Quasisimilarity and compact perturbation

Abstract

In this paper we show that quasisimilar n-tuples of tensor products of (p, k)-quasihyponormal operators have the same spectra, essential spectra and indices. The properties of single Fredholm operators possess [4] is related to an important property which has a leading role on the theory of Fredholm operators: Fredholm n-tuples of operators. It is well known that a Fredholm operator of index zero can be perturbed by a compact operator to an invertible operator. In [5, Problem 3] the author asked if this property holds in several variables. R. Gelca in [10] gave an example showing that this perturbation property fails in several variables. In this paper we give a positive answer to this question in case of tensor products of some classes of operators.