Vector-valued holomorphic functions revisited

Abstract. Let $\Omega \subset \C$ be open,X a Banach space and $W\subset X^\prime$. We show that every $\sigma (X,W)\mbox{-holomorphic function } f: \Omega \to X$ is holomorphic if and only if every $\sigma(X,W)\mbox{-bounded}$ set inX is bounded. Things are different if we assume f to be locally bounded. Then we show that it suffices that $\varphi \circ f$ is holomorphic for all $\varphi \in W$, where W is a separating subspace of $X^\prime$ 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 $\Omega$, 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).