A continuous model for quasinilpotent operators

Eva A Gallardo-Gutiérrez,Daniel J Rodríguez,Jonathan R Partington

doi:10.1007/s00209-016-1673-2

Abstract

A classical result due to Foias and Pearcy establishes a discrete model for every quasinilpotent operator acting on a separable, infinite-dimensional complex Hilbert space HH . More precisely, given a quasinilpotent operator T on HH , there exists a compact quasinilpotent operator K in HH such that T is similar to a part of K⊕K⊕⋯⊕K⊕⋯K⊕K⊕⋯⊕K⊕⋯ acting on the direct sum of countably many copies of HH . We show that a continuous model for any quasinilpotent operator can be provided. The consequences of such a model will be discussed in the context of C0C0 -semigroups of quasinilpotent operators.

Highlights

Whether every quasinilpotent operator on a separable, infinite-dimensional complex Hilbert space has a non-trivial closed invariant subspace is, a long-standing open question which have called the attention of many operator theorists in last half-century
In [4, Theorem 1], they prove that given any bounded linear quasinilpotent operator T on a separable, infinite-dimensional complex Hilbert space H, there exists a compact quasinilpotent weighted backward shift K in H such that T is similar to a part of K ⊕ K ⊕ · · · ⊕ K ⊕ · · · acting on the direct sum of countably many copies of H
In Theorem 2.1, Sect. 2, we will prove the existence of a continuous model for a given quasinilpotent operator T in the sense that there exists a C∞ increasing positive function w in R+ such that T is similar to the restriction of the backward shift operator S∗ to one of its closed invariant subspaces M in L2(R+, w(t) dt)

Summary

Introduction

Whether every quasinilpotent operator on a separable, infinite-dimensional complex Hilbert space has a non-trivial closed invariant subspace is, a long-standing open question which have called the attention of many operator theorists in last half-century. In [4, Theorem 1], they prove that given any bounded linear quasinilpotent (but not nilpotent) operator T on a separable, infinite-dimensional complex Hilbert space H, there exists a compact quasinilpotent weighted backward shift K in H such that T is similar to a part of K ⊕ K ⊕ · · · ⊕ K ⊕ · · · acting on the direct sum of countably many copies of H. 2, we will prove the existence of a continuous model for a given quasinilpotent operator T in the sense that there exists a C∞ increasing positive function w in R+ such that T is similar to the restriction of the backward shift operator S∗ to one of its closed invariant subspaces M in L2(R+, w(t) dt).

The model

C0-semigroups of quasinilpotent operators