Sharp two-weight inequalities for singular integrals, with applications to the Hilbert transform and the Sarason conjecture

We prove two-weight norm inequalities for Calderón-Zygmund singular integrals that are sharp for the Hilbert transform and for the Riesz transforms. In addition, we give results for the dyadic square function and for commutators of singular integrals. As an application we give new results for the Sarason conjecture on the product of unbounded Toeplitz operators on Hardy spaces.     

Lifting strong commutants of unbounded subnormal operators

Various theorems on lifting strong commutants of unbounded subnormal (as well as formally subnormal) operators are proved. It is shown that the strong symmetric commutant of a closed symmetric operatorS lifts to the strong commutant of some tight selfadjoint extension ofS. Strong symmetric commutants of orthogonal sums of subnormal operators are investigated. Examples of (unbounded) irreducible subnormals, pure subnormals with rich strong symmetric commutants and cyclic subnormals with highly nontrivial strong commutants are discussed.This work was supported by the KBN grant # 2P03A 041 10.     

Nonlinear mean ergodic theorems for nonexpansive semigroups in Banach spaces

We prove two nonlinear ergodic theorems for noncommutative semigroups of nonexpansive mappings in Banach spaces. Using these results, we obtain some nonlinear ergodic theorems for discrete and one-parameter semigroups of nonexpansive mappings. Dedicated to Professors Albrecht Dold and Ed Fadell     

A harmonic-type maximal principle in the three chains completion problem

In this note, we prove a harmonic-type maximal principle for the Schur parametrization of all contractive interpolants in the three chains completion problem (see [4]), which is analogous to the maximal principle proven in [2] in case of the Schur parametrization of all contractive intertwining liftings in the commutant lifting theorem.     

Approximation of approximation numbers by truncation

LetA be a bounded linear operator onsome infinite-dimensional separable Hilbert spaceH and letA _n be the orthogonal compression ofA to the span of the firstn elements of an orthonormal basis ofH. We show that, for eachk1, the approximation numberss _k(A_n) converge to the corresponding approximation numbers _k(A) asn. This observation implies almost at once some well known results on the spectral approximation of bounded selfadjoint operators. For example, it allows us to identify the limits of all upper and lower eigenvalues ofA _n in the case whereA is selfadjoint. These limits give us all points of the spectrum of a selfadjoint operator which lie outside the convex hull of the essential spectrum. Moreover, it follows that the spectrum of a selfadjoint operatorA with a connected essential spectrum can be completely recovered from the eigenvalues ofA _n asn goes to infinity.     

k-Hyponormality of multivariable weighted shifts

We characterize joint k-hyponormality for 2-variable weighted shifts. Using this characterization we construct a family of examples which establishes and illustrates the gap between k-hyponormality and (k+1)-hyponormality for each k?1. As a consequence, we obtain an abstract solution to the Lifting Problem for Commuting Subnormals.     

Subnormal operators, self-commutators, and pseudocontinuations

We study pure subnormal operators whose self-commutators have zero as an eigenvalue. We show that various questions in this are closely related to questions involving approximation by functions satisfying and to the study ofgeneralized quadrature domains.First some general results are given that apply to all subnormal operators within this class; then we consider characterizing the analytic Toeplitz operators, the Hardy operators and cyclic subnormal operators whose self-commutators have zero as an eigenvalue.     

Sarason interpolation and Toeplitz corona theorem for almost periodic matrix functions

Sarason interpolation and Toeplitz corona problems are studied for almost periodic matrix functions. Recent results on almost periodic factorization and related generalized Toeplitz operators are the main tools in the study.Supported in part by NSF Grant DMS 9500912Supported in part by NATO Collaborative Research Grant 950332Supported by NSF Grant DMS 9500924     

A trace formula for multiplication operators on invariant subspaces of the Bergman space

Supposef is a bounded analytic function in the unit disk andM _f is the multiplication operator on an invariant subspaceI of the Bergman space. We show that wheren=dim(IzI).     

On statistical convergence of difference sequences of fractional order and related Korovkin type approximation theorems

Abstract

Prior to investigating on sequence spaces and their convergence, we study the notion of statistical convergence of difference sequences of fractional order α ∈ ?. As generalizations of previous works, this study includes several special cases under different limiting conditions of α, such as the notion of statistical convergence of difference sequences of zeroth and mth (integer) order. In fact, we study certain new results on statistical convergence via the difference operator Δ^α and interpret them to those of previous works. Also, by using the convergence of Δ^α-summable sequences which is stronger than statistical convergence of difference sequences, we apply classical Bernstein operator and a generalized form of Meyer-Konig and Zeller operator to construct an example in support of our result. Also, we study the rates of Δ^α-statistical convergence of positive linear operators.     

Recursively generated weighted shifts and the subnormal completion problem,II

Both authors were partially supported by grants from NSF Raúl Curto was partially supported by a University of Iowa faculty scholar award     

Quadratically hyponormal weighted shifts and their examples

We discuss some characterizations for the quadratical hyponormal unilateral weighted shiftW with a weight sequence , which give a distinction example for quadratical hyponormality and positively quadratical hyponormality. In addition, we consider a recursively quadratically hyponormal weighted shift with a recursive weight : {ie480-1} which is a back step extension of subnormal completion ofu,v, andw with0, and prove that the recursively weighted shiftW is quadratically hyponormal if and only if it is positively quadratically hyponormal.Research partially supported by KOSEF 971-0102-006-2 and the Basic Science Research Institute Program, Ministry of Education, 1997, BSRI-97-1401.     

The weighted commutant lifting theorem in the coupling approach

A simple coupling argument is seen to provide an alternate proof of the weighted commutant lifting theorem of Biswas, Foias and Frazho (which includes, as a particular case, the abstract Nehari theorem of Treil and Volberg).     

Bernstein functions, complete hyperexpansivity and subnormality-II

Multivariable Bernstein functions are used to discover some interesting connections between multivariable completely hyperexpansive weighed shifts and multivariable subnormal weighted shifts.     

Toeplitz operators hyponormal with the unilateral shift

For the unilateral shift operator U on the Hardy space H²(T), we describe conditions on operators T, acting on H²(T), that are necessary and sufficient for the pair (U, T) to be jointly hyponormal. One necessary condition is that T be a Toeplitz operator. Consequently, we study certain nonanalytic symbols that give rise to Toeplitz operators hyponormal with the shift, and thereby obtain examples of noncommuting, jointly hyponormal pairs.Supported in part by a research grant from NSERC     

Quadratically hyponormal weighted shifts with two equal weights

We present an extensive analysis ofpositively quadratically hyponormal weighted shiftsW with ₀ = ₁ = 1. Our main result states that such weighted shifts abound! Specifically, by focusing on recursively generated weighted shifts of the form, we establish that the planar set is positively quadratically hyponormal} is a closed convex, set with nonempty interior. In addition, we are able to describe in detail the boundary of Research partially supported by NSF grants DMS-9401455 and DMS-9800931Research partially supported by KOSEF grant 971-0102-006-2 and by TGRC-KOSEF     

Brezis-Nirenberg type theorems and multiplicity of positive solutions for a singular elliptic problem

We study Brezis-Nirenberg type theorems for the equation

     

Ergodic Properties for Regular A-Contractions

The current article pleads for the possibility to obtain an orthogonal decomposition of a Hilbert space which is induced by a regular A-contraction defined in [9, 10], A being a positive operator on . The decomposition generalizes the well-known decomposition related to a contraction T of , which gives the ergodic character of T. This decomposition is being used to prove certain versions for regular A-contractions of the mean ergodic theorem, as well as a version of Patil’s theorem from [8]. Also, we characterize the solutions of corresponding functional equations in the range of A^1/2, by analogy with the result of Lin-Sine in [7].     

Recursively generated weighted shifts and the subnormal completion problem

Both authors were partially supported by grants from NSF Raúl Curto was partially supported by a University of Iowa faculty scholar award