Elisabetta Barletta and Sorin Dragomir: Vector valued holomorphic functions, p.211-226

Abstract:

We survey results on holomorphic functions (of one complex variable) with values in a complex topological vector space hinting to their extension to the case of several complex variables. We give a version of the Hartogs theorem on separate analyticity for weakly holomorphic functions with values in a complex Fréchet space. The theory of $\alpha$-differentiable functions (due to N. Teodorescu, [27], and extended by F-H. Vasilescu, [28], to functions with values in a Fréchet space) is briefly reviewed as related to areolar derivatives. We present a selection of results on holomorphic functions with values in a complex Banach space with an emphasis on the boundary behavior of vector-valued holomorphic functions. We announce an extension of work by M.S. Baouendi & F. Treves, [3] (on the approximation of CR functions by holomorphic functions) to the case of CR functions with values in a complex Fréchet space.

Key Words: Vector-valued holomorphic function, $\alpha$-differentiability, vector valued CR function.

2000 Mathematics Subject Classification: Primary: 30A05,
Secondary: 32V10, 46A04.

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