An integral representation for strictly continuous linear operators

Let B denote the algebra of bounded analytic functions on the open unit disc D in the complex plane. Let (B, ) denote B endowed with the topology r, where r is chosen from K, or , respectively, the topology of uniform convergence on compact subsets of D, the strict topology and the topology of uniform convergence on D. This note obtains an integral representation of the form Tf(z) = I f(w)K(z, w)dw where = {z : I = 1} Jr for the linear operators which are continuous from (B,) into (B,). This representation is then used to study the convergence of operators in the full algebra of all continuous linear operators from (B,) into (B,). 1. Introduction. Let M(D) denote the set of bounded complex valued Borel measures on D. R. C. Buck [5] showed that L is a continuous linear functional on (C(D),) if and only if Lf = fd, JD VfGC(D) for some GM(D). L. A. Rubel and A. L. Shields [7] showed that for any G M(D) there exists a function h in V() such that fd = I f(x)h(x)dx, V/ G B and conversely, that any h G L) JD JY