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Shuaibing Luo

Publications and source records attributed to Shuaibing Luo.

16 recordsLinked to original sources
arXiv2026

When do Kernels Admit Characteristic Functions?

A general framework for deriving characteristic functions for reproducing kernels that do not necessarily possess the complete Pick property was recently established by Bhattacharyya and Jindal. We show that, in this setting, the existence of a characteristic function is equivalent to a Beurling-type invariant subspace condition. Combined with recent results characterizing kernels satisfying this condition, our theorem implies that the existence of a characteristic function is equivalent to a concrete Agler-type decomposition of the underlying kernels.

math.FA↗
arXiv2026

On the holomorphic differential operator $\frac{d}{dz}: Q_K\to L^q(WdA)$

In this paper, we obtain non-testing characterizations, in terms of dyadic capacity gauges, of the boundedness and compactness of the differentiation operator $$ \frac{d}{dz}:Q_K\longrightarrow L^q(W\,dA), \qquad 0<q<\infty. $$ We also characterize the limiting case as $q\to0^+$, formulated in terms of a logarithmic geometric mean, while the endpoint $q=\infty$ is treated separately using a standard testing argument. These results greatly extend the previous work on ${\mathcal Q}_p$-spaces to the general setting of $Q_K$-spaces. As applications, we characterize composition operators and Volterra-type integral operators between different $Q_K$-spaces. In particular, the off-diagonal characterization established here, together with the previously established diagonal case, completely resolves Zhao's 2009 open question on composition operators between ${\mathcal Q}_p$-spaces.

math.CV↗
arXiv2026

Beurling Criteria for Reproducing Kernels

A classical theorem due to Beurling-Lax-Halmos characterizes the invariant subspaces of the unilateral shift as ranges of isometric multiplication operators acting on the Hardy space. The class of Beurling-Lax-Halmos (BLH) pairs of reproducing kernels is introduced, consisting of those pairs that admit an analogue of this theorem. The BLH class is, under varying hypotheses, characterized in several equivalent ways: dilation-theoretically, via a complete Leech interpolation property, and through a sums-of-squares-inspired Agler-style decomposition. These characterizations unify and extend a variety of related results in the literature. Examples are given to illustrate the theory, the hypotheses and compare and contrast with the recent developments in the study of complete Pick pairs. In particular, while it is anticipated, perhaps under some mild assumptions, that the class of complete Pick pairs of kernels is contained in the class of BLH pairs of kernels, it is shown that the reverse inclusion fails in a strong sense.

math.FA↗
arXiv2026

Tangential boundary behavior in Hilbert spaces of analytic functions

Sarason's Hilbert space version of Carath\'eodory-Julia Theorem connects the non-tangential boundary behavior of functions in de Branges-Rovnyak space $H(b)$ with the existence of angular derivatives in the sense of Carath\'eodory for $b$, an analytic self-mapping of the unit disk. In this article, we continue the study of higher order extensions of this result that deal with derivatives of functions in $H(b)$, and we consider notions of approach regions more general than the non-tangential ones. Our main result generalizes the recent work of Duan-Li-Mashreghi on boundary behavior in model spaces to $H(b)$-spaces and to higher order derivatives, and we give a new self-contained proof of that result. It also generalizes earlier radial results of Fricain-Mashreghi. In relation to existence of angular derivatives, we show that in the classical Carath\'eodory-Julia Theorem one cannot replace the non-tangential approach region by any essentially larger region.

math.FA↗
arXiv2024

De Branges-Rovnyak spaces generated by row Schur functions with mate

In this paper, we study the de Branges-Rovnyak spaces $\mathcal{H}(B)$ generated by row Schur functions $B$ with mate $a$. We prove that the polynomials are dense in $\mathcal{H}(B)$, and characterize the backward shift invariant subspaces of $\mathcal{H}(B)$. We then describe the cyclic vectors in $\mathcal{H}(B)$ when $B$ is of finite rank and $\dim (aH^2)^\perp < \infty$.

math.FA↗
arXiv2024

On Dirichlet-type and $n$-isometric shifts in finite rank de~Branges--Rovnyak spaces

This paper studies the function spaces $\mathcal{D}(μ)$ by Richter and Aleman, and $\mathcal{D}_{\vecμ}$ by the second author. It is known that the forward shift $M_z$ is bounded and expansive on $\mathcal{D}(μ)$, and therefore $\mathcal{D}(μ)$ coincides with a de~Branges--Rovnyak space $\mathcal{H}[B]$. We show that such a $B$ is rational if and only if $μ$ is finitely atomic, and this happens exactly when the corresponding defect operator has finite rank. We also outline a method for calculating the reproducing kernel of $\mathcal{D}(μ)$ for finitely atomic $μ$. Similarly, we characterize the allowable tuples $\vecμ = (\frac{|dz|}{2π}, μ_1, \ldots, μ_{n-1})$ such that $M_z$ on $\mathcal{D}_{\vecμ}$ is expansive with finite rank defect operator. This investigation provides many interesting examples of normalized allowable tuples $\vecμ$.

math.FA↗
arXiv2023

Invariant subspaces of the direct sum of forward and backward shifts on vector-valued Hardy spaces

Let $S_{E}$ be the shift operator on vector-valued Hardy space $H_{E}^{2}.$ Beurling-Lax-Halmos Theorem identifies the invariant subspaces of $S_{E}$ and hence also the invariant subspaces of the backward shift $S_{E}^{\ast}.$ In this paper, we study the invariant subspaces of $S_{E}\oplus S_{F}^{\ast}.$ We establish a one-to-one correspondence between the invariant subspaces of $S_{E}\oplus S_{F}^{\ast}$ and a class of invariant subspaces of bilateral shift $B_{E}\oplus B_{F}$ which were described by Helson and Lowdenslager. As applications, we express invariant subspaces of $S_{E}\oplus S_{F}^{\ast}$ as kernels or ranges of mixed Toeplitz operators and Hankel operators with partial isometry-valued symbols. Our approach greatly extends and gives different proofs of the results of Câmara and Ross, and Timotin where the case with one dimensional $E$ and $F$ was considered.

math.FA↗
arXiv2023

Corona theorem for the Dirichlet-type space

This paper utilizes Cauchy's transform and duality for the Dirichlet-type space $D(μ)$ with positive superharmonic weight $U_μ$ on the unit disk $\mathbb{D}$ to establish the corona theorem for the Dirichlet-type multiplier algebra $M\big(D(μ)\big)$ that: if $$\{f_1,...,f_n\}\subseteq M\big(D(μ)\big)\quad\text{and}\quad \inf_{z\in\mathbb{D}}\sum_{j=1}^n|f_j(z)|>0$$ then $$ \exists\,\{g_1,...,g_n\}\subseteq M\big(D(μ)\big)\quad\text{such that}\quad \sum_{j=1}^nf_jg_j=1, $$ thereby generalizing Carleson's corona theorem for $M(H^2)=H^\infty$ and Xiao's corona theorem for $M(\mathscr{D})\subset H^\infty$ thanks to $$ D(μ)=\begin{cases} \text{Hardy space}\ H^2\quad &\text{as}\quad dμ(z)=(1-|z|^2)\,dA(z)\ \ \forall\ z\in\mathbb{D};\\ \text{Dirichlet space}\ \mathscr{D}\ &\text{as}\quad dμ(z)=|dz|\ \ \forall\ z\in\mathbb{T}=\partial{\mathbb{D}}. \end{cases} $$

math.FA↗
arXiv2023

Higher order isometric shift operator on the de Branges-Rovnyak space

The de Branges-Rovnyak space $H(b)$ is generated by a bounded analytic function $b$ in the unit ball of $H^\infty$. When $b$ is a nonextreme point, the space $H(b)$ is invariant by the forward shift operator $M_z$. We show that the $H(b)$ spaces provide model spaces for expansive quasi-analytic $2n$-isometric operators $T$ with $T^*T - I$ being rank one. Then we describe the invariant subspaces of the $2n$-isometric forward shift operator $M_z$ on $H(b)$.

math.FA↗
arXiv2023

Sub-Bergman Hilbert spaces on the unit disk III

For a bounded analytic function $\varphi$ on the unit disk $\D$ with $\|\varphi\|_\infty\le1$ we consider the defect operators $D_\varphi$ and $D_{\overline\varphi}$ of the Toeplitz operators $T_\varphi$ and $T_{\overline\varphi}$, respectively, on the weighted Bergman space $A^2_\alpha$. The ranges of $D_\varphi$ and $D_{\overline\varphi}$, written as $H(\varphi)$ and $H(\overline\varphi)$ and equipped with appropriate inner products, are called sub-Bergman spaces. We prove the following three results in the paper: for $-1<\alpha\le0$ the space $H(\varphi)$ has a complete Nevanlinna-Pick kernel if and only if $\varphi$ is a M\"{o}bius map; for $\alpha>-1$ we have $H(\varphi)=H(\overline\varphi)=A^2_{\alpha-1}$ if and only if the defect operators $D_\varphi$ and $D_{\overline\varphi}$ are compact; and for $\alpha>-1$ we have $D^2_\varphi(A^2_\alpha)= D^2_{\overline\varphi}(A^2_\alpha)=A^2_{\alpha-2}$ if and only if $\varphi$ is a finite Blaschke product. In some sense our restrictions on $\alpha$ here are best possible.

math.CV↗
arXiv2020

Higher order local Dirichlet integrals and de Branges-Rovnyak spaces

We investigate expansive Hilbert space operators $T$ that are finite rank perturbations of isometric operators. If the spectrum of $T$ is contained in the closed unit disc $\overline{\mathbb{D}}$, then such operators are of the form $T= U\oplus R$, where $U$ is isometric and $R$ is unitarily equivalent to the operator of multiplication by the variable $z$ on a de Branges-Rovnyak space $\mathcal{H}(B)$. In fact, the space $\mathcal{H}(B)$ is defined in terms of a rational operator-valued Schur function $B$. In the case when $\dim \ker T^*=1$, then $\mathcal{H}(B)$ can be taken to be a space of scalar-valued analytic functions in $\mathbb{D}$, and the function $B$ has a mate $a$ defined by $|B|^2+|a|^2=1$ a.e. on $\partial \mathbb{D}$. We show the mate $a$ of a rational $B$ is of the form $a(z)=a(0)\frac{p(z)}{q(z)}$, where $p$ and $q$ are appropriately derived from the characteristic polynomials of two associated operators. If $T$ is a $2m$-isometric expansive operator, then all zeros of $p$ lie in the unit circle, and we completely describe the spaces $\mathcal{H}(B)$ by use of what we call the local Dirichlet integral of order $m$ at the point $w\in \partial \mathbb{D}$.

math.FA↗
arXiv2018

Multiplication and composition operators on the derivative Hardy space $S^{2}({\mathbb{D}})$

In this paper we propose a different (and equivalent) norm on $S^{2} ({\mathbb{D}})$ which consists of functions whose derivatives are in the Hardy space of unit disk. The reproducing kernel of $S^{2}({\mathbb{D}})$ in this norm admits an explicit form, and it is a complete Nevanlinna-Pick kernel. Furthermore, there is a surprising connection of this norm with $3$ -isometries. We then study composition and multiplication operators on this space. Specifically, we obtain an upper bound for the norm of $C_φ$ for a class of composition operators. We completely characterize multiplication operators which are $m$-isometries. As an application of the 3-isometry, we describe the reducing subspaces of $M_φ$ on $S^{2}({\mathbb{D}})$ when $φ$ is a finite Blaschke product of order 2.

math.FA↗
arXiv2018

Hilbert-Schmidtness of some finitely generated submodules in $H^2(\mathbb{D}^2)$

A closed subspace $\mathcal{M}$ of the Hardy space $H^2(\mathbb{D}^2)$ over the bidisk is called a submodule if it is invariant under multiplication by coordinate functions $z_1$ and $z_2$. Whether every finitely generated submodule is Hilbert-Schmidt is an unsolved problem. This paper proves that every finitely generated submodule $\mathcal{M}$ containing $z_1 - φ(z_2)$ is Hilbert-Schmidt, where $φ$ is any finite Blaschke product. Some other related topics such as fringe operator and Fredholm index are also discussed.

math.FA↗
arXiv2018

Reducing subspaces of multiplication operators on the Dirichlet space

In this paper, we study the reducing subspaces for the multiplication operator by a finite Blaschke product $ϕ$ on the Dirichlet space $D$. We prove that any two distinct nontrivial minimal reducing subspaces of $M_ϕ$ are orthogonal. When the order $n$ of $ϕ$ is $2$ or $3$, we show that $M_ϕ$ is reducible on $D$ if and only if $ϕ$ is equivalent to $z^n$. When the order of $ϕ$ is $4$, we determine the reducing subspaces for $M_ϕ$, and we see that in this case $M_ϕ$ can be reducible on $D$ when $ϕ$ is not equivalent to $z^4$. The same phenomenon happens when the order $n$ of $ϕ$ is not a prime number. Furthermore, we show that $M_ϕ$ is unitarily equivalent to $M_{z^n} (n > 1)$ on $D$ if and only if $ϕ= az^n$ for some unimodular constant $a$.

math.FA↗
arXiv2018

Reducing subspaces for multiplication operators on the Dirichlet space through local inverses and Riemann surface

This paper is devoted to the study of reducing subspaces for multiplication operator $M_ϕ$ on the Dirichlet space with symbol of finite Blaschke product. The reducing subspaces of $M_ϕ$ on the Dirichlet space and Bergman space are related. Our strategy is to use local inverses and Riemann surface to study the reducing subspaces of $M_ϕ$ on the Bergman space, and we discover a new way to study the Riemann surface for $ϕ^{-1}\circϕ$. By this means, we determine the reducing subspaces of $M_ϕ$ on the Dirichlet space when the order of $ϕ$ is $5$; $6$; $7$ and answer some questions of Douglas-Putinar-Wang \cite{DPW12}.

math.FA↗