Proper contractions and invariant subspaces

Abstract

Abstract. Let T be a contraction and A the strong limit of {T * n T n } n≥1 . We prove the following theorem: if a hyponormal contraction T does not have a nontrivial invariant subspace, then T is either a proper contraction of class Ꮿ 00 or a nonstrict proper contraction of class Ꮿ 10 for which A is a completely nonprojective nonstrict proper contraction. Moreover, its self-commutator [T * ,T ] is a strict contraction.2000 Mathematics Subject Classification. 47A15, 47B20.

Contractions Satisfying the Absolute Value Property $$ |A|^2 \leq |A^2| $$

“…Recall that a contraction A is said to be a proper contraction if Ax < x for every nonzero x in H. A strict contraction (i.e., a contraction A such that A < 1) is a proper contraction, but a proper contraction is not necessarily a strict contraction (although the concepts of strict and proper contraction coincide for compact operators). It was recently proved in [11] that if a hyponormal contraction A has no nontrivial invariant subspace, then (a) A is a proper contraction and (b) its self-commutator [A * , A] is a strict contraction. We start by extending item (a), and giving a counterpart of item (b), to contractions A in U; but first we need the following auxiliary result.…”

Applications of Hilbert Space Dissipative Norm

“…We refer to [2] for a recent and comprehensive discussion. The present paper is a sequel to our effort to go after strong stability of continuous and discrete Hilbert space contraction semigroups [1,7,8,9,10].…”

Contractions of Class Q and Invariant Subspaces

“…It was recently proved in [10] that if a hyponormal contraction T has no nontrivial invariant subspace, then T is a proper contraction and its self-commutator [T * , T ] is a strict contraction. This was extended in [5] to contractions of class U (if a contraction T in U has no nontrivial invariant subspace, then both T and the nonnegative operator |T 2 | − |T | 2 are proper contractions), and to paranormal contractions in [6]: If a paranormal contraction T has no nontrivial invariant subspace, then T is a proper contraction and so is the nonnegative operator |T 2 | 2 − 2|T | 2 + I.…”