Solid bases and functorial constructions for (p-)Banach spaces of analytic functions

Abstract

Motivated by new examples of functional Banach spaces over the unit disk, arising as the symbol spaces in the study of random analytic functions, for which the monomials $\{z^n\}_{n\geq 0}$ exhibit features of an unconditional basis yet they often don’t even form a Schauder basis, we introduce a notion called solid basis for Banach spaces and p-Banach spaces and study its properties. Besides justifying the rich existence of solid bases, we study their relationship with unconditional bases, the weak-star convergence of Taylor polynomials, the problem of a solid span and the curious roles played by c0. The two features of this work are as follows: (1) during the process, we are led to revisit the axioms satisfied by a typical Banach space of analytic functions over the unit disk, leading to a notion of $\mathcal{X}^\mathrm{max}$ (and $\mathcal{X}^\mathrm{min}$ ), as well as a number of related functorial constructions, which are of independent interests; (2) the main interests of solid basis lie in the case of non-separable (p-)Banach spaces, such as BMOA and the Bloch space instead of VMOA and the little Bloch space.