Injective factorization of holomorphic mappings

Colloquium Mathematicum, 2009

We describe a part of the recent developments in the theory of separately holomorphic mappings between complex analytic spaces. Our description focuses on works using the technique of holomorphic discs.

Collectanea Mathematica, 2012

Given A a multiplicative sequence of polynomial ideals, we consider the associated algebra of holomorphic functions of bounded type, H bA (E). We prove that, under very natural conditions satisfied by many usual classes of polynomials, the spectrum M bA (E) of this algebra "behaves" like the classical case of

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.

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.

Journal of Mathematical Analysis and Applications

In this paper we study the behavior of holomorphic mappings on A-compact sets. Motivated by the recent work of Aron, Ç alişkan, García and Maestre (2016), we give several conditions (on the holomorphic mappings and on the λ-Banach operator ideal A) under which A-compact sets are preserved. Appealing to the notion of tensorstability for operator ideals, we first address the question in the polynomial setting. Then, we define a radius of (A; B)compactification that permits us to tackle the analytic case. Our approach, for instance, allows us to show that the image of any (p, r)-compact set under any holomorphic function (defined on any open set of a Banach space), is again (p, r)-compact.

The Quarterly Journal of Mathematics, 1999

Abstract and Applied Analysis, 2014

For two complex Banach spaces and , in this paper, we study the generalized spectrum M ( , ) of all nonzero algebra homomorphisms from H ( ), the algebra of all bounded type entire functions on , into H ( ). We endow M ( , ) with a structure of Riemann domain over L( * , * ) whenever is symmetrically regular. The size of the fibers is also studied. Following the philosophy of (Aron et al., 1991), this is a step to study the set M ,∞ ( , ) of all nonzero algebra homomorphisms from H ( ) into H ∞ ( ) of bounded holomorphic functions on the open unit ball of and M ∞ ( , ) of all nonzero algebra homomorphisms from H ∞ ( ) into H ∞ ( ). Abstract and Applied Analysis M ,∞ ( , ) of all nonzero algebra homomorphisms from H ( ) to H ∞ ( ) of bounded holomorphic functions on the open unit ball of . Finally, in Section 6, we deal with the generalized spectrum M ∞ ( , ) of all nonzero algebra homomorphisms from H ∞ ( ) to H ∞ ( ). Φ, ∩ Ψ, = 0 for = ‖ (Φ) − (Ψ)‖/2. This gives that the topology generated by { Φ, } >0 is Hausdorff.

Journal of Mathematical Analysis and Applications, 1992

Journal of Mathematical Analysis and Applications, 2000

It is shown that for every quasi-normed ideal Q of n-homogeneous continuous polynomials between Banach spaces there is a quasi-normed ideal A of n-linear continuous mappings A such that q ∈ Q if and only if the associated n-linear mappingˇq of q is in A. Mathematics subject classification 2000: 46G25, 47H60