A theorem on holomorphic mappings into Banach spaces with basis

The Quarterly Journal of Mathematics, 1999

1997

Givena holomorphicmapping of bounded typeg E Hb(u, F), where U E is a balanced open subset, and E , F are complex B a n d spaces, let A : Hb(F) 4 Hb(U) be the homomorphism defined by A ( f ) = f o g for all f E Hb(F). We prove that: (a) for F having the Dunford-Pettis property, A is weakly compact if and only if g is weakly compact; (b) A is completely continuous if and only if g(W) is a Dunford-Pettis set for every U-bounded subset W C U. To obtain these results, we prove that the class of Dunford-Pettis sets is stable under projective tensor products. Moreover, we diaracterize the reflexivity of the space Hb(U,F) and prove that E' and F have the Schur property if and only if H b ( u , F ) has the Schur property. As an application, we obtain some results on linearization of holomorphic mappings. 1991 Maihemaiics Subjeci Classificaiion. Primary 46 E 25; Secondary 46 G 05, 46 G 20,47 B 38. Keyword8 and phrases. Algebras of analytic fullctions of bounded type, holomorphic mappings, :omposition operators, weakly compact homomorphisms, linearization, Dunford-Pettis set.

Transactions of the American Mathematical Society, 1988

C(X) which are holomorphically barrelled and holomorphically quasibarrelled (cf. [5]). Dineen has recently proved in [16] that every weakly holomorphic mapping defined on any open subset of C^ x C^' is holomorphic. This result provides the first example of a holomorphically ...

Carpathian Mathematical Publications, 2015

In this paper we investigate the boundedness of holomorphic functionals on a Banach space with a normalized basis $\{e_n\}$ which have a very special form $f(x)=f(0)+\sum_{n=1}^\infty c_nx_n^n$ and which we call strictly diagonal. We consider under which conditions strictly diagonal functions are entire and uniformly continuous on every ball of a fixed radius.

MATH. SCAND, 2009

In this paper, we study composition operators on some Möbius invariant Banach function spaces like Bloch and F (p, q, s) spaces. We give a Carleson measure characterization on F (p, q, s) spaces, then we use this Carleson measure characterization of the compact compositions on F (p, q, s) spaces to show that every compact composition operator on F (p, q, s) spaces is compact on a Bloch space. Also, we give conditions to clarify when the converse holds.

Proceedings of the Steklov Institute of Mathematics, 2017

This expository paper is concerned with the properties of proper holomorphic mappings between domains in complex affine spaces. We discuss some of the main geometric methods of this theory, such as the Reflection Principle, the scaling method, and the Kobayashi-Royden metric. We sketch the proofs of certain principal results and discuss some recent achievements. Several open problems are also stated.

Transactions of the American Mathematical Society, 1984

The theory of complex interpolation of Banach spaces is viewed as a branch of the theory of vector valued holomorphic functions. Versions of the Schwarz lemma, Liouville’s theorem, the identity theorem and the reflection principle are proved and are interpreted from the point of view of interpolation theory.

The purpose of this article is to show a second main theorem with the explicit truncation level for holomorphic mappings of $ \mathbb{C} $ (or of a compact Riemann surface) into a compact complex manifold sharing divisors in subgeneral position.

Journal of Mathematical Analysis and Applications, 1996

Expositiones Mathematicae, 2003

The authors lay the foundations for the study of normal families of holomorphic functions and mappings on an infinitedimensional normed linear space. Characterizations of normal families, in terms of value distribution, spherical derivatives, and other geometric properties are derived. Montel-type theorems are established. A number of different topologies on spaces of holomorphic mappings are considered. Theorems about normal families are formulated and proved in the language of these various topologies. Normal functions are also introduced. Characterizations in terms of automorphisms and also in terms of invariant derivatives are presented. 1 Both authors were guests at the American Institute of Mathematics (AIM) during a portion of this work. AIM sponsored a workshop on holomorphic mappings that was particularly useful to these studies. 2 Author supported in part by grant R01-1999-00005 from The Korean Science and Engineering Foundation.