Holomorphic Functions and polynomial ideals on Banach spaces

Abstract

Given $\u$ a multiplicative sequence of polynomial ideals, we consider the associated algebra of holomorphic functions of bounded type, $H_{b\u}(E)$. We prove that, under very natural conditions verified by many usual classes of polynomials, the spectrum $M_{b\u}(E)$ of this algebra "behaves" like the classical case of $M_b(E)$ (the spectrum of $H_b(E)$, the algebra of bounded type holomorphic functions). More precisely, we prove that $M_{b\u}(E)$ can be endowed with a structure of Riemann domain over $E"$ and that the extension of each $f\in H_{b\u}(E)$ to the spectrum is an $\u$-holomorphic function of bounded type in each connected component. We also prove a Banach-Stone type theorem for these algebras.

19 pages

References in corpus (3)

• The symmetric Radon-Nikodým property for tensor norms
• Natural symmetric tensor norms
• Extending polynomials in maximal and minimal ideals

Cited by in corpus (7)

• Every Banach ideal of polynomials is compatible with an operator ideal
• Polynomial ideals from a nonlinear viewpoint
• Multiple Cohen strongly summing operators, ideals, coherence and compatibility
• On multi-ideals and polynomial ideals of Banach spaces: a new approach to coherence and compatibility
• Strongly mixing convolution operators on Fréchet spaces of holomorphic functions
• A Lindenstrauss theorem for some classes of multilinear mappings
• Almost Summing Polynomials: A Coherent and Compatible Approach the from the Infinite-Dimensional Holomorphy Viewpoint