Invariant subspaces of powers of some unicellular operators

TL;DR

This paper investigates the structure of subspaces invariant under powers of unicellular backward weighted shift operators, providing characterizations for finite and certain infinite-dimensional cases.

Contribution

It offers a comprehensive analysis of invariant subspaces under powers of unicellular operators, including explicit characterizations for all weights in finite dimensions.

Findings

Finite-dimensional invariant subspaces characterized for all weights.

Infinite-dimensional subspaces characterized for two classes of weights.

Invariant subspaces under squares and cubes are analyzed both separately and jointly.

Abstract

In this paper we study subspaces which are invariant under squares and cubes (separately as well as jointly) of unicellular backward weighted shift operators on a separable Hilbert space. The finite-dimensional subspaces are characterized for all weights and the infinite-dimensional subspaces are characterized for two classes of weights.