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

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 ${\mathcal{H}}^{\prime }$ are not identified, and equivalent norms are distinguished, and show that in this setting the investigation of ${\ell }^2$-Banach frames make sense.

23 pages; accepted for publication in 'Numerical Functional Analysis and Optimization'