On perturbations of the isometric semigroup of shifts on the semiaxis

TL;DR

This paper investigates how perturbations within Schatten-von Neumann classes affect the spectral properties of isometric semigroups of shifts on the positive real axis, demonstrating the ability to achieve any spectral type through such perturbations.

Contribution

It provides explicit constructions showing that Schatten class perturbations can produce any desired spectral type in the unitary part of the semigroup's cogenerator.

Findings

Any singular spectral type can be achieved by $ ext{S}_1$ perturbations.

Explicit construction of perturbations with prescribed spectral types.

Any spectral type can be obtained with $ ext{S}_p$ perturbations for $p>1$.

Abstract

We study perturbations $(\tilde\tau_t)_{t\ge 0}$ of the semigroup of shifts $(\tau_t)_{t\ge 0}$ on $L^2(\R_+)$ with the property that $\tilde\tau_t - \tau_t$ belongs to a certain Schatten-von Neumann class $\gS_p$ with $p\ge 1$. We show that, for the unitary component in the Wold-Kolmogorov decomposition of the cogenerator of the semigroup $(\tilde\tau_t)_{t\ge 0}$, {\it any singular} spectral type may be achieved by $\gS_1$ perturbations. We provide an explicit construction for a perturbation with a given spectral type based on the theory of model spaces of the Hardy space $H^2$. Also we show that we may obtain {\it any} prescribed spectral type for the unitary component of the perturbed semigroup by a perturbation from the class $\gS_p$ with $p>1$.