The invariant subspace problem and Rosenblum operators I

TL;DR

This paper investigates invariant subspaces of invertible operators by analyzing the growth of combined powers, introducing shift representation operators based on Rosenblum operators, and linking the problem to Hankel operator injectivity.

Contribution

It introduces shift representation operators derived from Rosenblum operators to study invariant subspaces and connects the invariant subspace problem to Hankel operator injectivity.

Findings

Growth estimates of T^n + \u03bb T^{-n} for invariant subspace analysis

Introduction of shift representation operators based on Rosenblum operators

Equivalence of the invariant subspace problem to Hankel operator injectivity

Abstract

Let $T\in B(\mathcal{H})$ be an invertible operator. From the 1940's, Gelfand, Hille and Wermer investigated the invariant subspaces of $T$ by analyzing the growth of $\|T^n\|$, where $n\in \mathbb{Z}$. In this paper, we study the invariant subspaces of $T$ by estimating the growth of $\|T^n+\lambda T^{-n}\|$, where $n\in \mathbb{N}$ and $\lambda$ is a nonzero complex constant. The key ingredient of our approach is introducing the notion of shift representation operators, which is based on the Rosenblum operators. In addition, by employing shift representation operators, we provide an equivalent of the Invariant Subspace Problem via the injectivity of certain Hankel operators.