Decompositions of Singular Continuous Spectra of \mathcal{H}_{ - 2} -class Rank One Perturbations

The decomposition theory for the singular continuous spectrum of rank one singular perturbations is studied. A generalization of the well-known Aronszajn-Donoghue theory to the case of decompositions with respect to α-dimensional Hausdorff measures is given and a characterization of the supports of the α-singular, α-absolutely continuous, and strongly α-continuous parts of the spectral measure of - class rank one singular perturbations is given in terms of the limiting behaviour of the regularized Borel transform.     

Non- (Quantum) Differentiable C 1-Functions in the Spaces with Trivial Boyd Indices

If E is a separable symmetric sequence space with trivial Boyd indices and is the corresponding ideal of compact operators, then there exists a C¹-function f_E, a self-adjoint element and a densely defined closed symmetric derivation δ on such that , but      

Spectral theory of difference operators with almost constant coefficients II

We show that the operators whose coefficients are approximately constant in a general sense have an absolutely continuous spectrum which is equal to that of the corresponding constant coefficient operator. For such operators, the absolutely continuous spectrum can be read off from the associated characteristic polynomial. This generalizes the classical results on second-order operators and extends those of higher order differential operators to the difference setting. Our approach relies on an analysis of the associated difference equation with the help of uniform asymptotic summation techniques.     

Normal boundary dilations and rationally invariant subspaces

The first named author was supported by grants from the National Science Foundation.     

Asymptotics of spectrum under infinitesimally form-bounded perturbation

LetA1 be a selfadjoint operator with discrete spectrum and known distribution function of its spectrumN(r,A). SupposeB is a (nonselfadjoint) operator that is form-bounded with respect toA with relative bound zero. If in addition thenN(r,A+B)=N(r,A)(1+o(1)), whereA+B is the operator defined as form sum. The applications to the Schrödinger operator with polynomially growing potential and to the third boundary value problem for the second order elliptic operator are given.Research supported by the Israel Ministries of Science and Absorption     

A construction of unitary linear systems

A construction is made of a unitary linear system whose transfer function is a given power seriesB(z) with operator coefficients such that multiplication byB(z) is an everywhere defined transformation in the space of square summable power series with vector coefficients. A condition is also given for the existence of an observable linear system with such a transfer function. For both constructions properties of the spaces are given which imply essential uniqueness of linear systems with given transfer functions. A canonical conjugate-isometric linear system is uniquely determined by its transfer function whenever the state space is a Pontryagin space.     

Multilinear Commutators of Singular Integrals with Non Doubling Measures

Let be a Radon measure on which may be non-doubling. The only condition that must satisfy is for all and for some fixed In this paper, under this assumption, the L^p()-boundedness (1 < p < ) and certain weak type endpoint estimate are established for multilinear commutators, which are generated by Calderón-Zygmund singular integrals with RBMO() functions or with functions for r 1, where is a space of Orlicz type satisfying that if r = 1 and if r > 1.     

Regular Functions of Operators on Tensor Products of Hilbert Spaces

A class of linear operators on tensor products of Hilbert spaces is considered. Estimates for the norm of operator-valued functions regular on the spectrum are derived. These results are new even in the finite-dimensional case. By virtue of the obtained estimates, we derive stability conditions for semilinear differential equations. Applications of the mentioned results to integro-differential equations are also discussed.     

Commutant Lifting Theorem for the Bergman Space

The central question of this paper is the one of finding the right analogue of the Commutant Lifting Theorem for the Bergman space L_a². We also analyze the analogous problem for weighted Bergman spaces L_a,α², − 1 < α < ∞.     

The Quasiapproximate $(\mathcal{U} + \mathcal{K})$ –Invariants of Essentially Normal Operators

For a class of essentially normal operators, we characterize their norm closures of –orbits. Moreover, we introduce a notion of the quasiapproximate – equivalence of essentially normal operators and determine completely the quasiapproximate –invariants. Finally, we give the canonical forms of essentially normal operators under this quasiapproximate –equivalence.     

One-dimensional perturbations of the shift

The structure of isometries on a Hilbert space are well studied. In this paper we study contractions which are one-dimensional perturbations of isometries, in particular, perturbations of the shift operator onH ².     

On Fredholm index in Banach spaces

We prove the stability of the index of a Fredholm complex of Banach spaces under those compact perturbations which are uniform limits of finite-rank operators (Theorem 3.3). This result is a consequence of some similar statements (Theorems 3.1 and 3.2) concerning more general objects, namely the Fredholm pairs (Definition 1.1).     

The Aronszajn–Donoghue Theory for Rank One Perturbations of the $$\mathcal{H}_{-2} {\text{-Class}}$$

A singular rank one perturbation of a self-adjoint operator A in a Hilbert space is considered, where and but with the usual A–scale of Hilbert spaces. A modified version of the Aronszajn-Krein formula is given. It has the form where F denotes the regularized Borel transform of the scalar spectral measure of A associated with . Using this formula we develop a variant of the well known Aronszajn–Donoghue spectral theory for a general rank one perturbation of the class.Submitted: March 14, 2002 Revised: December 15, 2002     

Commutant lifting theorem and interpolation in discrete nest algebras

Ball in [Ba] showed that the commutant lifting theorem for the nest algebras due to Paulsen and Power gives a unified approach to a wide class of interpolation problems for nest algebras. By restricting our attention to the case when nest algebras associated with the problems are discrete we derive a variant of the commutant lifting theorem which avoids language of representation theory and which is sufficient to treat an analog of the generalized Schur-Nevannlinna-Pick (SNP) problem in the setting of upper triangular operators.     

Some Sharp Norm Estimates in the Subspace Perturbation Problem

We discuss the spectral subspace perturbation problem for a self-adjoint operator. Assuming that the convex hull of a part of its spectrum does not intersect the remainder of the spectrum, we establish an a priori sharp bound on variation of the corresponding spectral subspace under off-diagonal perturbations. This bound represents a new, a priori, tan Θ Theorem. We also extend the Davis–Kahan tan 2Θ Theorem to the case of some unbounded perturbations.     

Isometric representations of some quotients ofH ∞ of an annulus

Let denote an annulus,E a finite subset of with at least three elements, and the ideal of functions in which vanish at the points ofE. The quotient does not have a completely isometric representation on a finite dimensional Hilbert space. This complements a result of [11] which implies that the quotient has an isometric representation on a Hilbert space of dimension twice the cardinality ofE.     

Intertwining liftings and Schur contractions

In their paper [5], C. Foias and A. Frazho have found explicit formulas for the Schur contraction associated with a given contractive intertwining lifting. In this Note we give a more direct and probably the simplest way to find the Schur contraction for a given contractive intertwining lifting.     

Dilation to the unilateral shifts

The classical result of Foias says that an operator power dilates to a unilateral shift if and only if it is aC ₀ contraction. In this paper, we consider the corresponding question of dilating to a unilateral shift. We show tht for contractions with at least one defect index finite, dilation and power dilation to some unilateral shift amount to the same thing. The only difference is on the minimum multiplicity of the unilateral shift to which the contraction can be (power) dilated. We also obtain a characterization of contractions which are finite-rank perturbations of a unilateral shift, generalizing the rank-one perturbation result of Nakamura.     

Caratheodory interpolation kernels

Kernels over the unit disk for which a version of Carathéodory interpolation is true are characterized in a simple computationally verifiable manner.     

Compressions of Stable Contractions

The stability of compressions of stable contractions is studied and a sufficient orbit condition is given. On the other hand, it is shown that there are non-stable compressions of the 1-dimensional backward shift and a complete characterization of weighted unilateral shifts with this property is provided. Dilations of bilateral weighted shifts to backward shifts are also considered.