Duality results in Banach and quasi-Banach spaces of homogeneous polynomials and applications

TL;DR

This paper introduces a method to convert quasi-normed spaces of homogeneous polynomials into normed spaces while preserving duality, enabling new applications in convolution equations and hypercyclic operators.

Contribution

It develops a dual-preserving technique to replace quasi-norms with norms in polynomial spaces, facilitating analysis and applications.

Findings

Dual of the space with the new norm matches the original dual.

Applications to convolution equations and entire function operators.

Enhanced tools for approximation and existence problems in polynomial spaces.

Abstract

Spaces of homogeneous polynomials on a Banach space are frequently equipped with quasinorms instead of norms. In this paper we develop a technique to replace the original quasi-norm by a norm in a dual preserving way, in the sense that the dual of the space with the new norm coincides with the dual of the space with the original quasi-norm. Applications to problems on the existence and approximation of solutions of convolution equations and on hypercyclic convolution operators on spaces of entire functions are provided.