Compact operators on Hilbert modules

We prove that an adjointable contraction acting on a countably generated Hilbert module over a separable unital C*-algebra is compact if and only if the set of its second contractive perturbations is separable.

     

Subspace hypercyclicity for Toeplitz operators

In this short note we give a sufficient condition for a coanalytic Toeplitz operator on the Hardy–Hilbert space to be subspace-hypercyclic.     

Reflection symmetry and symmetrizability of Hilbert space operators

A general factorization theorem for symmetrizable operators relating their spectra to spectra of selfadjoint operators induced by minimal factorizations is established. Its modified version essentially improves and completes a theorem of Jorgensen, which concerns diagonalizing operators with reflection symmetry.

     

A note on supercyclic vectors of Hilbert space operators

Let H be an infinite dimensional complex Hilbert space and T be a bounded linear operator on H. We show that if there exists $x\in H$ such that the closure of $\{\alpha T^{n}x:\alpha \in \mathbb{C},n?0\}$ is H, then there is a subsequence ${(n_k)}_{k=1}^{\infty }$ such that the closed linear span of $\{T^{n_k}x:k?1\}$ is not the whole space H.     

Diagonalization of compact operators in Hilbert modules over finiteW*-algebras

It is known that a continuous family of compact self-adjoint operators can be diagonalized pointwise. One can consider this fact as a possibility of diagonalization of the compact operators on Hilbert modules over a commutativeW*-algebra. The aim of the present paper is to generalize this fact to a finiteW*-algebraA not necessarily commutative. We prove that for a compact operatorK acting on the right HilbertA-moduleH* _A dual toH _A under slight restrictions one can find a set of eigenvectorsx _i H* _A and a non-increasing sequence of eigenvalues _i A such thatK x _i=x _{i i} and the selfdual HilbertA-module generated by these eigenvectors is the wholeH* _A. As an application we consider the Schrödinger operator in a magnetic field with irrational magnetic flow as an operator acting on a Hilbert module over the irrational rotation algebraA and discuss the possibility of its diagonalization.     

On lifting of operators to Hilbert spaces induced by positive selfadjoint operators

We introduce the notion of induced Hilbert spaces for positive unbounded operators and show that the energy spaces associated to several classical boundary value problems for partial differential operators are relevant examples of this type. The main result is a generalization of the Krein-Reid lifting theorem to this unbounded case and we indicate how it provides estimates of the spectra of operators with respect to energy spaces.     

The circular Bedrosian identity for translation‐invariant operators: existence and characterization

The analytic signal method via the circular Hilbert transform is a critical tool in the time–frequency analysis of signals of finite duration. The circular Bedrosian identity is of major theoretical and practical value in the method. The identity holds whenever the Fourier coefficients of f,g∈L²([?π,π]) are respectively supported on A = [?n,m] and for some non‐negative integers 0≤n,m≤+∞. In this note, we investigate the existence of such an identity for a general‐bounded linear translation‐invariant operator on L²([?π,π]^d) and for general support sets . We give an insightful geometric characterization of the support sets for the existence. In addition, we find all the support sets for the partial Hilbert transforms. Copyright © 2015 John Wiley & Sons, Ltd.     

Norm estimates for functions of two operators on tensor products of Hilbert spaces

Analytic operator valued functions of two operators on tensor products of Hilbert spaces are considered. A precise norm estimate is established. Applications to operator differential equations are also discussed. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Generalized random linear operators on a Hilbert space

In this paper, we are concerned with generalized random linear operators on a separable Hilbert space. Generalized random linear bounded operators, generalized random linear normal operators and generalized random linear self-adjoint operators are defined and investigated. The spectral theorems for generalized random linear normal operators and generalized random linear self-adjoint operators are obtained.     

Sampling in a Hilbert space

An analog of the Whittaker-Shannon-Kotel'nikov sampling theorem is derived for functions with values in a separable Hilbert space. The proof uses the concept of frames and frame operators in a Hilbert space. One of the consequences of this theorem is that it allows us to derive sampling theorems associated with boundary-value problems and some homogeneous integral equations, which in turn gives us a generalization of another sampling theorem by Kramer.

     

Elementary operators on algebras of unbounded operators

Summary Structural results about elementary operators of length one, local elementary operators and injectivity preserving maps are proved. These are generalizations of results concerning algebras of bounded operators on Banach spaces to algebras of unbounded operators on Hilbert spaces.     

Riesz-type representation for positive linear operators preserving continuity

A Riesz-type representation theorem is given for all positive linear endomorphisms of the space of continuous functions on a compact interval.     

拟相似半控制算子的本质谱

讨论两个拟相似半控制算子的本质谱之间的关系,应用算子谱的精密结构的分析方法,给出拟相似半控制算子的本质谱相等的若干充分条件,所得结果改进和发展了L.R.W illiam s在文[7-8]中的有关结果.     

Uniform primeness of the Jordan algebra of symmetric operators

In this note we establish the best possible constant for the general lower estimate for the Jacobson - McCrimmon operator on the algebra of symmetric operators acting on a Hilbert space.

     

Estimates for norms of resolvents of~operators on tensor products of Hilbert spaces

A class of linear operators on tensor products of Hilbert spaces is considered. That class contains integro-differential operators arising in various applications. Estimates for the norm of the resolvent of considered operators are derived. By virtue of the obtained estimates, the spectrum of perturbed operators is investigated. These results are new even in the finite-dimensional case. Applications to integro-differential operators are also discussed. This revised version was published online in August 2006 with corrections to the Cover Date.     

Diagonalization of compact operators on Hilbert modules overC *-algebras of real rank zero

The classical Hilbert-Schmidt theorem can be extended to compact operators on HilbertA-modules overW ^*-algebras of finite type; i.e., with minor restrictions, compact operators onH* _A can be diagonalized overA. We show that ifB is a weakly denseC ^*-subalgebra ofA with real rank zero and if some additional condition holds, then the natural extension fromH _B toH* _A ⊃H _B of a compact operator can be diagonalized so that the diagonal elements belong to the originalC ^*-algebraB. Translated fromMatematicheskie Zametki, Vol. 62, No. 6, pp. 865–870, December, 1997. Translated by O. V. Sipacheva     

A characterization of positive semidefinite operators on a Hilbert space

In this paper, we show that, under certain conditions, a Hilbert space operator is positive semidefinite whenever it is positive semidefinite plus on a closed convex cone and positive semidefinite on the polar cone (with respect to the operator). This result is a generalization of a result by Han and Mangasarian on matrices.This paper was presented at the 90th Annual Meeting of the American Mathematical Society, Louisville, Kentucky, January 25–28, 1984.     

Representations of Lie algebras by non-skewselfadjoint operators in Hilbert space

We study non-skewselfadjoint representations of a finite dimensional real Lie algebra $\mathfrak{g}$. To this end we embed a non-skewselfadjoint representation of $\mathfrak{g}$ into a more complicated structure, that we call a $\mathfrak{g}$-operator vessel and that is associated to an overdetermined linear conservative input/state/output system on the corresponding simply connected Lie group $\mathfrak{G}$. We develop the frequency domain theory of the system in terms of representations of $\mathfrak{G}$, and introduce the joint characteristic function of a $\mathfrak{g}$-operator vessel which is the analogue of the classical notion of the characteristic function of a single non-selfadjoint operator. As the first non-commutative example, we apply the theory to the Lie algebra of the $ax+b$ group, the group of affine transformations of the line.