On the direct sum of two bounded linear operators and subspace-hypercyclicity

TL;DR

This paper investigates the relationship between the subspace-hypercyclicity of operators and their direct sums, establishing conditions under which properties of the sum imply properties of individual operators.

Contribution

It proves that the subspace-hypercyclicity of a direct sum implies the same for each operator and extends criteria to the direct sum and individual operators, advancing understanding of subspace-hypercyclicity.

Findings

Subspace-hypercyclicity of a direct sum implies each operator is subspace-hypercyclic.

Operators satisfying subspace-hypercyclic criterion have their direct sum also satisfy it.

Under certain conditions, hypercyclicity of the direct sum implies subspace-hypercyclicity of the individual operator.

Abstract

In this paper, we show that if the direct sum of two operators is subspace-hypercyclic (satisfies subspace hypercyclic criterion), then both operators are subspace-hypercyclic (satisfy subspace hypercyclic criterion). Moreover, if an operator $T$ satisfies subspace-hypercyclic criterion, then so $T\oplus T$ does. Also, we obtain that under certain conditions, if $T\oplus T$ is hypercyclic then $T$ satisfies subspace-hypercyclic criterion and, the subspace-hypercyclic operators satisfy subspace-hypercyclic criterion which gives the "subspace-hypercyclic" analogue of Theorem 2.3. (in Hereditarily hypercyclic operators, J. Funct. Anal., 167:94--112, 1999 by P. B\'es and A. Peris).