Separable Complex Hilbert Space Research Articles

THE STABILITY OF PROPERTY (gw) UNDER COMPACT PERTURBATION

Let \(\mathcal {H}\) be a complex separable infinite dimensional Hilbert space. In this paper, a necessary and sufficient condition is given for an operator T on \(\mathcal {H}\) to satisfy that f(T) obeys property (gw) for each function f analytic on some neighborhood of σ(T). Also, we investigate the stability of property (gw) under (small) compact perturbations.

Estimates of k-hyperreflexivity constants

The notion of k-hyperreflexivity for spaces of operators between separable complex Hilbert spaces is studied. We consider a few cases when a k-hyperreflexive space S is transformed into a k′-hyperreflexive space S′. We give estimates for the k′-hyperreflexivity constant of the transformed space.

Some unitary similarity invariant sets preservers of skew Lie products

Let H and K be complex separable Hilbert spaces with dimensions at least three, and B(H) the Banach algebra of all bounded linear operators on H. Let Δ(⋅) denote W(⋅) or σε(⋅), where, for A, W(A) stands for the numerical range of A∈B(H) and σε(A) the ε-pseudospectrum of A. It is shown that a bijective map (no algebraic structure assumed) Φ:B(H)→B(K) satisfies that Δ(AB−BA⁎)=Δ(Φ(A)Φ(B)−Φ(B)Φ(A)⁎) for all A,B∈B(H) if and only if there exists a unitary operator U∈B(H,K) such that Φ(A)=μUAU⁎ for all A∈B(H), where μ∈{−1,1}. If Δ(⋅)=W(⋅), then the injectivity assumption on Φ can be omitted.

SELF-ADJOINT INTERPOLATION ON AX=Y IN A TRIDIAGONAL ALGEBRA ALG𝓛

Given operators X and Y acting on a separable Hilbert space <TEX>$\mathcal{H}$</TEX>, an interpolating operator is a bounded operator A such that AX = Y. In this article, we investigate self-adjoint interpolation problems for operators in a tridiagonal algebra : Let <TEX>$\mathcal{L}$</TEX> be a subspace lattice acting on a separable complex Hilbert space <TEX>$\mathcal{H}$</TEX> and let X = (<TEX>$x_{ij}$</TEX>) and Y = (<TEX>$y_{ij}$</TEX>) be operators acting on <TEX>$\mathcal{H}$</TEX>. Then the following are equivalent: (1) There exists a self-adjoint operator A = (<TEX>$a_{ij}$</TEX>) in <TEX>$Alg{\mathcal{L}}$</TEX> such that AX = Y. (2) There is a bounded real sequence {<TEX>${\alpha}_n$</TEX>} such that <TEX>$y_{ij}={\alpha}_ix_{ij}$</TEX> for <TEX>$i,j{\in}\mathbb{N}$</TEX>.

Automorphisms of corona algebras, and group cohomology

In 2007 Phillips and Weaver showed that, assuming the Continuum Hypothesis, there exists an outer automorphism of the Calkin algebra. (The Calkin algebra is the algebra of bounded operators on a separable complex Hilbert space, modulo the compact operators.) In this paper we establish that the analogous conclusion holds for a broad family of quotient algebras. Specifically, we will show that assuming the Continuum Hypothesis, if $A$ is a separable algebra which is either simple or stable, then the corona of $A$ has nontrivial automorphisms. We also discuss a connection with cohomology theory, namely, that our proof can be viewed as a computation of the cardinality of a particular derived inverse limit.

HEREDITARILY HYPERCYCLICITY AND SUPERCYCLICITY OF WEIGHTED SHIFTS

In this paper we first characterize the hereditarily hypercyclicity of the unilateral (or bilateral) weighted shifts on the spaces <TEX>$L^2(\mathbb{N},\mathcal{K})$</TEX> (or <TEX>$L^2(\mathbb{Z},\mathcal{K})$</TEX>) with weight sequence {<TEX>$A_n$</TEX>} of positive invertible diagonal operators on a separable complex Hilbert space <TEX>$\mathcal{K}$</TEX>. Then we give the necessary and sufficient conditions for the supercyclicity of those weighted shifts, which extends some previous results of H. Salas. At last, we give some conditions for the supercyclicity of three different weighted shifts.

Quasinilpotent Part of w-Hyponormal Operators

For a ω-hyponormal operator T acting on a separable complex Hilbert space , we prove that: 1) the quasi-nilpotent part H0 (T－λI) is equal to ker(T－λI); 2) T has Bishop’s property β; 3) if σω (T)={0} , then it is a compact normal operator; 4) If T is an al-gebraically ω-hyponormal operator, then it is polaroid and reguloid. Among other things, we prove that if Tn and Tn* are ω-hyponormal, then T is normal.

AN EXTENSION OF PENROSE’S INEQUALITY ON GENERALIZED INVERSES TO THE SCHATTEN p-CLASSES

Abstract Let B(H) be the algebra of all bounded linear operators on a complex separable infinite dimensional Hilbert space H. In this paper we minimize the Schatten Cp-norm of suitable affine mappings from B(H) to Cp, using convex and differential analysis (Gâteaux derivative) as well as input from operator theory. The mappings considered generalize Penrose’s inequality which asserts that if A+ and B+ denote the Moore-Penrose inverses of the matrices A and B, respectively, then ||AXB − C||2 ≥ ||AA+CB+B − C||2, with A+CB+ being the unique minimizer of minimal ||:||2 norm. The main results obtained characterize the best Cp-approximant of the operator AXB.

Bilocal *-automorphisms of B(H)

In this note we show that the bilocal *-automorphisms of the C*-algebra B(H) of all bounded linear operators acting on a complex infinite dimensional separable Hilbert space H are precisely the unital algebra *-endomorphisms of B(H).

Recurrent Linear Operators

We study the notion of recurrence and some of its variations for linear operators acting on Banach spaces. We characterize recurrence for several classes of linear operators such as weighted shifts, composition operators and multiplication operators on classical Banach spaces. We show that on separable complex Hilbert spaces the study of recurrent operators reduces, in many cases, to the study of unitary operators. Finally, we study the notion of product recurrence and state some relevant open questions.

Unitary representations of finite abelian groups realizable by an action

Let Γ be a finite abelian group and let H be an infinite-dimensional separable complex Hilbert space. We prove that the set of realizable by an action unitary representations of Γ on H is comeager in the space of all unitary representations of Γ on H.

SOME REMARKS ON $n$-POWER CLASS(Q) OPERATORS

The purpose of the present paper is to give some results relate to a class of linear bounded operators, known as n-power class(Q) operators acting on infinite complex separable Hilbert space H. n-power class(Q) operators is extension of class(Q) operators and class of n-normal operators The class of n-power class (Q) operators was defined by S. Panayappan and N. Sivamani in [1], where they have given some of their properties. Based, mainly, on [1], [3], [4], we contribute with some others properties of such operators. AMS Subject Classification: 47B20

Algebraic reflexivity of isometry groups and automorphism groups of some operator structures

We establish the algebraic reflexivity of three isometry groups of operator structures: the group of all surjective isometries on the unitary group, the group of all surjective isometries on the set of all positive invertible operators equipped with the Thompson metric, and the group of all surjective isometries on the general linear group of B(H), the operator algebra over a complex infinite dimensional separable Hilbert space H. We show that those isometry groups coincide with certain groups of automorphisms of corresponding structures and hence we also obtain the reflexivity of some automorphism groups.

Subscalarity of operators for which T *T and T + T * commute

In this paper, we study operators for which T ∗ T and T + T ∗ commute. Let Θ denote the class of such operators in L(H). We show that every operator in Θ is subscalar of order 4. From this result, we give partial solutions to the invariant subspace problem. In addition, we examine some extensions of operators in Θ. Finally, we prove that if T belongs to Θ, then Weyl’s theorem holds for T and the spectral mapping theorem corresponding to the Weyl spectrum is satisfied for T . Let H be a complex separable Hilbert space and let L(H) denote the algebra of all bounded linear operators on H .I fT ∈L (H), then we write σ(T ), σp(T ), σap(T ), and σe(T ) for the spectrum, the point spectrum, the approximate point spectrum, and the essential spectrum of T , respectively. The notation r(T ) is used for the spectral radius of T . An operator S ∈L (H) is called scalar of order m (0 m ∞) if it possesses a spectral distribution of order m, that is, if there exists a continuous unital homomorphism of topological algebras Φ: C m 0 (C) −→ L (H)

Lie Triple Derivations of CSL Algebras

Let Open image in new window be a commutative subspace lattice generated by finite many commuting independent nests on a complex separable Hilbert space Open image in new window with Open image in new window, and Open image in new window the associated CSL algebra. It is proved that every Lie triple derivation from Open image in new window into any σ-weakly closed algebra Open image in new window containing Open image in new window is of the form X→XT−TX+h(X)I, where Open image in new window and h is a linear mapping from Open image in new window into ℂ such that h([[A,B],C])=0 for all Open image in new window.

Singular Value Inequalities for Compact Normal Operators

We give singular value inequality to compact normal operators, which states that if is compact normal operator on a complex separable Hilbert space, where is the cartesian decomposition of , then Moreover, we give inequality which asserts that if is compact normal operator, then .Several inequalities will be proved.

More Results on Singular Value Inequalities for Compact Operators

The well-known arithmetic-geometric mean inequality for singular values, according to Bhatia and Kittaneh, says that if and are compact operators on a complex separable Hilbert space, then Hirzallah has proved that if are compact operators, then We give inequality which is equivalent to and more general than the above inequalities, which states that if are compact operators, then

On Quasi-Class A Operators

Abstract Let H be a separable infinite dimensional complex Hilbert space, and let B(H) denote the algebra of all bounded linear operators on H. Let A;B be operators in B(H). In this paper we prove that if A is quasi-class A and B* is invertible quasi-class A and AX = XB, for some X ∈ C2 (the class of Hilbert-Schmidt operators on H), then A*X = XB*. We also prove that if A is a quasi-class A operator and f is an analytic function on a neighborhood of the spectrum of A, then f(A) satisfies generalized Weyl's theorem. Other related results are also given.

On Realization of Generalized Effect Algebras

A well-known fact is that there is a finite orthomodular lattice with an order determining set of states which is not representable in the standard quantum logic, the lattice L(H) of all closed subspaces of a separable complex Hilbert space. We show that a generalized effect algebra is representable in the operator generalized effect algebra GD(H) of effects of a complex Hilbert space H iff it has an order determining set of generalized states. This extends the corresponding results for effect algebras of Riečanová and Zajac. Further, any operator generalized effect algebra GD(H) possesses an order determining set of generalized states.

GENERALIZED WEYL'S THEOREM FOR FUNCTIONS OF OPERATORS AND COMPACT PERTURBATIONS

Let H be a complex separable infinite dimensional Hilbert space. In this paper, a necessary and sufficient condition is g for an operator T on H to satisfy that f(T) obeys generalized Weyl's the- orem for each function f analytic on some neighborhood of �(T). Also we investigate the stability of generalized Weyl's theorem under (small) compact perturbations. Throughout this paper, H will always denote a complex separable infinite dimensional Hilbert space. We denote by B(H) the algebra of all bounded linear operators on H, and by K(H) the ideal of compact operators in B(H). Let T 2 B (H). We denote by �(T) andp(T) the spectrum of T and the point spectrum of T respectively. Denote by kerT and ranT the kernel of T and the range of T respectively. T is called a semi-Fredholm operator, if ranT is closed and either dimkerT or dimkerTis finite; in this case, indT := dimkerT −dimkerTis called the index of T. In particular, if −1 < indT <