Quasisimilarity

Carlos S. Kubrusly
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Carlos S. Kubrusly: Catholic University — PUC/RJ

Chapter Chapter 4 in An Introduction to Models and Decompositions in Operator Theory, 1997, pp 61-74 from Springer

Abstract: Abstract Recall that X ∈ B[H, K] is invertible if it is injective and surjective (i.e. if N (X) = {0} and R(X) = K - cf. Proposition 0.1). X ∈ B[H K] is quasiinvertible or a quasiaffinity if it is an injective operator with dense range (i.e. N(X) = {0} and R(X) - = K; equivalently, N(X) = {0} and N (X*) = {0} - thus X ∈ B[H, K] is quasiinvertible if and only if X* ∈ B[K, H] is quasiinvertible). An operator T ∈ B[H] is a quasiaffine transform of L ∈ B[K] if there exists a quasiinvertible X ∈ B[H, K] such that XT = LX. Two operators T ∈ B[H] and L ∈ B[K] are quasisimilar (denoted by T ~ L) if they are quasiaffine transforms of each other. That is, if there exist quasiinvertible operators X ∈ B[H, K] and Y ∈ B[K, H] such that XT = LX and YL = TY. It is readily verified that quasisimilarity is an equivalence relation, and also that T* ~ L* whenever T~L. Similar operators are, of course, quasisimilar. We shall see below that, in a sense, similarity preserves nontrivial invariant subspaces, while quasisimilarity preserves nontrivial hyperinvariant subspaces. This chapter will focus on these classical results, as well as on the characterization of contractions quasisimilar to a unitary operator.

Keywords: Hilbert Space; Operator Theory; Unitary Operator; Invariant Subspace; Positive Answer (search for similar items in EconPapers)
Date: 1997
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Persistent link: https://EconPapers.repec.org/RePEc:spr:sprchp:978-1-4612-1998-9_5

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DOI: 10.1007/978-1-4612-1998-9_5

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