Subspaces of $\displaystyle H^{p}$ linearly homeomorphic to $l^{p}.$

TL;DR

This paper introduces two rapid methods for constructing weak*-copies of ^ in H^, explores their properties, and connects them through a stability theorem, with broader implications for dual spaces and ^p spaces.

Contribution

It provides new constructions of weak*-copies of ^ in H^ and links these via a stability theorem, extending to general dual spaces and ^p spaces.

Findings

Constructed weak*-copies of ^ in H^ using two fast methods.

Showed these copies are necessarily weak*-complemented.

Connected the constructions through a Paley-Wiener type stability theorem.

Abstract

We present two fast constructions of weak*-copies of $\ell ^\infty$ in $H^{\infty}$ and show that such copies are necessarily weak*-complemented. Moreover, via a Paley-Wiener type of stability theorem for bases, a connection can be made in some cases between the two types of construction, via interpolating sequences (in fact these are at the basis of the second construction). Our approach has natural generalizations where H $\infty$ is replaced by an arbitrary dual space and $\ell ^{\infty }$ by $\ell^{p}$ (1 $\le$ p $\le$ $\infty$) relying on the notions of generalized interpolating sequence and bounded linear extension. An old (very simple but unpublished so far) construction of bases which are Besselian but not Hilbertian finds a natural place in this development.