2. Preliminaries and Notation 2.1. Contractions of class Co. If {Mi}iei is a family of subsets of the Hilbert space H, we denote by \J ieI Mi the closed linear span generated by Uie/ Mi- Moreover, M~ denotes the closure of M., for any subset M of H. DEFINITION 2.1.1. A contraction T G C(H) is said to be completely nonuni- tary if there is no invariant subspace M for T such that T\M is a unitary operator. If K is a Hilbert space, H C X is a subspace, 5 G C(K) and T G C(H), then we say that 5 is a dilation of T (and T is a power-compression of 5) provided that Tn = Ptf S[k for n G {0,1,... }. If in addition S is an isometry (unitary operator) then S will be called an isometric (unitary) dilation of T. 5 is a minimal isometric (unitary) dilation of T if and only if V~=0 ^H = K (V^-o o SnH = K). THEOREM 2.1.2. Sz.-Nagy Dilation Theorem([24]) (i) Every contraction T G C{H) has a minimal isometric dilation. This dilation is unique in the following sense: if S G C(K) and Sf G C{Kf) are two minimal isometric dilations for T, then there exists an isometry U of K onto K' such that Ux x, x G H, and S'U = US. (ii) Every contraction T G C(H) has a minimal unitary dilation. This dilation is unique in the sense specified in (i). THEOREM 2.1.3. ([25]) The spectral measure of the minimal unitary dilation of a completely nonunitary contraction is mutually absolutely continuous with respect to arc-length measure on the unit circle T. Let T G C(H) be a completely nonunitary contraction with minimal unitary dilation U G C(K), and let PH denote the orthogonal projection onto H. For every polynomial p(z) = 2j= o a jzj w e t n e n n a v e P(?) = PHP(U)\H, and this re- lation suggests that the functional calculus p » p(T) might be extended to more general functions. More precisely, since the spectral measure of U is absolutely continuous with respect to arc-length on T, the expression f(U) makes sense for every / in L°°(T). Therefore it is possible to define f(T) by f(T) = PHf(U)\H for all / G I/°°(T). Even if the mapping / f(T) is obviously linear, it is not in general multiplicative. It turns out that there is a unique maximal algebra such that, for all operators T, the map is multiplicative. This algebra is if00, i.e., the algebra of bounded analytic functions on D. A representation of H°° into C(H) is an algebra homomorphism of H°° into C(H). In the following proposition we mention some important properties of this functional calculus [27]. 4
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