2)
2. Definitions and preliminaries	313	(5)
3. Symmetric factorizations	318	(8)
4. Factorization and Toeplitz operators	326	(6)
5. Factorization indices under small perturbations; left vs right indices	332	(4)
6. Application of (non)canonical factorization to normalizing bases	336	(5)
References	341	(4)
Criterion for the Fredholmness of Singular Operators with Piecewise Continuous Coefficients in Generalized Hölder Spaces with Weight
by NATASHA SAMKO
1. Introduction	345	(1)
2. The results on Fredholmness	345	(4)
3. The proof of necessity	349	(6)
References	355	(2)
On Inversion of Fractional Spherical Potentials by Spherical Hypersingular Operators
by STEFAN SAMKO
1. Introduction	357	(1)
2. Preliminaries	357	(4)
3. Justification of the inversion in Lp(Sn-1) and identity approximation on the sphere	361	(4)
4. Inversion of the spherical Riesz potential in the case Ra > 2	365	(2)
References	367	(2)
An Axiomatic Approach to Stability of Approximation Sequences
by PEDRO A. SANTOS AND BERND SILBERMANN
1. Introduction	369	(2)
2. Algebraic results	371	(1)
3. Abstract Toeplitz operators	372	(1)
4. Approximation methods for the operator a(V), a E PC(T)	373	(6)
5. Finite subsection method for Toeplitz operators as example	379	(1)
References	380