Similarity of Operators in a Hilbert Space

Abstract

One of the earliest non-trivial results about similarity is Nagy's converse: if A is invertible and both A and A -I are power bounded, then A is similar to a unitary operator. If A and B are operators. and if at least one of them is invertible, then AB and BA are similar. In this paper we show that similar operators have the same spectrum, the same point spectrum. the same approximate point spectrum. and the same compression spectrum. This implies that if at least one of A and B is invertible. then AB and BA have the same spectrum. In the finite- dimensional case more is known: with no invertibility assumptions. AB and BA always have the same characteristic polynomial. If neither A nor B is invertible. then, in the infinite-dimensional case. the two products need not have the same spectrum, but their spectra cannot differ by much. This paper also discusses the necessary and sufficient condition for two unilateral weighted shifts A and B with non-zero weights {a J and {,o,,} to be similar. Similarity is a less severe restriction than unitary equivalence. Bya modification of the argument for one-sided shifts, a modification whose difficulties are more notational than conceptual, it is possible to get a satisfactory condition