Reflexivity of N-tuples of contractions with rich joint left essential spectrum

On the Reflexivity of Pairs of Contractions

“…The representation generated by {Tx, T^} is, by Lemma 2.2, absolutely continuous. We apply the technique used in [KP,Chapter 4 and 5]. Note that [KP,Theorem 4.5 It is well known [BFP] that any dual algebra with property Xo, 1 is reflexive.…”

“…We apply the technique used in [KP,Chapter 4 and 5]. Note that [KP,Theorem 4.5 It is well known [BFP] that any dual algebra with property Xo, 1 is reflexive. Also the set of functionals [L] £ Q satisfying (*) is absolutely convex and closed.…”

“…By [G,VI. 1.2,II.7.5], the above definition is equivalent to that given in [KP,Chapters 3 and 4], which uses the terminology of bands of measures.…”

“…Recall that the joint left essential spectrum o¡e(Tx, ... , Tn) of Tx, ... , TN is defined as the joint left spectrum of 7r(70, ..., n(TN) . Let us recall (see [KP,Lemma 2.1]) that X = (Xx, ... , Xfi) £ oie(Tx, ... , 7jv), if and only if there exists an orthonormal sequence {xn} such that lim,,-^ \\(T¡ -A,)jc"|| = 0, for i = I, ... , N. Moreover, using standard techniques we can prove Lemma 1.1. Let Tx, ... ,Tn be commuting operators on ß?…”

“…), let us construct, using Ando Theorem [SNF,1.6.4] like in [KP,Chapter 5], the representation <t> of the algebra ^4(D2) generated by {Tx, T2}, i.e., Ö(z,-) = T¡ , i = 1,2. We say that the pair {Tx, T2} is a.c. if O is a.c.…”