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 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.

Keywords:

• Banach frames
• discretization of operators
• frames
• invertibility
• matrix representation
• Stevenson frames

Acknowledgment

The authors like to thank Stephan Dahlke, Wolfgang Kreuzer, and Diana Stoeva for fruitful discussions.

Additional information

Funding

This research was supported by the START project FLAME Y551-N13 of the Austrian Science Fund (FWF) and the DACH project BIOTOP I-1018-N25 of the Austrian Science Fund (FWF) and 200021E-142224 of the Swiss National Science Foundation (SNSF).