Null-orbit reflexive operators

TL;DR

This paper introduces null-orbit reflexivity, a new operator property that slightly modifies orbit-reflexivity, and explores its implications and classes of operators that possess this property.

Contribution

It defines null-orbit reflexivity, extends existing results to this new notion, and characterizes various classes of operators that are null-orbit reflexive.

Findings

Null-orbit reflexivity extends orbit reflexivity results.

Certain classes like hyponormal and compact operators are null-orbit reflexive.

Polynomially bounded operators are both orbit and null-orbit reflexive.

Abstract

We introduce and study the notion of null-orbit reflexivity, which is a slight perturbation of the notion of orbit-reflexivity. Positive results for orbit reflexivity and the recent notion of $\mathbb{C}$-orbit reflexivity both extend to null-orbit reflexivity. Of the two known examples of operators that are not orbit-reflexive, one is null-orbit reflexive and the other is not. The class of null-orbit reflexive operators includes the classes of hyponormal, algebraic, compact, strictly block-upper (lower) triangular operators, and operators whose spectral radius is not 1. We also prove that every polynomially bounded operator on a Hilbert space is both orbit-reflexive and null-orbit reflexive.