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Mathematische Zeitschrift

, Volume 234, Issue 4, pp 777–805 | Cite as

Vector-valued holomorphic functions revisited

  • W. Arendt
  • N. Nikolski
Original article

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).

Mathematics Subject Classification (1991): 46G20 

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Copyright information

© Springer-Verlag Berlin Heidelberg 2000

Authors and Affiliations

  • W. Arendt
    • 1
  • N. Nikolski
    • 2
  1. 1.Universität Ulm, Abteilung Mathematik V, Helmholtzstr. 18, 89081 Ulm, Germany (e-mail: arendt@mathematik.uni-ulm.de) DE
  2. 2.Université Bordeaux I, UFR de Mathématiques et Informatique, 351, cours de la Libération, 33404 Talence, France, (e-mail: nikolski@math.u-bordeaux.fr) FR

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