OPERATORS OF CLASS C0 5 PROPOSITION 2.1.4. Let T £ C(H) be a completely nonunitary contraction. Then there exists a unique representation 3 of H°° into C(H) such that: (i) $(1) = 1H, where 1# £ C(H) is the identity operator and l(z) = 1 for all zeD (ii) &(g) = T, where g(z) z for all z £ D (Hi) I is continuous when H°° and C(H) are given the weak*-topology. We also have that fr is contractive, i.e. \\$(u)\\ \\u\\ for all u £ H°°(£l). We simply denote by u(T) the operator $(u). PROPOSITION 2.1.5. If for every u £ H°° we denote by u e H°° the function defined by u(z) = u(~z), we then have u(T)* = u(T*). DEFINITION 2.1.6. A completely nonunitary contraction T £ C(H) is said to be of class Co if there exists u £ if00, u •=£ 0, such that u(T) = 0. Let T be an operator of class Co. Then the set J = {u £ H°° : u(T) = 0} is a weak*-closed ideal, and hence it is of the form J = vH°° for some inner function v ([10]). DEFINITION 2.1.7. The inner function v such that vH°° = {u £ H°° : u{T) = 0} is called the minimal function of T and is denoted by my. Let us note that the function mr is determined only up to a constant scalar multiple of absolute value one. DEFINITION 2.1.8. A contraction T £ C(H) is said to be of class C*0 if for all x £ H limn^oo ||T*n^|| = 0 T is of class C0* if T* is of class C*0. Finally, T is of class Coo if it is both of class C*o and Co*. PROPOSITION 2.1.9. ([26]) IfT is of class Co, then it is of class Coo- 2.2. # p ( ^ ) Spaces. The theory of Hardy spaces over multiply connected regions has been first studied by Rudin [21] cf. also [13]. Much of what is known for the unit disk D extends to a region whose boundary consists of finitely many disjoint, analytic, simple closed curves, since many questions can be reduced to the simply connected case by means of a decomposition theorem. Let 1 p oo. A holomorphic function / on 17 is in Hp(tt) if the subharmonic function \f\p has a harmonic major ant on D. For a fixed ZQ £ f£, there is a norm on Hp(n) defined by ll/H = mi{u(zo)l/p : u is a harmonic majorant of | / | p } . Let g(-,zo) be the Green's function for Q, with pole at zo, and uo be the harmonic measure on V for the point z$. In our case, since the region ft has a nice boundary, the harmonic measure is easy to understand as the following result ([13] Theorem 1.6.4) shows. THEOREM 2.2.1. Letzo be any point in Q. Then we have that duo = -^-§^g(',zo where -J^ is the derivative in the direction of the outward normal at T, and ds is the arc-length measure on T.
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