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.
We develop a general framework for $B$-multiplier spaces; these are vector spaces $M = M(V,W)$ obtained from a bilinear operator $B \colon M \times V \longrightarrow W$, where $V$ and $W$ are Banach spaces. We focus on their normability and completeness, mainly in the setting of spaces consisting of functions with valu...
Rafael Correa-Morales, F. Galaz-Fontes· 0 citations
We characterize the positive Borel measures $\mu$ on the unit disc $\mathbb{D}$ for which the M\"obius-invariant space $Q_K$ embeds continuously or compactly into $L^2(\mu)$. The characterization is given in terms of a discrete dyadic capacity $C^{(b)}_{K,\mathcal{R}}(\mu)$ built from a polar dyadic resolution of $\mat...
We prove the second commutation lemma for the Lebesgue spaces $L^p({\bf R}^d)$, $1<p<\infty$, on the whole unbounded domain, extending the $L^2$ theory of Tartar (1990) to the Banach-space framework of H-distributions. Unlike the $L^2$ setting, where the Plancherel isometry and Hilbert-space compactness are available,...
Let $LS(\mathcal{M})$ be the algebra of locally measurable operators affiliated with a von Neumann algebra $\mathcal{M}$, equipped with an $F$-norm defined via a dimension function and a probability measure. We prove that every bijective linear isometry between $LS(\mathcal{M})$ admits a canonical representation of the...
Jing-Hao Huang, K. Kudaybergenov, Bing Yan· 0 citations
Let $\HH$ be a real Hilbert space and let $(f_i)_{i\in I}$ be a finite family of proper, lower semicontinuous, and convex functions on $\HH$. This study investigates the existence and uniqueness of exact solutions to systems involving proximity operators of the form: \ $(\forall i\in I)\ \pr{f_i}(x)=p_i,$ where $(p_i)_...
Let $\omega$ be a radial $\widehat{\mathcal D}$-weight and $u$ be a bounded function on the unit disk $\mathbb D$. We prove that the Toeplitz operator \(T_{\omega,u}\) is compact on \(A_\omega^2\) if and only if its Berezin transform vanishes at the boundary. Our approach is based on a polynomial frame for $A_\omega^2$...
Yuerang Li, Zi-Peng Wang· 1 citation
We use cookies to run the site and, with your consent, for analytics and to show ads.
See our Cookie Policy.
