C-orbit reflexive operators

TL;DR

This paper introduces the concept of C-orbit reflexivity for operators, characterizes it in finite dimensions via Jordan form, and extends results to infinite dimensions using algebraic methods.

Contribution

It defines C-orbit reflexivity, provides a complete characterization in finite-dimensional spaces, and develops algebraic techniques for infinite-dimensional cases.

Findings

Finite-dimensional operators are C-orbit reflexive iff the two largest Jordan blocks differ in size by at most one.

Most infinite-dimensional results are derived from algebraic analogs of the finite-dimensional case.

The paper establishes a clear criterion for C-orbit reflexivity based on Jordan form structure.

Abstract

We introduce the notion of C-orbit reflexivity and study its properties. An operator on a finite-dimensional space is C-orbit reflexive if and only if the two largest blocks in its Jordan form corresponding to nonzero eigenvalues with the largest modulus differ in size by at most one. Most of the proofs of our results in infinite dimensions are obtained from purely algebraic results we obtain from linear-algebraic analogs of C-orbit reflexivity.