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arXiv · 2608.20718

Characterization of Frame-Related Sequences on Semi-Hilbert Spaces

Hemalatha M·P. Sam Johnson
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Abstract

Let $A\in\mathcal{B}(H)^+$ and $B\in\mathcal{B}(\ell^2)^+$ be positive bounded operators. We investigate reduced weighted adjoints of densely defined closable operators and use them to develop frame-type systems in the semi-Hilbert spaces induced by $A$ and $B$. Closedness, boundedness, range behavior, and algebraic properties of the $A$-adjoint are established, with particular attention to sums, products, and double adjoints in the unbounded setting. We then study the associated $AB$-analysis, $AB$-synthesis, and $AB$-frame operators. Operator-theoretic characterizations of $AB$-Bessel sequences, $AB$-lower semi-frames, and $AB$-frames are obtained, and examples show that several natural weighted-adjoint identities may hold only as proper inclusions. A Douglas-type range characterization for $AB$-frames is also derived.

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BibTeXRIS

Hemalatha M, P. Sam Johnson. 2026-08-21. Characterization of Frame-Related Sequences on Semi-Hilbert Spaces. https://arxiv.org/abs/2608.20718

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