By Gawronski W., Shawyer B. L., Trautner R.

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**Extra info for A Banach space version of Okada's theorem on summability of power series**

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In this case X is separable, but Y need not be. , The spectral multiplicity of a multiplication operator, Indiana Math. J. 22 (1973), 845-847. , Elements de mathematiques, Livre VI, Hermann et Cie, 1959. , The Theory of Subnormal Operators, Amer. Math. , 1991. , Measure Theory, Springer-Verlag, 1974. , Asymptotie Methods in Statistieal Deeision Theory, Springer-Verlag, 1986. , Disintegration and compact measures, Math. Seand. 43 (1978), 157-168. , Probability Measures on Metrie Spaees, Academic Press, 1967.

5. The multipliers (C, Lb') are precisely those of the form cP = P, where J1 in M(X) satisfies 44 M. O. Gebuhrer Moreover, the function cP has the property that Ln CP(Xn)](Xn)xnw{Xn} has uniformly bounded partial sums for each continuous function f on X if and only if cP E (C, L'{') and in that case (i) the series Ln (Xn)}(Xn)Xnw{Xn} is uniformly convergent for each continuous function fand (ii) the series Ln (Xn)](Xn)xnw{Xn} converges in the LP norm for every f E LP(X, 0-) for 1 ~ p < 00.

Specifically, we construct a nonextreme point b such that bitself belongs to M(ä) but b2 does not. The property that we will use to identify whether or not functions belong to M(ä) is given in the following definition. Definition 1. Suppose that 1 and 9 belong to Hoo. We say that (1, g) is an H 2-corona pair i1 there exist 1unctions

### A Banach space version of Okada's theorem on summability of power series by Gawronski W., Shawyer B. L., Trautner R.

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