arXiv subjects
Publications and source records attributed to Songxiao Li.
Several upper bounds as well as one lower bound for the operator norm of the generalized Hilbert operator $\mathcal{H}_b$ acting on weighted Bergman spaces $A_{\alpha}^p$ are established. Moreover, under some mild assumptions, we obtain the exact norm of $\mathcal{H}_b$ on $ A_{\alpha}^p$.
We study the generalized Hilbert operator \[ \mathcal{H}_b f(z)=\int_0^1 f(t)\,\frac{(1-t)^b}{(1-tz)^{b+1}}\,dt, \qquad b>0, \] acting on the Hardy spaces $H^p$ for $1\leq p\leq \infty$. We establish the precise operator norm \[ \|\mathcal{H}_b\|_{H^p\to H^p}=B\!\left(\frac1p,b+1-\frac1p\right) \] for every $1 0$, in contrast with the classical Hilbert operator, and we obtain the sharp restricted norm estimate \[ \|\mathcal{H}_b\|_{H^1_0\to H^1}=B(1,b). \] We also determine the exact norm \[ \|\mathcal{H}_b\|_{H^\infty\to \mathcal B}=\frac{1}{b+1}+2. \]
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).
For a bounded operator \(T\) between Banach spaces, we introduce the \emph{fixed-copy exponent sets} \[ \begin{aligned} \operatorname{Fix}_{\ell}(T) &:= \{r\ge1:T\text{ fixes a copy of }\ell^r\},\\ \operatorname{Fix}_{L}(T) &:= \{r\ge1:T\text{ fixes a copy of }L^r(0,1)\}. \end{aligned} \] For \(1\le p,q<\infty\), we completely determine both sets for every bounded composition operator \(C_\varphi:H^p\to H^q\). In particular, \(C_\varphi\) is strictly singular if and only if \(\operatorname{Fix}_{\ell}(C_\varphi)=\varnothing\). The classification also gives complete characterizations of the \(\ell^r\)-singular and \(L^r(0,1)\)-singular subclasses for every \(r\ge1\). As a further consequence, it completely resolves Problems~4.3\textup{(1)} and~4.3\textup{(2)} posed by Laitila, Nieminen, Saksman, and Tylli. The proofs introduce a new localization method for producing fixed copies from boundary lower estimates. Its main ingredient is a localization theorem independent of composition operators: for every measurable \(E\subset\mathbb T\) with \(m(E)>0\) and \(1\le p<r\le2\), it constructs a single copy of \(L^r(0,1)\) in \(H^p\) on which lower \(L^s(E)\) estimates hold simultaneously for all \(1\le s\le p\), with constants depending on \(E\) only through \(m(E)\). The classification combines this method with pullback measure criteria, known fixed-copy results, and classical subspace restrictions. For \(r<2\), the localization theorem is proved using stable integrals and analytic lifting; the endpoint \(r=2\) is handled by an \(E\)-adapted lacunary construction.
In this paper, we investigate the boundedness and compactness of generalized integration operators $T_g^{n,k}$ and $S_g^{n,0}$ from analytic tent spaces $AT_p^\infty(α)$ to $AT_q^\infty(β)$ when $0 -2$.
Using some estimates in [J. Funct. Anal. {\bf 278}(2020), Article No. 108401], we completely characterized the boundedness and compactness of the Stević-Sharma type operators with different weights and different composition symbols between Bergman spaces induced by two-sides doubling weights.
In this paper, we characterize the boundedness and compactness of differences of weighted composition operators from weighted Bergman spaces $A^p_ω$ induced by a doubling weight $ω$ to Lebesgue spaces $L^q_μ$ on the unit ball for full $0<p,q<\infty$, which extend many results on the unit disk. As a byproduct, a new characterization of $q$-Carleson the measure for $A^p_ω$ in terms of the Bergman metric ball is also presented.
A conjecture posed by Chalmoukis in 2020 states that if $T_{g,a}:H^p\to H^q(0<q<p<\infty)$ is bounded, then $g$ must be in $H^{\frac{pq}{p-q}}$. In this article, we provide a positive answer to the aforementioned conjecture. We also consider the compactness of $T_{g,a}:H^p\to H^q(0<q<p<\infty)$.
In this paper, the boundedness and compactness of generalized integration operators $T_g^{n,k}$ between different analytic tent spaces in the unit disc are completely characterized.
Using Khinchin's inequality, Ger$\check{\mbox{s}}$gorin's theorem and the atomic decomposition of Bergman spaces, we estimate the norm and essential norm of Stević-Sharma type operators from weighted Bergman spaces $A_ω^p$ to $A_μ^q$ and the sum of weighted differentiation composition operators with different symbols from weighted Bergman spaces $A_ω^p$ to $H^\infty$.The estimates of those between Bergman spaces remove all the restrictions of a result in [Appl. Math. Comput.,{\bf 217}(2011),8115--8125]. As a by-product, we also get an interpolation theorem for Bergman spaces induced by doubling weights.
In this paper, we give a universal description of the boundedness and compactness of Toeplitz operator $\mathcal{T}_μ^ω$ between Bergman spaces $A_η^p$ and $A_\upsilon^q$ when $μ$ is a positive Borel measure, $1<p,q<\infty$ and $ω,η,\upsilon$ are regular weights. By using Khinchin's inequality and Kahane's inequality, we get a new characterization of the Carleson measure for Bergman spaces induced by regular weights.
In this paper, we investigate the complex symmetric structure of generalized weighted composition operators $D_{m,ψ,φ}$ on the weighted Hardy space $H^2(β)$. We obtain explicit conditions for $ D_{m,ψ,φ}$ to be complex symmetric with the conjugation $J_w$. Under the assumption that $ D_{m,ψ,φ}$ is $J_w$-symmetric, some sufficient and necessary conditions for $D_{m,ψ,φ}$ to be Hermitian and normal are given.
A bounded linear operator $T$ on a separable complex Hilbert space $H$ is called $C$-normal if there is a conjugation $C$ on $H$ such that $ CT^\ast TC=TT^\ast$. Let $φ$ be a linear fractional self-map of $\mathbb{D}$. In this paper, we characterize the necessary and sufficient condition for the composition operator $C_φ$ and weighted composition operator $W_{ψ,φ}$ to be $C$-normal with some conjugations $C$ and a function $ψ$.
In this paper, the boundedness and compactness of the difference of composition-differentiation operators $D_φ-D_ψ$ acting from Hardy spaces $H^p$ to weighted Bergman spaces $A^q_α$ are completely characterize for all $0<p,q<\infty$.
In this paper, we study 2-complex symmetric composition operators with the conjugation $J$ on the Hardy space $H^2$. More precisely, we obtain the necessary and sufficient condition for the composition operator $C_ϕ$ to be 2-complex symmetric when the symbols $ϕ$ is an automorphism of $\mathbb D$. We also characterize the 2-complex symmetric composition operator $C_ϕ$ on the Hardy space $H^2$ when $ϕ$ is a linear fractional self-map of $\mathbb D$.
We study the boundedness and compactness of the generalized Volterra integral operator on weighted Bergman spaces with doubling weights on the unit disk. A generalized Toeplitz operator is defined and the boundedness, compactness and Schatten class of this operator are investigated on the Hilbert weighted Bergman space. As an application, Schatten class membership of generalized Volterra integral operators are also characterized. Finally, we also get the characterizations of Schatten class membership of generalized Toeplitz operator and generalized Volterra integral operators on the Hardy space $H^2$.
The purpose of this paper is concerned with the approximate solution of split equality problems. We introduce two types of algorithms and a new self-adaptive stepsize without prior knowledge of operator norms. The corresponding strong convergence theorems are obtained under mild conditions. Finally, some numerical experiments demonstrate the efficiency of our results and compare them with the existing results.
This paper is to analyze the approximation solution of a split variational inclusion problem in the framework of infinite dimensional Hilbert spaces. For this purpose, several inertial hybrid and shrinking projection algorithms are proposed under the effect of self-adaptive stepsizes which does not require information of the norms of the given operators. Some strong convergence properties of the proposed algorithms are obtained under mild constraints. Finally, an experimental application is given to illustrate the performances of proposed methods by comparing existing results.