## Abstract and Applied Analysis

### Homomorphisms between Algebras of Holomorphic Functions

#### Abstract

For two complex Banach spaces $X$ and $Y$, in this paper, we study the generalized spectrum ${\scr M}_{b}(X,Y)$ of all nonzero algebra homomorphisms from ${\scr H}_{b}(X)$, the algebra of all bounded type entire functions on $X$, into ${\scr H}_{b}(Y)$. We endow ${\scr M}_{b}(X,Y)$ with a structure of Riemann domain over $\scr L({X}^{\mathrm{\ast}},{Y}^{\mathrm{\ast}})$ whenever $X$ 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 ${\scr M}_{b,\mathrm{\infty }}(X,{B}_{Y})$ of all nonzero algebra homomorphisms from ${\scr H}_{b}(X)$ into ${\scr H}_{\mathrm{\infty }}({B}_{Y})$ of bounded holomorphic functions on the open unit ball of $Y$ and ${\scr M}_{\mathrm{\infty }}({B}_{X},{B}_{Y})$ of all nonzero algebra homomorphisms from ${\scr H}_{\mathrm{\infty }}({B}_{X})$ into ${\scr H}_{\mathrm{\infty }}({B}_{Y})$.

#### Article information

Source
Abstr. Appl. Anal., Volume 2014, Special Issue (2014), Article ID 612304, 12 pages.

Dates
First available in Project Euclid: 2 October 2014

https://projecteuclid.org/euclid.aaa/1412278777

Digital Object Identifier
doi:10.1155/2014/612304

Mathematical Reviews number (MathSciNet)
MR3212436

Zentralblatt MATH identifier
07022722

#### Citation

Dimant, Verónica; García, Domingo; Maestre, Manuel; Sevilla-Peris, Pablo. Homomorphisms between Algebras of Holomorphic Functions. Abstr. Appl. Anal. 2014, Special Issue (2014), Article ID 612304, 12 pages. doi:10.1155/2014/612304. https://projecteuclid.org/euclid.aaa/1412278777

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