Reflexivity of Operators Intertwining Weak Contractions
by
M. Zajac
Department of Mathematics, FEI, Slovak University of Technology, Bratislava

1. Introduction

Let \bopH be the algebra of all continuous linear operators on a complex separable Hilbert space H and let T in \bopH. We denote by {T}' the commutant of T. If {T}' is reflexive the operator T is called hyperreflexive. The reflexivity of {T}' was generalized to the reflexivity of the set of all operators intertwining two given operators . Let T in \bop H, T' in \bopH'. I(T, T')={A in \bopH, H' AT=T'A} is called reflexive if

I(T, T')=\al I(T, T')= Ç x in H  {X in \bopH, H' Xx in Ú A in I(T, T')  Ax} .

In the characterization of reflexive I(T, T') was given in the case of isometries. In it was shown that if dimH < \infty then T is hyper-reflexive iff it is similar to a normal operator or equivalently, iff all roots of the minimal polynomial of T are simple. In we showed that in finite-dimensional spaces I(T, T') is reflexive iff all roots of the greatest common divisor of the minimal polynomials of T and T' are simple. In we showed that a weak contraction is hyper-reflexive if and only if its C₀ part is hyperreflexive.

We extend these results to pairs of C₀ contractions and weak contractions.

2. Results

The following result is an easy consequence of Theorem 1.16 of , S(\Theta) means the Jordan block corresponding to the inner function \Theta.

Theorem 1 Let v₁, v₂, d be inner functions, v₁ /\ v₂=1. Put \Theta₁=v₁d, \Theta₂=v₂d. Then

(i)

X in I(S(\Theta₁), S(\Theta₂)) if and only if there exists a function j in H^\infty such that

X=PH(\Theta2)u(S) | H(\Theta1) ,     where    u=v2j .

Moreover, X=0 if d|j.

(ii)

An operator A in \al I(S(\Theta₁), S(\Theta₂)) if and only if

Date received: June 5, 2000

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Copyright © 2000 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # caeo-53.