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On the non-hypercyclicity of scalar type spectral operators and collections of their exponentials

Markin Marat V.

Demonstratio Mathematica

2020

Vol. 53, nr 1

352--359

Generalizing the case of a normal operator in a complex Hilbert space, we give a straightforward proof of the non-hypercyclicity of a scalar type spectral operator A in a complex Banach space as well as of the collection {etA}t≥0 of its exponentials, which, under a certain condition on the spectrum of the operator A, coincides with the C0-semigroup generated by A. The spectrum of A lying on the imaginary axis, we also show that non-hypercyclic is the strongly continuous group {etA}t∈R of bounded linear operators generated by A. From the general results, we infer that, in the complex Hilbert space L2(R), the anti-self-adjoint differentiation operator A≔ddx with the domain D(A)≔W12(R) is non-hypercyclic and so is the left-translation strongly continuous unitary operator group generated by A.

Fractional evolution equation nonlocal problems with noncompact semigroups

Zhang X, Chen P.

Opuscula Mathematica

2016

Vol. 36, no. 1

123--137

This paper is concerned with the existence results of mild solutions to the nonlocal problem of fractional semilinear integro-differential evolution equations. New existence theorems are obtained by means of the fixed point theorem for condensing maps. The results extend and improve some related results in this direction.

Sturm-Liouville Systems Are Riesz-Spectral Systems

Delattre C., Dochain D., Winkin J.

International Journal of Applied Mathematics and Computer Science

2003

Vol. 13, no 4

481-484

The class of Sturm-Liouville systems is defined. It appears to be a subclass of Riesz-spectral systems, since it is shown that the negative of a Sturm-Liouville operator is a Riesz-spectral operator on L2(a,b) and the infinitesimal generator of a C0-semigroup of bounded linear operators.