Frames for the solution of operator equations in Hilbert spaces with fixed dual pairing

TL;DR

This paper extends the concept of Stevenson frames to Banach spaces, exploring their properties without identifying a Hilbert space with its dual, and shows their relevance in non-Hilbert settings like Sobolev spaces.

Contribution

It introduces and analyzes Stevenson frames for Banach spaces, generalizing the concept beyond Hilbert spaces and clarifying their role when dual spaces are not identified.

Findings

Stevenson frames can be extended to Banach spaces.

In this setting, Banach spaces are isomorphic to Hilbert spaces via new inner products.

The investigation of $ ext{ell}^2$-Banach frames is meaningful without dual space identification.

Abstract

For the solution of operator equations, Stevenson introduced a definition of frames, where a Hilbert space and its dual are {\em not} identified. This means that the Riesz isomorphism is not used as an identification, which, for example, does not make sense for the Sobolev spaces $H_0^1(\Omega)$ and $H^{-1}(\Omega)$. In this article, we are going to revisit the concept of Stevenson frames and introduce it for Banach spaces. This is equivalent to $\ell^2$-Banach frames. It is known that, if such a system exists, by defining a new inner product and using the Riesz isomorphism, the Banach space is isomorphic to a Hilbert space. In this article, we deal with the contrasting setting, where $\mathcal H$ and $\mathcal H'$ are not identified, and equivalent norms are distinguished, and show that in this setting the investigation of $\ell^2$-Banach frames make sense.