Strictly Continuous Linear Operators on the Bounded Analytic Functions on the Disk |
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algebra ANALYTIC FUNCTIONS Borel measure bounded set bounded subsets characterize closure Co[D commute continuity classes continuous linear functional CONTINUOUS LINEAR OPERATORS converge uniformly converges strictly converges to zero Corollary correspondence define diagonal operators disk dual e¹º eigenvalues exists f converges f in H(D FLORAL PARK functions f given by Lf implies f K_(z K₂ L(U f L¹(r Lemma Let f let h(z lim sup linear fractional transformations n n n=0 norm closed obtain open set operators in ß pointwise polynomials power series previous theorem Proof r₁ Sf(z ß ß strict topology strictly continuous functional strictly continuous linear strictly to f strictly to zero sup f(a supremum T(zk T₂ tanh Tf(z Theorem 3.8 thesis uniform boundedness principle uniform convergence uniformly bounded uniformly on bounded w)dw ΙΣ σ σ ск ᎥᎾ