Commutators and accretive operators

In this paper, singular values of commutators of Hilbert space operators are estimated. To this aim the accretivity of a transform of the operators is applied. Some recent results of Kittaneh [F. Kittaneh, Singular value inequalities for commutators of Hilbert space operators, Linear Algebra Appl. 430 (2009) 2362-2367] are extended.     

Noncommuting Domination in Krein Spaces Via Commutators of Block Operator Matrices

The commutators of 2 × 2 block operator matrices with (unbounded) operator entries are investigated. The matrix representation of a symmetric operator in a Krein space is exploited. As a consequence, the domination result due to Cichoń, Stochel and Szafraniec is extended to the case of Krein spaces.     

An elementary operator with log-hyponormal, p-hyponormal entries

A Hilbert space operator A∈B(H) is p-hyponormal, A∈(p-H), if |A^∗|^2p?|A|^2p; an invertible operator A∈B(H) is log-hyponormal, A∈(?-H), if log(TT^∗)?log(T^∗T). Let d_AB=δ_AB or ?_AB, where δ_AB∈B(B(H)) is the generalised derivation δ_AB(X)=AX-XB and ?_AB∈B(B(H)) is the elementary operator ?_AB(X)=AXB-X. It is proved that if A,B^∗∈(?-H)∪(p-H), then, for all complex λ, , the ascent of (d_AB-λ)?1, and d_AB satisfies the range-kernel orthogonality inequality ‖X‖?‖X-(d_AB-λ)Y‖ for all X∈(d_AB-λ)^-1(0) and Y∈B(H). Furthermore, isolated points of σ(d_AB) are simple poles of the resolvent of d_AB. A version of the elementary operator E(X)=A₁XA₂-B₁XB₂ and perturbations of d_AB by quasi-nilpotent operators are considered, and Weyl’s theorem is proved for d_AB.     

Little Hankel operators on the half plane

In this paper we consider a class of weighted integral operators onL ² (0, ) and show that they are unitarily equivalent to Hankel operators on weighted Bergman spaces of the right half plane. We discuss conditions for the Hankel integral operator to be finite rank, Hilbert-Schmidt, nuclear and compact, expressed in terms of the kernel of the integral operator. For a particular class of weights these operators are shown to be unitarily equivalent to little Hankel operators on weighted Bergman spaces of the disc, and the symbol correspondence is given. Finally the special case of the unweighted Bergman space is considered and for this case, motivated by approximation problems in systems theory, some asymptotic results on the singular values of Hankel integral operators are provided.     

A Note on <Emphasis Type="Italic">b</Emphasis>-Weakly Compact Operators

We consider a continuous operator T: E → X where E is a Banach lattice and X is a Banach space. We characterize the b-weak compactness of T in terms of its mapping properties.     

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.     

Singular values, norms, and commutators

Let and X_i, i=1,…,n, be bounded linear operators on a separable Hilbert space such that X_i is compact for i=1,…,n. It is shown that the singular values of are dominated by those of , where ‖·‖ is the usual operator norm. Among other applications of this inequality, we prove that if A and B are self-adjoint operators such that a₁?A?a₂ and b₁?B?b₂ for some real numbers and b₂, and if X is compact, then the singular values of the generalized commutator AX-XB are dominated by those of max(b₂-a₁,a₂-b₁)(X⊕X). This inequality proves a recent conjecture concerning the singular values of commutators. Several inequalities for norms of commutators are also given.     

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 the Range of Elementary Operators

Let B(H) denote the algebra of all bounded linear operators on a separable infinite dimensional complex Hilbert space H into itself. Let A = (A₁,A₂,.., A_n) and B = (B₁, B₂,.., B_n) be n-tuples in B(H), we define the elementary operator by In this paper we initiate the study of some properties of the range of such operators.     

Complete hyperexpansivity, subnormality and inverted boundedness conditions

Athavale introduced in [3] the notion of a completely hyperexpansive operator. In this paper some results concerning powers of completely (alternatingly) hyperexpansive operators (not necessarily bounded) are extended tok-hyperexpansive ones. A semispectral measure is associated with a subnormal contraction as well as with a completely hyperexpansive operator, and an operator version of the Levy-Khinchin representation is obtained. Passing to the Naimark dilation of the semispectral measure, such an operator is related to a positive contraction in a natural way. New characterizations of a completely hyperexpansive operator and a subnormal contraction are given. The power bounded completely hyperexpansive operators are characterized. All these are illustrated using weighted shifts.     

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).     

Commutators on Half-Spaces

We study the boundedness and compactness of commutators on , where and are defined by and respectively. If satisfies some upper and lower estimates, then we obtain a necessary and sufficient condition for to be bounded or compact on for . The reproducing kernel of the harmonic Bergman space of can be shown to satisfy all the required estimates. Our result is the real variable analogue of the complex variable one for commutators associated with an analytic reproducing kernel.     

Cyclicity results for Jordan and compressed Toeplitz operators

A vectorx in a Hilbert spaceH iscyclic for a bounded linear operatorTHH if the closed linear span of the orbit {T ⁿ xn0} ofx underT is all ofH. Operators which have a cyclic vector are said to be cyclic.Jordan operators are the infinite direct sums of Jordan cells acting on finite- dimensional Hilbert spaces. Necessary and sufficient conditions for a Jordan operator to be cyclic are given (see Corollary 6). In this case, a dense set of cyclic vectors is exhibited (see Corollary 4). Sufficient conditions for uncountable collections of cyclic Jordan operators to have a common cyclic vector are given and, in this case, a dense set of common cyclic vectors is exhibited (see Corollary 9).Analogues of these cyclicity results for Jordan operators are obtained for compressions of analytic Toeplitz operatorsT _A FAF on the Hardy spaceH ² to subspaces (BH ²) invariant for the backward shiftT _z ^* whereB is a Blaschke product by showing that such compressions are quasisimilar to Jordan operators.     

Paraproducts and Hankel operators of Schatten class via p-John-Nirenberg Theorem

We give an interpolation-free proof of the known fact that a dyadic paraproduct is of Schatten-von Neumann class S_p, if and only if its symbol is in the dyadic Besov space B_p^d. Our main tools are a product formula for paraproducts and a “p-John-Nirenberg-Theorem” due to Rochberg and Semmes.We use the same technique to prove a corresponding result for dyadic paraproducts with operator symbols.Using an averaging technique by Petermichl, we retrieve Peller's characterizations of scalar and vector Hankel operators of Schatten-von Neumann class S_p for 1<p<∞. We then employ vector techniques to characterise little Hankel operators of Schatten-von Neumann class, answering a question by Bonami and Peloso.Furthermore, using a bilinear version of our product formula, we obtain characterizations for boundedness, compactness and Schatten class membership of products of dyadic paraproducts.     

Differences of Composition Operators between Weighted Banach Spaces of Holomorphic Functions on the Unit Polydisk

We consider differences of composition operators between given weighted Banach spaces H ^∞ _v or H ⁰ _v of analytic functions defined on the unit polydisk D ^N with weighted sup-norms and give estimates for the distance of these differences to the space of compact operators. We also study boundedness and compactness of the operators. This paper is an extension of [6] where the one-dimensional case is treated. Received: May 15, 2007. Revised: October 8, 2007.