Frames for the Solution of Operator Equations in Hilbert Spaces with Fixed Dual Pairing

Area of Science:

• Functional Analysis
• Operator Theory
• Wavelet Theory

Background:

• Stevenson frames were initially defined for Hilbert spaces, not identifying the space with its dual.
• This approach avoids the Riesz isomorphism, which is unsuitable for Sobolev spaces like H^1 and H^-1.
• Existing literature suggests Banach spaces with frames are isomorphic to Hilbert spaces via the Riesz isomorphism.

Purpose of the Study:

• To revisit and extend the concept of Stevenson frames to Banach spaces.
• To introduce and investigate Stevenson frames in the context of Banach spaces, termed "generalized Stevenson frames" or "$$-Banach frames".
• To explore the utility of these frames in settings where equivalent norms are distinguished and the space and its dual are not identified.

Main Methods:

• Revisiting Stevenson's original definition of frames.
• Introducing a new definition for Stevenson frames applicable to Banach spaces.
• Analyzing the properties and implications of these frames in non-Hilbertian settings.

Main Results:

• The article successfully introduces Stevenson frames for Banach spaces ($$-Banach frames).
• It demonstrates that the investigation of $$-Banach frames is meaningful even when the Banach space and its dual are not identified.
• The study highlights the importance of distinguishing equivalent norms in this generalized framework.

Conclusions:

• Stevenson frames offer a valuable perspective for operator equations, particularly in Banach spaces.
• The concept of $$-Banach frames extends the applicability of frame theory to a broader class of mathematical spaces.
• This work provides a foundation for future research into the applications of generalized Stevenson frames in various mathematical and scientific domains.