Stable and Norm-stable Invariant Subspaces

TL;DR

This paper characterizes norm-stable invariant subspaces for certain operators on infinite-dimensional Hilbert spaces, showing that under specific spectral and index conditions, only finite-dimensional subspaces are norm-stable, and it explores stability in weighted shift operators.

Contribution

It provides a new characterization of norm-stable invariant subspaces for operators with connected spectrum and essential spectrum, and analyzes stability properties of weighted unilateral shift operators.

Findings

Only finite-dimensional subspaces are norm-stable under given spectral conditions.

Quasianalytic shift operators are points of norm continuity for invariant subspace lattices.

A necessary condition for strong stability of invariant subspaces is established.

Abstract

We prove that if T is an operator on an infinite-dimensional Hilbert space whose spectrum and essential spectrum are both connected and whose Fredholm index is only 0 or 1, then the only nontrivial norm-stable invariant subspaces of T are the finite-dimensional ones. We also characterize norm-stable invariant subspaces of any weighted unilateral shift operator. We show that quasianalytic shift operators are points of norm continuity of the lattice of the invariant subspaces. We also provide a necessary condition for strongly stable invariant subspaces for certain operators.