Boundary Value Problems for the Helmholtz Equation in an Octant

We consider a class of boundary value problems for the three-dimensional Helmholtz equation that appears in diffraction theory. On the three faces of the octant, which are quadrants, we admit first order boundary conditions with constant coefficients, linear combinations of Dirichlet, Neumann, impedance and/or oblique derivative type. A new variety of surface potentials yields 3 × 3 boundary pseudodifferential operators on the quarterplane that are equivalent to the operators associated to the boundary value problems in a Sobolev space setting. These operators are analyzed and inverted in particular cases, which gives us the analytical solution of a number of well-posed problems. Dedicated to Vladimir G. Maz’ya on the occasion of his 70th birthday     

Parameterization of all solutions of the three chains completion problem

The family of all solutions of the three chains completion problem with a prescribed tolerance is described explicitly. It is shown that this family can be parameterized by a natural set of contractive upper triangular operators. As an application all solutions to a suboptimal nonstationary Nehari problem are described.     

On the Besov regularity of conformal maps and layer potentials on nonsmooth domains

We study the Besov regularity of conformal mappings for domains with rough boundary based on the well-posedness for the Dirichlet problem with Besov data. Also, sharp invertibility results for the classical layer potential operators on Sobolev-Besov spaces on the boundary of curvilinear polygons are obtained.     

Nevanlinna-Pick interpolation with boundary data

Versions of the Nevanlinna-Pick interpolation problem with boundary interpolation nodes and boundary interpolated values are investigated.     

Robin boundary value problem for the Cauchy-Riemann operator

The aim of this article is to give explicit representations for solutions of the Robin boundary value problem for the Cauchy-Riemann operator [image omitted]. In the homogeneous cases we investigate the Robin boundary condition in a more general form. Finally, we give solutions of the corresponding higher-order operators.     

On algebras of two dimensional singular integral operators with homogeneous discontinuities in symbols

We describe the Fredholm symbol algebra for theC ^*-algebra generated by two dimensional singular integral operators, acting onL ₂(²), and whose symbols admit homogeneous discontinuities. Locally these discontinuities are modeled by homogeneous functions having slowly oscillating (and, in particular, piecewise continuous) discontinuities on a system of rays outgoing from the origin.These results extend the well-known Plamenevsky results for the two dimensional case. We present here an alternative and much clearer approach to the problem.Rostov State University RussiaPartially supported by Russian Fund for Fundamental Investigations, RFFI-98-01-01-023, and by CONACYT project 32424-EPartially supported by CONACYT Project 27934-E, México.     

JOINT SPECTRAL SETS

Abstract

We consider the problem: if S,TEB(H) are commuting operators with von Neumann spectral sets X and Y respectively, does it imply that X x Y is a joint spectral set for the pair (S,T)?     

On perturbations for oblique projection generalized inverses of closed linear operators in Banach spaces

The main concern of this paper is the perturbation problem for oblique projection generalized inverses of closed linear operators in Banach spaces. We provide a new stability characterization of oblique projection generalized inverses of closed linear operators under T-bounded perturbations, which improves some well known results in the case of the closed linear operators under the bounded perturbation or that the perturbation does not change the null space.     

Strongly definitizable linear pencils in Hilbert space

Selfadjoint linear pencils F–G are considered which have discrete spectrum and neither F nor G is definite. Several characterizations are given of a strongly definitizable property when F and G are bounded, and also when both operators are unbounded. The theory is applied to analysis of the stability of a linear second order initial-boundary value problem with boundary conditions dependent on the eigenvalue parameter.Research supported in part by a grant from the Natural Sciences and Engineering Research Council of Canada.     

Unitary mappings between multiresolution analyses of L2 (R) and a parametrization of low-pass filters

This article provides classes of unitary operators of L²(R) contained in the commutant of the Shift operator, such that for any pair of multiresolution analyses of L²(R) there exists a unitary operator in one of these classes, which maps all the scaling functions of the first multiresolution analysis to scaling functions of the other. We use these unitary operators to provide an interesting class of scaling functions. We show that the Dai-Larson unitary parametrization of orthonormal wavelets is not suitable for the study of scaling functions. These operators give an interesting relation between low-pass filters corresponding to scaling functions, which is implemented by a special class of unitary operators acting on L²([−π, π)), which we characterize. Using this characterization we recapture Daubechies' orthonormal wavelets bypassing the spectral factorization process. Acknowledgements and Notes. Partially supported by NSF Grant DMS-9157512, and Linear Analysis and Probability Workshop, Texas A&M University Dedicated to the memory of Professor Emeritus Vassilis Metaxas.     

Spectral stability of general non-negative self-adjoint operators with applications to Neumann-type operators

We prove a stability theorem for the eigenvalues of general non-negative self-adjoint linear operators with compact resolvents and by applying it we prove a sharp stability result for the dependence of the eigenvalues of second order uniformly elliptic linear operators with homogeneous Neumann boundary conditions upon domain perturbation.     

Slowing down and reflection of waves in truncated periodic media

We consider wave propagation and scattering governed by 1-D Schrödinger operators with truncated periodic potentials. The propagation of wave packets with narrow frequency supports is studied. The goal is to describe potentials for which the group velocity (for a periodic problem) is small and the transmission coefficient for the truncated potential is not too small, i.e. to find media where a slowing down of the wave packets coexists with a transparency.     

Bifurcation points of Hammerstein equations

We apply global bifurcation theorems to systems of nonlinear integral equations of Hammerstein type involving a scalar parameter. To this end, we give sufficient conditions for the continuous dependence, compactness, Fréchet differentiability, and asymptotic linearity of the corresponding operators, which are more general than in the classical setting. These properties are ensured only after passing to some equivalent operator equation which typically contains fractional powers of the linear part. Finally, we show that the abstract hypotheses on the operators correspond to natural hypotheses on the kernel function and the nonlinearity in the Hammerstein equation under consideration.     

Strongly omnipresent integral operators

An operatorT on the spaceH(G) of holomorphic functions on a domainG is strongly omnipresent whenever there is a residual set of functionsfH(G) such thatT f exhibits an extremely wild behaviour near the boundary. The concept of strong omnipresence was recently introduced by the first two authors. In this paper it is proved that a large class of integral operators including Volterra operators with or without a perturbation by differential operators has this property, completing earlier work about differential and antidifferential operators.The work of the first two authors has been partially supported by DGES grant PB96-1348 and the Junta de Andalucía.     

Some inequalities for the Euclidean operator radius of two operators in Hilbert spaces

Some sharp bounds for the Euclidean operator radius of two bounded linear operators in Hilbert spaces are given. Their connection with Kittaneh’s recent results which provide sharp upper and lower bounds for the numerical radius of linear operators are also established.     

Spectral stability of Dirichlet second order uniformly elliptic operators

We prove sharp stability results for the dependence of the eigenvalues of second order uniformly elliptic linear operators with homogeneous Dirichlet boundary conditions upon domain perturbation.     

w-hyponormal operators II

A continuation of the study of thew-hyponormal operators is presented. It is shown thatw-hyponormal operators are paranormal. Sufficient conditions which implyw-hyponormal operators are normal are given. The nonzero points of the approximate and joint approximate point spectra are shown to be identical forw-hyponormal operators. The square of an invertiblew-hyponormal operator is shown to bew-hyponormal.     

The Bitangential Inverse Input Impedance Problem for Canonical Systems,I: Weyl-Titchmarsh Classification,Existence and Uniqueness

The inverse input impedance problem is investigated in the class of canonical integral systems with matrizants that are strongly regular J-inner matrix valued functions in the sense introduced in [ArD1]. The set of solutions for a problem with a given input impedance matrix (i.e., Weyl- Titchmarsh function) is parameterized by chains of associated pairs of entire inner p × p matrix valued functions. In our considerations the given data for the inverse bitangential input impedance problem is such a chain and an input impedance matrix, i.e., a p × p matrix valued function in the Carathéodory class. Existence and uniqueness theorems for the solution of this problem are obtained by consideration of a corresponding family of generalized bitangential Carathéodory interpolation problems. The connection between the inverse bitangential input scattering problem that was studied in [ArD4] and the bitangential input impedance problem is also exploited. The successive sections deal with: 1. The introduction, 2. Domains of linear fractional transformations, 3. Associated pairs of the first and second kind, 4. Matrix balls, 5. The classification of canonical systems via the limit ball, 6. The Weyl-Titchmarsh characterization of the input impedance, 7. Applications of interpolation to the bitangential inverse input impedance problem. Formulas for recovering the underlying canonical integral systems, examples and related results on the inverse bitangential spectral problem will be presented in subsequent publications.D. Z. Arov thanks the Weizmann Institute of Science for hospitality and support, partially as a Varon Visiting Professor and partially through the Minerva Foundation. H. Dym thanks Renee and Jay Weiss for endowing the chair which supports his research and the Minerva Foundation.     

Estimates for norms of matrix-valued and operator-valued functions and some of their applications

A survey is presented of estimates for a norm of matrix-valued and operator-valued functions obtained by the author. These estimates improve the Gel'fand-Shilov estimate for regular functions of matrices and Carleman's estimates for resolvents of matrices and compact operators.From the estimates for resolvents, the well-known result for spectrum perturbations of self-adjoint operators is extended to quasi-Hermitian operators. In addition, the classical Schur and Brown's inequalities for eigenvalues of matrices are improved.From estimates for the exponential function (semigroups), bounds for solution norms of nonlinear differential equations are derived. These bounds give the stability criteria which make it possible to avoid the construction of Lyapunov functions in appropriate situations.