The algebraic structure ofH-dissipative operators in a finite-dimensional space

We study properties of Jordan representations ofH-dissipative operators in a finite-dimensional indefiniteH-space. An algebraic proof is given of the fact that such operators always have maximal semidefinite invariant subspaces. Translated fromMatematicheskie Zametki, Vol. 63, No. 2, pp. 163–169, February, 1998. The authors axe grateful to Professor A. A. Shkalikov for useful discussions. The research of the first author was supported by INTAS under grant No. 93-0249. The research of the second author was supported by the International Science Foundation and the Russian Government under grant No. NZP300.     

On supercyclicity of operators from a supercyclic semigroup

We show that for every supercyclic strongly continuous operator semigroup {Tt}_t?0 acting on a complex F-space, every Tt with t>0 is supercyclic. Moreover, the set of supercyclic vectors of each Tt with t>0 is exactly the set of supercyclic vectors of the entire semigroup.     

On the spectra of finite-dimensional quadratic Bose operators

A complete study of the spectrum of a finite-dimensional Bose operator is carried out in the paper. The cases in which the spectrum is discrete or continuous are studied. Translated fromMatematicheskie Zametki, Vol. 61, No. 6, pp. 835–854, June, 1997. Translated by A. M. Chebotarev     

On the asymptotic behaviour of a semigroup of linear operators

Let T = {T(t)}_{t ≥ 0} be a C₀-semigroup on a Banach space X. In this paper, we study the relations between the abscissa ω_Lp(T) of weak p-integrability of T (1 ≤ p < ∞), the abscissa ω^p_R(A) of p-boundedness of the resolvent of the generator A of T (1 ≤ p ≤ ∞), and the growth bounds ω_β(T), β ≥ 0, of T. Our main results are as follows.

1. (i) Let T be a C₀-semigroup on a B-convex Banach space such that the resolvent of its generator is uniformly bounded in the right half plane. Then ω_{1 − ε}(T) < 0 for some ε > 0.

2. (ii) Let T be a C₀-semigroup on L^p such that the resolvent of the generator is uniformly bounded in the right half plane. Then ω_β(T) < 0 for all β>¦1/p − 1/p′¦, 1/p + 1/p′ = 1.

3. (iii) Let 1 ≤ p ≤ 2 and let T be a weakly L^p-stable C₀-semigroup on a Banach space X. Then for all β>1/p we have ω_β(T) ≤ 0.

Further, we give sufficient conditions in terms of ω^q_R(A) for the existence of L^p-solutions and W^1,p-solutions (1 ≤ p ≤ ∞) of the abstract Cauchy problem for a general class of operators A on X.     

On the topology induced by the adjoint of a semigroup of operators

The adjoint of aC ₀-semigroup on a Banach spaceX induces a locally convex σ(X,X ^ℴ)-topology inX, which is weaker than the weak topology ofX. In this paper we study the relation between these two topologies. Among other things a class of subsets ofX is given on which they coincide. As an application, an Eberlein-Shmulyan type theorem is proved for the σ(X,X ^ℴ)-topology and it is shown that the uniform limit of σ(X,X ^ℴ)-compact operators is σ(X,X ^ℴ)-compact. Finally our results are applied to the problem when the Favard class of a semigroup equals the domain of the infinitesimal generator.     

On the spectra of finite-dimensional perturbations of matrix multiplication operators

Let be a closed rectifiable curve and a region in the complex plane. Suppose for each , R() represents multiplication by an nxn-matrix of rational functions and F() is a finite rank operator, both acting on the Hilbert space L ₂ ⁿ (). Sufficient conditions are given for the integer valued function dim ker (R()+F()) to be continuous at all but finitely many points in . This result is applied to singular integral operators.This work was partially supported by the National Science Foundation.     

On the boundary of numerical ranges of operators

In this note, a new characterization of an attained boundary point of the numerical range of an operator on a Hilbert space is given. As an application, we point out a gap in the proof of the main result in the paper [M.T. Chien, L. Yeh, On the boundary of the numerical range of a matrix, Appl. Math. Lett. 23 (2010) 725–727].     

On the sharpness of non-optimal approximation for semigroup operators

The sharpness of the rate of approximation by certain operators to the identity is considered in the sense that there exist smooth elements such that a rate (given by a direct theorem) cannot be improved. The operators dealt with are semigroup operators, the Cesàro means of these, as well as the associated resolvent operators.Dedicated to Professor H. G. Tillmann on the occasion of his 60th birthday on Sept. 30, 1984, in friendship and admiration     

Doubly stochastic operators on a finite-dimensional simplex

Generalizing the concept of a quadratic doubly stochastic operator, we introduce the concept of an arbitrary doubly stochastic operator. We give a necessary condition for double stochasticity. Moreover, we prove an ergodic theorem for doubly stochastic operators. Original Russian Text Copyright ? 2009 Shahidi F. A. __________ Tashkent. Translated from Sibirskiĭ Matematicheskiĭ Zhurnal, Vol. 50, No. 2, pp. 463–468, March–April, 2009.     

The semigroup generated by certain operators on the congruence lattice of a clifford semigroup

Communicated by N. R. Reilly