Hyperinvariant subspaces for trace class perturbations of normal operators and decomposability

TL;DR

The paper proves that a large class of trace-class perturbations $T = D_\Lambda + \sum_{k\ge1} u_k\otimes v_k$ of diagonalizable normal operators on a separable infinite-dimensional Hilbert space possess non-trivial closed hyperinvariant subspaces, and are decomposable when $\sigma_p(T)\cup\sigma_p(T^*)$ is at most countable. The authors develop an unconventional Dunford functional calculus along spectral curves, constructing spectral idempotents in the bicommutant ${\{T\}''}$ with ranges equal to spectral subspaces, and show that these idempotents generate a rich lattice of hyperinvariant subspaces. A local summability condition yields a local version of the result, and classical decomposability arguments extend to the trace-class perturbation setting via the spectral-idempotent framework. Together, these results extend prior finite-rank and Schatten-class perturbation findings and illuminate the spectral structure of trace-class normal perturbations.

Abstract

We prove that a large class of trace-class perturbations of diagonalizable normal operators on a separable, infinite dimensional complex Hilbert space have non-trivial closed hyperinvariant subspaces. Moreover, a large subclass consists of decomposable operators in the sense of Colojoară and Foiaş.

Figures (1)

• Figure 1: Decomposability in the rank-one perturbation case

Theorems & Definitions (30)

• Remark 1.1
• Definition 2.1
• Theorem 2.2
• proof
• Remark 2.3
• Lemma 2.4
• Definition 3.1
• Remark 3.2
• Proposition 3.3
• proof