On operators commuting with Toeplitz operators modulo the finite rank operators

Let X be a bounded linear operator on the Hardy space H² of the unit disk. We show that if is of finite rank for every inner function θ, then X=T_?+F for some Toeplitz operator T_? and some finite rank operator F on H². This solves a variant of an open question where the compactness replaces the finite rank conditions.     

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.     

On log-hyponormal operators

LetTB(H) be a bounded linear operator on a complex Hilbert spaceH.TB(H) is called a log-hyponormal operator itT is invertible and log (TT ^*)log (T ^* T). Since log: (0, )(–,) is operator monotone, for 0<p1, every invertiblep-hyponormal operatorT, i.e., (TT ^*) ^p (T ^* T) ^p , is log-hyponormal. LetT be a log-hyponormal operator with a polar decompositionT=U|T|. In this paper, we show that the Aluthge transform is . Moreover, ifmeas ((T))=0, thenT is normal. Also, we make a log-hyponormal operator which is notp-hyponormal for any 0<p.This research was supported by Grant-in-Aid Research No. 10640185     

Cellular-indecomposable subnormal operators III

A bounded operatorT is called cellular-indecomposable ifL M {0} wheneverL andM are nonzero invariant subspaces forT. We prove that a cyclic subnormal operator is cellular-indecomposable if and only if it is quasi-similar to an analytic Toeplitz operator whose symbol is a weak-star generator ofH . This completes our previous work [5], [6].     

Angular cutting for log-hyponormal operators

We define an angular section for a log-hyponormal operator on a Hilbert space, cut by an arc on the unit circle, and establish various spectral properties for it, which correspond to the properties of an angular section for ap-hyponormal opertor.Dedicated to Professor Norio Shimakura on the occasion of his sixtieth birthdayThis research was supported by Grant-in-Aid Research 1 No. 12640187.     

Quasi-similarp-hyponormal operators

IfA _i i=1, 2 are quasi-similarp-hyponormal operators such thatUi is unitary in the polar decompositionA _i =U _i |A _i |, then (A ₁)=(A ₂) and _c(A₁) = _e(A₂). Also a Putnam-Fuglede type commutativity theorem holds for p-hyponomral operators.     

Polaroid operators satisfying Weyl’s theorem

A Banach space operator T is polaroid and satisfies Weyl’s theorem if and only if T is Kato type at points λ ∈ iso σ(T) and has SVEP at points λ not in the Weyl spectrum of T. For such operators T, f(T) satisfies Weyl’s theorem for every non-constant function f analytic on a neighborhood of σ(T) if and only if f(T^∗) satisfies Weyl’s theorem.     

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.     

On self-dual subnormal operators

We improve the result of C. C. Huang about self-dual subnormal operators, and consider the converse of this result.     

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     

Inequalities of Putnam and Berger-Shaw forp-quasihyponormal operators

For an operatorT satisfying thatT ^*(T ^* T–TT ^*)T0, we shall show that and, moreover, tr itT isn-multicyclic.For an operatorT satisfying thatT ^* {(T ^* T) ^p –(TT ^*) ^p }T0 for somep (0, 1], we shall show that and, moreover, ifT isn-multicyclic.     

Extensions of normal operators

We prove that every one dimensional extension of a separably acting normal operator has a cyclic commutant, and that every non-algebraic normal operator has a two-dimensional extension which fails to have a cyclic commutant. Contrasting this, we prove that ifT is an extension of a normal operator by an algebraic operator then the weakly closed algebraW(T) has a separating vector.Partially supported by NSF Grant DMS-9107137     

Aluthge transforms of operators

Associated with every operatorT on Hilbert space is its Aluthge transform (defined below). In this note we study various connections betweenT and , including relations between various spectra, numerical ranges, and lattices of invariant subspaces. In particular, we show that if has a nontrivial invariant subspace, then so doesT, and we give various applications of our results.     

Generalization of the Newman-Shapiro isometry theorem and Toeplitz operators

We give a generalization of the Newman-Shapiro Isometry Theorem to the case of Hilbert space-valued entire functions, which are square-summable with respect to the Gaussian measure on ⁿ , together with some applications in the theory of Toeplitz operators with operator-valued symbols. The study of various properties (such as density of domains, cores, closedness and boundedness from below) of these operators in illustrated with many relevant examples.Research supported by KBN under grant no. 2 P03A 041 10.     

Some generalized theorems onp-hyponormal operators

A bounded linear operatorT is calledp-Hyponormal if (T ^*T)^p(TT ^*)^p, 0<p1. In Aluthge [1], we studied the properties of p-hyponormal operators using the operator . In this work we consider a more general operator , and generalize some properties of p-hyponormal operators obtained in [1].     

Square roots of semihyponormal operators have scalar extensions

In this paper, we study some properties of , i.e., square roots of semihyponormal operators. In particular we show that an operator has a scalar extension, i.e., is similar to the restriction to an invariant subspace of a (generalized) scalar operator (in the sense of Colojoar?-Foia?). As a corollary, we obtain that an operator has a nontrivial invariant subspace if its spectrum has interior in the plane.     

Hyponormality of Toeplitz operators with rational symbols

In this article we introduce a notion of `division' for rational functions and then give a criterion for hyponormality of (f, g are rational functions) in the cases where g divides f. Furthermore we show that we may assume, without loss of generality, that g divides f when we consider the hyponormality of . Supported in part by a grant from Faculty Research Fund, Sungkyunkwan University, 2004. Supported in part by a grant (R14-2003-006-01000-0) from the Korea Research Foundation.     

Subnormality and composition operators on the Bergman space

Following the work of C. Cowen and T. Kriete on the Hardy space, we prove that under a regularity condition, all composition operators with a subnormal adjoint on A²(D) have linear fractional symbols of the form. Moreover, we show that all composition operators on the Bergman space having these symbols have a subnormal adjoint, with larger range for the parameterr than found in the Hardy space case.