## Mathematics Ph.D. Dissertations

# A Strictly Weakly Hypercyclic Operator with a Hypercyclic Subspace

## Date of Award

2023

## Document Type

Dissertation

## Degree Name

Doctor of Philosophy (Ph.D.)

## Department

Mathematics

## First Advisor

Kit Chan (Committee Chair)

## Second Advisor

Christopher Kluse (Other)

## Third Advisor

Mihai Staic (Committee Member)

## Fourth Advisor

Juan Bes (Committee Member)

## Abstract

An interesting topic of study for a hypercyclic operator T on a topological vector space X has been whether X has an infinite-dimensional, closed subspaces consisting entirely, except for the zero vector, of hypercyclic vectors. These subspaces are called hypercyclic subspaces. The existence of a strictly weakly hypercyclic operator T, which is a weakly hypercyclic operator that is not norm hypercyclic on a Hilbert space H has been shown by Chan and Sanders. However, it is not known whether there exists a strictly weakly hypercyclic subspace of H. We first show that the left multiplication operator LT with the aforementioned strictly weakly hypercyclic operator T is a strictly WOT-hypercyclic operator on the operator algebra B(H). Then we obtain a sufficient condition for an operator T on a Hilbert space H to have a strictly weakly hypercyclic subspace. After that we construct an operator that satisfies these conditions and therefore prove the existence of a strictly weakly hypercyclic subspace.

## Recommended Citation

Madarasz, Zeno, "A Strictly Weakly Hypercyclic Operator with a Hypercyclic Subspace" (2023). *Mathematics Ph.D. Dissertations*. 100.

https://scholarworks.bgsu.edu/math_diss/100