Integral Equations and Operator Theory

A complex number λ is an extended eigenvalue of an operator A if there is a nonzero operator X such that AX = λ XA. We characterize the set of extended eigenvalues, which we call extended point spectrum, for operators acting on finite dimensional spaces, finite rank operators, Jordan blocks, and C₀ contractions. We also describe the relationship between the extended eigenvalues of an operator A and its powers. As an application, we show that the commutant of an operator A coincides with that of Aⁿ, n ≥ 2, n ∈ N if the extended point spectrum of A does not contain any n–th root of unity other than 1. The converse is also true if either A or A^* has trivial kernel.     

On the decomposition theorem for meromorphic Fredholm resolvents

In this note results of B. Gramsch and W. Kaballo [8] on the decomposition of meromorphic (semi-) Fredholm resolvents are sharpened. A condition on an Orlicz function is given, under which the singular part in this decomposition can be chosen meromorphic inN , the ideal of -nuclear operators. Then the necessity of this condition is studied. Moreover, it is shown that for the rather steep Orlicz functions relevant to this question,N equalsS , the ideal of -approximable operators.Dedicated to Professor Albert Schneider on the occasion of his 60 ^th birthdayresearch supported by a grant from DAAD     

Applications of the Discrete Weiss Conjecture in Operator Theory

In this paper, we study a discrete version of the Weiss Conjecture. In Section 1 we discuss the Reproducing Kernel Thesis and in Section 2 we introduce the operators which concern us. Section 3 shows how to relate these operators to Carleson embeddings and weighted composition operators, so that we can apply the Carleson measure theorem to obtain conditions for boundedness and compactness of many weighted composition operators. Section 4 contains Theorem 4.4 which is a discrete version of the Weiss Conjecture for contraction semigroups, and finally Section 5 shows how the usual (continuous time) Weiss Conjecture is related to the discrete version studied here; in fact they are equivalent (for scalar valued observation operators). The main advantage of the discrete version is that it is technically simpler – the observation operators are automatically bounded and the functional calculus can be achieved using power series.     

The Mosaic and Principal Function of a Subnormal Operator

In this paper, we use the mosaic of a subnormal operator given by Daoxing Xia to give an alternate definition of the Pincus principal function for pure subnormal operators. This allows us to provide much simplified proofs of some of the basic properties of the principal function and of the Carey-Helton-Howe-Pincus Theorem in the subnormal case.     

On the Range of the Aluthge Transform

Let be the algebra of all bounded linear operators on a complex separable Hilbert space For an operator let be the Aluthge transform of T and we define for all where T = U|T| is a polar decomposition of T. In this short note, we consider an elementary property of the range of Δ. We prove that R(Δ) is neither closed nor dense in However R(Δ) is strongly dense if is infinite dimensional. An erratum to this article is available at .     

The Density of the Hypercyclic Operators in the Strong Operator Topology

In this note we give a simple proof of the fact that the set of all hypercyclic operators on a separable Hilbert space is dense in the strong operator topology.     

Riesz Idempotent and Algebraically M-hyponormal Operators

Let T be an M-hyponormal operator acting on infinite dimensional separable Hilbert space and let be the Riesz idempotent for λ₀, where D is a closed disk of center λ₀ which contains no other points of σ (T). In this note we show that E is self-adjoint and As an application, if T is an algebraically M-hyponormal operator then we prove : (i) Weyl’s theorem holds for f(T) for every (ii) a-Browder’s theorem holds for f(S) for every and f ∈ H(σ(S)); (iii) the the spectral mapping theorem holds for the Weyl spectrum of T and for the essential approximate point spectrum of T.     

Extremal Problems for Operators in Banach Spaces Arising in the Study of Linear Operator Pencils

Inspired by some problems on fractional linear transformations the authors introduce and study the class of operators satisfying the condition where stands for the spectral radius; and the class of Banach spaces in which all operators satisfy this condition, the authors call such spaces V-spaces. It is shown that many well-known reflexive spaces, in particular, such spaces as L_p(0,1) and C_p, are non-V-spaces if p 2; and that the spaces l_p are V-spaces if and only if 1 < p < . The authors pose and discuss some related open problems.     

The Virozub–Matsaev Condition and Spectrum of Definite Type for Self-adjoint Operator Functions

We establish sufficient conditions for the so-called Virozub–Matsaev condition for twice continuously differentiable self-adjoint operator functions. This condition is closely related to the existence of a local spectral function and to the notion of positive type spectrum. Applications to self-adjoint operators in Krein spaces and to quadratic operator polynomials are given. Received: September 22, 2007. Accepted: September 29, 2007.     

Positive Operator Majorization and <Emphasis Type="Italic">p</Emphasis>-hyponormality

In this note we examine the relationships between p-hyponormal operators and the operator inequality . This leads to a method for generating examples of p-hyponormal operators which are not q-hyponormal for any . Our methods are also shown to have implications for the class of Furuta type inequalities.     

Sesquitransitive and Localizing Operator Algebras

An algebra of operators on a Banach space X is said to be transitive if X has no nontrivial closed subspaces invariant under every member of the algebra. In this paper we investigate a number of conditions which guarantee that a transitive algebra of operators is “large” in various senses. Among these are the conditions of algebras being localizing or sesquitransitive. An algebra is localizing if there exists a closed ball B ∌ 0 such that for every sequence (x _n ) in B there exists a subsequence and a bounded sequence (A _k ) in the algebra such that converges to a non-zero vector. An algebra is sesquitransitive if for every non-zero z ∈ X there exists C > 0 such that for every x linearly independent of z, for every non-zero y ∈ X, and every there exists A in the algebra such that and ||Az|| ≤ C||z||. We give an algebraic version of this definition as well, and extend Jacobson’s density theorem to algebraically sesquitransitive rings. The second and the third authors were supported by NSERC.     

On the Spectrum of Invertible Semi-hyponormal Operators

It is known that for a semi-hyponormal operator, the spectrum of the operator is equal to the union of the spectra of the general polar symbols of the operator. The original proof of this theorem involves the so-called singular integral model. The purpose of this paper is to give a different proof of the same theorem for the case of invertible semi-hyponormal operators without using the singular integral model.      

On the peripheral spectrum of monic operator polynomials with positive coefficients

I wish to thank R. Nagel for his guidance and suggestions in the preparation of this paper. Also, I would like to thank G. Greiner and F. Räbiger for many interesting and helpful discussions.     

Algebraic Order Bounded Disjointness Preserving Operators and Strongly Diagonal Operators

Let T be an order bounded disjointness preserving operator on an Archimedean vector lattice. The main result in this paper shows that T is algebraic if and only if there exist natural numbers m and n such that n ≥ m, and T^n!, when restricted to the vector sublattice generated by the range of T^m, is an algebraic orthomorphism. Moreover, n (respectively, m) can be chosen as the degree (respectively, the multiplicity of 0 as a root) of the minimal polynomial of T. In the process of proving this result, we define strongly diagonal operators and study algebraic order bounded disjointness preserving operators and locally algebraic orthomorphisms. In addition, we introduce a type of completeness on Archimedean vector lattices that is necessary and sufficient for locally algebraic orthomorphisms to coincide with algebraic orthomorphisms.     

On the Isolated Points of the Spectrum of Paranormal Operators

For paranormal operator T on a separable complex Hilbert space we show that (1) Weyl’s theorem holds for T, i.e., σ(T) \ w(T) = π₀₀(T) and (2) every Riesz idempotent E with respect to a non-zero isolated point λ of σ(T) is self-adjoint (i.e., it is an orthogonal projection) and satisfies that ranE = ker(T − λ) = ker(T − λ)^*.     

Ergodic Properties for Regular A-Contractions

The current article pleads for the possibility to obtain an orthogonal decomposition of a Hilbert space which is induced by a regular A-contraction defined in [9, 10], A being a positive operator on . The decomposition generalizes the well-known decomposition related to a contraction T of , which gives the ergodic character of T. This decomposition is being used to prove certain versions for regular A-contractions of the mean ergodic theorem, as well as a version of Patil’s theorem from [8]. Also, we characterize the solutions of corresponding functional equations in the range of A^1/2, by analogy with the result of Lin-Sine in [7].     

Bicircular projections on some Banach spaces

In this paper we show that the bicircular projections are precisely the Hermitian projections and prove some immediate consequences of this result.