On a class of operators with finiterank self-commutators

This paper studies the class of pure operatorsA on a Hilbert space satisfying dimK _A <, where . The main tool is a pair of matrices and . A reproducing kernel Hilbert space model is introduced for a subclass of this class of operators. Some theorems are established for some subnormal operators as well as hyponormal operators in this class.     

On Compact Perturbations of Locally Definitizable Selfadjoint Relations in Krein Spaces

The aim of this paper is to prove two perturbation results for a selfadjoint operator A in a Krein space which can roughly be described as follows: (1) If is an open subset of and all spectral subspaces for A corresponding to compact subsets of have finite rank of negativity, the same is true for a selfadjoint operator B in for which the difference of the resolvents of A and B is compact. (2) The property that there exists some neighbourhood of such that the restriction of A to a spectral subspace for A corresponding to is a nonnegative operator in is preserved under relative perturbations in form sense if the resulting operator is again selfadjoint. The assertion (1) is proved for selfadjoint relations A and B. (1) and (2) generalize some known results.     

Towards a model theory for 2-hyponormal operators

We introduce the notion ofweak subnormality, which generalizes subnormality in the sense that for the extension ofT we only require that hold forf ; in this case we call a partially normal extension ofT. After establishing some basic results about weak subnormality (including those dealing with the notion of minimal partially normal extension), we proceed to characterize weak subnormality for weighted shifts and to prove that 2-hyponormal weighted shifts are weakly subnormal. Let { _n } _n=0 be a weight sequence and letW denote the associated unilateral weighted shift on . IfW is 2-hyponormal thenW is weakly subnormal. Moreover, there exists a partially normal extension on such that (i) is hyponormal; (ii) ; and (iii) . In particular, if is strictly increasing then can be obtained as

Equations arising from Jordan *-derivation pairs

Summary. We study certain functional equations derived from the definition of a Jordan *-derivation pair. More precisely, if A is a complex *-algebra and M is a bimodule over A, having the structure of a complex vector space compatible with the structure of A, such that implies m = 0 and implies m = 0 and if are unknown additive mappings satisfying then E and F can be represented by double centralizers. The obtained result implies that one of the equations in the definition of a Jordan *-derivation pair is redundant. Furthermore, a remark on the extension of this result to unknown additive mappings such that is given in a special case.     

Non-semibounded sesquilinear forms and left-indefinite Sturm-Liouville problems

By Kato's First and Second Representation Theorem a closed densely defined semibounded hermitian sesquilinear form [·,·] in a Hilbert spaceH can be represented by a selfadjoint operatorT inH and if, in particular, the formt[·,·] is nonnegative thenD(t)=D(T ^1/2 ). In the present note this result is generalized to non-semibounded forms by means of Krein space methods. An application to the form where the functionp changes its sign leads to an expansion theorem for the Sturm-Liouville problem –(pu)=u,u(a)=u(b)=0 with respect to the weighted Sobolev norm .     

On Bergman-Toeplitz operators with commutative symbol algebras

Let be the unit disk in, be the Bergman space, consisting of all analytic functions from , and be the Bergman projection of onto . We constructC ^*-algebras , for functions of which the commutator of Toeplitz operators [T _a ,T _b ]=T _a T _b –T _b T _a is compact, and, at the same time, the semi-commutator [T _a ,T _b )=T _a T _b –T _ab is not compact.It is proved, that for each finite set =n ₀,n ₁, ...,n _m , where 1=n _{0 1} <... _m , andn _k {}, there are algebras of the above type, such that the symbol algebras Sym of Toeplitz operator algebras arecommutative, while the symbol algebras Sym of the algebras , generated by multiplication operators and , haveirreducible representations exactly of dimensions n ₀,n ₁,..., n _m .This work was partially supported by CONACYT Project 3114P-E9607, México.     

On causality in commutant lifting theorem. I

Two functionals (A) and for an operatorA were introduced in [11] for the study of causality in commutant lifting theory. In this paper we give sufficient and necessary conditions for in a special case. We prove that in this case , and we show by some examples related to nonlinear system control that is the best constant in our inequality.     

Adjoints of slant Toeplitz operators II

Let be the unit circle {z|z|=1} and _n c _n e ⁱⁿ be a bounded measurable function on . Theslant Toeplitz operator A onL ² ( ) is defined by A e _n ,e _m =c _2m–n for allm, n wheree _n (z)=z ⁿ , . In this paper, we continue the study initiated in [6] onA ^* , the adjoint ofA . Specifically, we will show that for a certain dense set of continuous functions on ,A ^* is similar to some constant multiple of either a shift, or a shift plus a rank one operator.     

The Kreîn-Langer problem for Hilbert space operator valued functions on the band

We deal with the Kreîn-Langer problem for -valued functions on the band (–2a, 2a)×, where is the algebra of continuous linear operators on a Hilbert space ,a a finite positive number and a topological Abelian group. We show that every weakly continuous -indefinite function admits a strongly continuous -indefinite continuation to × with the same indefiniteness index . We give a parametrization of the extensions in terms of operator-valued Schur functions.     

Semi-commutators of Toeplitz operators on the Bergman space

In this paper several necessary and sufficient conditions are obtained for the semi-commutator of Toeplitz operators andT _g with bounded pluriharmonic symbols on the unit ball to be compact on the Bergman space. Using -harmonic function theory on the unit ball we show that with bounded pluriharmonic symbolsf andg is zero on the Bergman space of the unit ball or the Hardy space of the unit sphere if and only if eitherf org is holomorphic.The author was supported in part by the National Science Foundation.     

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.     

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.     

The boundedness below of 2×2 upper triangular operator matrices

Wen and are given we denote byM _C an operator acting on the Hilbert space of the form

Characterizations of logA≥logB and normaloid operators via Heinz inequality

In 1951, Heinz showed the following useful norm inequality:If A, B0and XB(H), then AXB ^r X^1–r A ^r XB ^r holds for r [0, 1]. In this paper, we shall show the following two applications of this inequality:Firstly, by using Furuta inequality, we shall show an extension of Cordes inequality. And we shall show a characterization of chaotic order (i.e., logAlogB) by a norm inequality.Secondly, we shall study the condition under which , where is Aluthge transformation ofT. Moreover we shall show a characterization of normaloid operators (i.e.,r(T)=T) via Aluthge transformation.     

Spectral pictures of Aluthge transforms of operators

In this paper we continue our study, begun in [12], of the relationships between an arbitrary operatorT on Hilbert space and its Aluthge transform . In particular, we show that in most cases the spectral picture ofT coincides with that of , and we obtain some interesting connections betweenT and as a consequence.     

Isometric dilations in nest algebras

LetT be a contraction acting in a separable Hilbert space and leaving invariant a nest of subspaces of . We answer the question: when doesT have an isometric extension to which leaves invariant the nest = {N N :N ;}.     

On the spectrum determined growth assumption and the perturbation ofC 0 semigroups

The spectrum determined growth property ofC ₀ semigroups in a Banach space is studied. It is shown that ifA generates aC ₀ semigroup in a Banach spaceX, which satisfies the following conditions: 1) for any >s(A), sup{R(;A) | Re}<; 2) there is a ₀>(A) such that , xX, and , fX ^*, then (A=s(A). Moreover, it is also shown that ifA=A ₀+B is the infinitesimal generator of aC ₀ semigroup in Hilbert space, whereA ₀ is a discrete operator andB is bounded, then (A)=s(A). Finally the results obtained are applied to wave equation and thermoelastic system.     

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.