AI 中文总结
本文针对半希尔伯特空间，研究稠定可闭算子的约化加权伴随，构造框架型系统并刻画AB-贝塞尔序列、AB-框架等，导出AB-框架的道格拉斯型值域刻画。
AI 中文摘要
设A∈ℬ(H)⁺与B∈ℬ(ℓ²)⁺为正有界算子，我们研究稠定可闭算子的约化加权伴随并将其用于构造由A、B诱导的半希尔伯特空间上的框架型系统。我们确立了A-伴随的闭性、有界性、值域性质与代数性质，特别关注无界情形下的和、积及双重伴随。随后研究相关的AB-分析、AB-合成与AB-框架算子，得到AB-贝塞尔序列、AB-下半框架及AB-框架的算子论刻画，实例表明若干自然的加权恒等式可能仅为真包含关系，还导出了AB-框架的道格拉斯型值域刻画。
英文摘要
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.
