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a c z algebra analytic functions Borel measure bounded set bounded subsets closure commute compact set continuity classes continuous linear functional CONTINUOUS LINEAR OPERATORS converges strictly converges to zero converges uniformly correspondence define diagonal operators disk dual eigenvalues exists f converges f in H(D functions f given by Lf given by Tf(x implies f J K(z L(U f Lemma Let f let h(z lim sup linear fractional transformation norm closed obtain open set polynomials power series previous theorem Proof r+l n=0 strict topology strictly continuous functional strictly continuous linear strictly continuous operators strictly to f strictly to zero sup f sup f(a supremum T(zk T(zn tanh Tf(z theorem 5.l9 thesis u. b. closed u. b. convergence u. b. topology uniform boundedness principle uniform convergence uniformly bounded uniformly on bounded w)dw