On supercyclicity of operators from a supercyclic semigroup

TL;DR

This paper proves that in a supercyclic semigroup of operators on a complex space, each individual operator for positive time is also supercyclic, sharing the same supercyclic vectors as the entire semigroup.

Contribution

It establishes that all operators in a supercyclic semigroup are individually supercyclic and share the same supercyclic vectors, extending the understanding of supercyclicity in operator semigroups.

Findings

Every operator in the semigroup with t>0 is supercyclic.

Supercyclic vectors of each T_t are identical to those of the semigroup.

Supercyclicity property is preserved across all operators in the semigroup.

Abstract

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