# Diskcyclicity of sets of operators and applications

## TL;DR

This paper generalizes the concept of diskcyclicity from single operators to sets of operators, establishing criteria and exploring applications to semigroups and groups, revealing conditions for their diskcyclicity.

## Contribution

It introduces a diskcyclicity criterion for sets of operators and links this property to semigroups and groups, providing new insights into their structure and behavior.

## Key findings

- Diskcyclic $C_0$-semigroups exist only on 1D or infinite-dimensional spaces.
- Diskcyclicity and disk transitivity are equivalent for certain operator groups.
- The paper establishes a relationship between diskcyclicity criteria and actual diskcyclicity.

## Abstract

In this paper, we extend the notion of diskcyclicity and disk transitivity of a single operator to a subset of $\mathcal{B}(X)$. We establish a diskcyclicity criterion and we give the relationship between this criterion and the diskcyclicity. As applications, we study the diskcyclicty of $C_0$-semigroups and $C$-regularized groups of operators. We show that a diskcyclic $C_0$-semigroup exists on a complex topological vector space $X$ if and only if dim$(X)=1$ or dim$(X)=\infty$ and we prove that diskcyclicity and disk transitivity of a $C_0$-semigroups and $C$-regularized groups are equivalent.