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 values in a Banach space $X$, particularly sequences. Our approach relies on the underlying Banach spaces satisfying the BK property, that is, having continuous evaluations. Classical multiplier spaces arise when $B$ is a pointwise product of scalar functions and we make the point for the case when $V$ or $W$ consists of vector functions. Special attention is given to the sequence spaces $\Sigma \ell_{\infty}(X)$ (bounded partial sums), $\Sigma c(X)$ (summable), and $\ell_u(X)$ (unconditionally summable). Given a BK scalar sequence space $V$, we introduce the multiplier space $M_{\Sigma}(V,X)$ and establish conditions under which it determines a closed subspace of bounded linear operators from $V$ into $X$. The notion of associate space is precised for BK-spaces, linking this construction with classical K\"othe duality. We consider what we named strong vectorialization $Y(X)$ and weak vectorialization $Y_w(X)$ of a Banach sequence ideal $Y$. The weak vectorialization is obtained as a multiplier space and employed to describe classical sequence spaces as $\ell_{p,w}(X)$, $1 \leq p \leq \infty$. We introduce the $b$-ideal part of a space and show that the $b$-ideal part of $\Sigma c(X)$ is $\ell_u(X)$ and that of $\Sigma \ell_{\infty}(X)$ is $\ell_{1,w}(X)$. We also study the sequence space $bv(X)$ (bounded variation), proving that $M(\Sigma c(X),\Sigma c(X)) = bv(\mathbb K)$.
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 $\mathbb{D}$, and of an equivalent capacity $D^{(b)}_{K,\mathcal{R}}(\mu)$ expressed as a semidefinite program. The equivalence of the two capacities is established through conic duality and the complex Grothendieck inequality. As an application, we characterize the boundedness and compactness of the Volterra integral operator $T_g$ on $Q_K$, bridging and completely resolving the gap between the sufficient and necessary conditions established by Li and Wulan (2010). We also obtain a complete non-testing characterization of the pointwise multipliers $\mathcal{M}(Q_K)$ on $Q_K$, thereby answering an open problem posed in the survey of Bao and Wulan (2021).
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, the global $L^p$ setting has neither. We control the non-local tail through Calder\'on--Zygmund kernel estimates, an explicit Taylor-remainder identity, and spatial truncation, obtaining order-$(-\epsilon)$ smoothing of the remainder into the Besov scale. Pairing weakly convergent sequences with their canonical Nemyckij duals, we then use the lemma to derive $L^p$ transport equations. For first-order scalar equations we establish the phase-space bicharacteristic (Vlasov) flow of the associated H-distribution when $p \ge 2$; for the quasilinear $p$-wave system we lift the local energy identity to the microlocal level, obtaining a Poynting-flux transport of microlocal energy in the linear core $p = 2$ and isolating the structural obstruction to its closure when $p \neq 2$. These results provide functional-analytic tools for tracking the propagation of singularities in degenerate nonlinear and non-local partial differential equations.
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 form $\Phi(x)=wJ(x)$, where $w$ is a unitary element and $J$ is a Jordan $^*$-isomorphism, which extends classical results such as the Banach--Stone theorem and Kadison's theorem. Under several structural assumptions on the underlying von Neumann algebras (including all type $\mathrm{II}_\infty$ and type $\mathrm{III}$ algebras, and all factors, and algebras with atomless centers), we prove the one-to-one correspondence between the $F$-norm and the pair $(\mu, D)$ of a probability measure and a dimension function, which fails for algebras with atomic centers.
Jinghao 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 $(p_i)_{i\in I}$ is a prescribed collection of proximal points in $\HH$, which naturally generalize classical projection problems. We establish necessary and sufficient conditions for the existence of approximate solutions to such systems. Moreover, we introduce and derive several characterizations of the inverse proximal property (IPP), as a generalization of the inverse best approximation property (IBAP). Applications of the obtained results are presented in the contexts of a feasibility problem and signal recovery problem, demonstrating the relevance of the proposed framework to optimization and inverse problems in Hilbert spaces.
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 $A_\omega^2$ and a detailed localization analysis of the resulting infinite matrix representation of $T_{\omega,u}$. Even in the unweighted Bergman space \(A^2\), our argument is new and does not rely on the classical translation operators. We further show that this Axler--Zheng compactness characterization does not extend, in general, to products of Toeplitz operators on \(\widehat{\mathcal D}\)-weighted Bergman spaces, and hence to the corresponding Toeplitz algebra generated by bounded symbols. More precisely, we construct a radial log-subharmonic \(\widehat{\mathcal D}\)-weight \(\omega\) and bounded symbols \(u,v\) such that the product \(T_{\omega,v}T_{\omega,u}\) is noncompact, whereas its Berezin transform vanishes at the boundary.
Yuerang Li, Zi-Peng Wang· 0 citations
We use cookies to run the site and, with your consent, for analytics and to show ads.
See our Cookie Policy.
