# Every Pure Quasinormal Operator Has a Supercyclic Adjoint

2020

Dissertation

## Degree Name

Doctor of Philosophy (Ph.D.)

## Department

Mathematics/Mathematics (Pure)

Jong Kwan Lee (Other)

Jonathan Bostic (Committee Member)

So-Hsiang Chou (Committee Member)

Mihai Staic (Committee Member)

## Abstract

We prove that every pure quasinormal operator T : H→H on a separable, infinite-dimensional, complex Hilbert space H has a supercyclic adjoint (see Theorem 3.3.2 and Corollary 3.3.12). It follows that if an operator has a pure quasinormal extension then the operator has a supercyclic adjoint. Our result improves a result of Wogen [52] who proved in 1978 that every pure quasinormal operator has a cyclic adjoint.

Feldman [26] proved in 1998 that every pure subnormal operator has a cyclic adjoint. Continuing with our result, it implies in particular that every pure subnormal operator having a pure quasinormal extension has a supercyclic adjoint (see Corollary 3.3.15). Hence improving Feldman’s result in this special case.

Indeed, we show that the adjoint T* of every pure quasinormal operator T is unitarily equivalent to an operator of the form Q : ⊕0∞L2(μ)→⊕0∞L2(μ) defined by Q(f0, f1, f2, . . .) = (A1f1 , A2f2 , A3f3 , . . .) for all vectors (f0, f1, f2, . . .)∈⊕0∞L2(μ), where each An : L2(μ)→L2(μ) is a left multiplication operator Mφn with symbol φn∈ L ∞(μ) satisfying φn≠0 a.e. We constructively obtain a supercyclic vector for the operator Q and this then yields our result by the fact that unitary equivalence preserves supercyclicity. Furthermore, we prove that the adjoint T* of a pure quasinormal operator T : H→H is hypercyclic precisely when T is bounded below by a scalar α> 1 (see Theorem 2.6.4 and Corollary 2.6.8).

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