Digital Object Identifier Vector-valued holomorphic functions revisited

Abstract. Let Ω ⊂ C be open, X a Banach space and W ⊂ X . We show that every σ(X, W )-holomorphic function f : Ω → X is holomorphic if and only if every σ(X, W )-bounded set in X is bounded. Things are different if we assume f to be locally bounded. Then we show that it suffices that ϕ • f is holomorphic for all ϕ ∈ W , where W is a separating subspace of X to deduce that f is holomorphic. Boundary Tauberian convergence and membership theorems are proved. Namely, if boundary values (in a weak sense) of a sequence of holomorphic functions converge/belong to a closed subspace on a subset of the boundary having positive Lebesgue measure, then the same is true for the interior points of Ω, uniformly on compact subsets. Some extra global majorants are requested. These results depend on a distance Jensen inequality. Several examples are provided (bounded and compact operators; Toeplitz and Hankel operators; Fourier multipliers and small multipliers). Mathematics Subject Classification (1991):46G20 Introduction Vector-valued holomorphic functions are very useful, for example, in the theory of one-parameter semigroups or in spectral theory. But even for proving theorems about scalar-valued holomorphic functions, it is sometimes a useful trick to consider functions with values in a Banach space (we give two examples in Sect. 2)