A continuous model for quasinilpotent operators

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 H. More precisely, given a quasinilpotent operator T on H, there exists a compact quasinilpotent operator 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. 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 C 0 -semigroups of quasinilpotent operators.

Nearly invariant subspaces for shift semigroups

“…, with an infinite Blaschke product θ, is universal (see, e.g., [5]) and it is similar to the backward shift S(1) * on L 2 (0, ∞), given by S(1) * f (t) = f (t + 1). For the shift on weighted space L 2 ((0, ∞), w(t)dt), one may refer to [9]. S(1) * is a special case of the adjoint semigroup {S(t) * } t≥0 defined by (S(t…”