Emmanuel Fricain, Javad Mashreghi, Rishika Rupam
In this note, we describe the backward shift invariant subspaces for a large class of reproducing kernel Hilbert spaces. This class includes in particular de Branges-Rovnyak spaces (the non-extreme case) and the range space of co-analytic Toeplitz operators.
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In this paper, we study random sampling on reproducing kernel space $V$, which is a range of an idempotent integral operator. Under certain decay condition on the integral kernel, we show that any element in $V$ can be approximated by an element in a finite-dimensional subspace of $V$. Moreover, we prove with overwhelming probability that random points uniformly distributed over a cube $C$ is stable sample for the set of functions concentrated on $C$
In this article, we consider the random sampling in the image space $V$ of mixed Lebesgue space $L^{p,q}(\mathbb{R}^{n+1})$ under an idempotent integral operator. We assume some decay and regularity conditions of the kernel and approximate the unit sphere in $V$ on a bounded cube $C_{R,S}$ by a finite-dimensional subspace of $V$. Consequently, the set of concentrated functions is totally bounded. We prove with an overwhelming probability that the random sample set uniformly distributed over $C_{R,S}$ is a stable set of sampling for the set of concentrated functions on $C_{R,S}$. Moreover, we p…
Contractively included subspaces of Pick spaces
Pick空间是一类再生核希尔伯特空间，它推广了经典的Hardy空间和Drury-Arveson再生核空间.给出了Pick空间的某些压缩包含子空间的特征。这些推广了封闭的不变子空间的特伦特和麦卡洛的特征，以及球，Bolotnikov和方获得的德鲁里-Arveson空间的结果。
We obtaine the full characterization of proper closed invariant subspaces of a generalized backward shift operator (Pommiez operator) in the Frechet space of all holomorphic functions on a simply connected domain $Ω$ of the complex plane, containing the origin. In the case when the function, which defines this operator, is not has zeros in $Ω$ all such subspaces are finite-dimensional. If additionally $Ω$ coincides with the complex plane, then the considered operator is unicellular. If this function has zeros in $Ω$, then the family of mentioned invariant subspaces splits into two classes: the…
Random sampling of signals concentrated on compact set in localized reproducing kernel subspace of $L^p({\mathbb R}^n)$
本文研究局部再生核空间中随机采样的稳定性，证明紧集上满足积分范数离散化条件的采样集对集中函数稳定，并给出随机采样点数量的上界。
In this paper, we mainly study the random sampling stability for signals in a weighted reproducing kernel subspace of $L_ν^p(\mathbb{R}^d)$ without the additional requirement that the kernel function has symmetry. The sampling set is independently and randomly drawn from a general probability distribution over $\mathbb{R}^d$. Based on the frame characterization of weighted reproducing kernel subspaces, we first approximate the weighted reproducing kernel space by a finite dimensional subspace on any bounded domains. Then, we prove that the random sampling stability holds with high probability …
The techniques developed by Popescu, Muhly-Solel and Good for the study of algebras generated by weighted shifts are applied to generalize results of Sarkar and of Bhattacharjee-Eschmeier-Keshari-Sarkar concerning dilations and invariant subspaces for commuting tuples of operators. In that paper the authors prove Beurling-Lax-Halmos type results for commuting tuples $T=(T_1,\ldots,T_d)$ operators that are contractive and pure; that is $Φ_T(I)\leq I$ and $Φ_T^n(I)\searrow 0$ where $$Φ_T(a)=Σ_i T_iaT_i^*.$$ Here we generalize some of their results to commuting tuples $T$ satisfying similar condi…
Almost invariant subspaces of shift operators and products of Toeplitz and Hankel operators
本文通过Toeplitz与Hankel算子乘积的值域或核，给出了后向移位算子几乎不变子空间的更显式刻画，并证明其与前向移位算子的几乎不变子空间相同。
An Invariant Subspace Theorem and Invariant Subspaces of Analytic Reproducing Kernel Hilbert Spaces - I
设T是Hilbert空间H上的一个C_{\cdot 0}-压缩，S是H的一个非平凡闭子空间.本文证明了S是H的T-不变子空间当且仅当存在Hilbert空间D和部分等距算子H ^2_D（\mathbb{D}）\raro H使得H ^M_z = T_z且S = ran_z，或等价地P_S = T_z ^*.作为应用，我们对单位圆盘上C_{\cdot 0}-压缩和解析再生核希尔伯特空间的平移不变子空间进行了完全分类。我们的结果也包括单位圆盘上加权Bergman空间的情形。
The Feichtinger conjecture for reproducing kernels in model subspaces
对于Hardy空间$H^2$的模型子空间$K_Θ= H^2\ominus ΘH^2$中的归一化再生核系统，得到了关于Feichtinger猜想的两个结果，其中$Θ$是一个内函数.首先，我们在假设$\sup_n |Θ(λ_n)|<1$的情况下，验证了核$ \tilde k_{λ_n} = k_{λ_n}/\|k_{λ_n}\|$的Feichtinger猜想。其次，我们证明了在$Θ$是单分量内函数的情况下的Feichtinger猜想，这意味着集合$\{z:|Θ(z)|<\varepsilon\}$对于某些$\varepsilon\in(0,1)$是连通的。
Sampling and Reconstruction of Signals in a Reproducing Kernel Subspace of $L^p({\Bbb R}^d)$
本文研究了在$L^p(\Rd), 1\le p\le \infty$的再生核子空间中信号的采样与重构问题，该再生核子空间中的积分算子具有非对角衰减性和正则性. $p$ -可积非均匀样条空间和由多个局部化函数生成的平移不变空间是$L^p(\Rd)$再生核子空间的典型例子。我们表明，在这样的再生核子空间中的信号可以重建在一个稳定的方式，从它的样本上采取一个相对分离的集足够小的差距。我们还研究了指数收敛性，一致性，和渐近逐点误差估计的迭代近似投影算法和迭代框架算法重建的信号在这些再生核空间从它的样本具有足够小的差距。
Nearly invariant subspaces in the real Hardy space
本文研究实 Hardy 空间中 backward shift 算子的近不变子空间，探讨有限缺陷近不变子空间，并为 backward shift 算子的近似不变子空间提供特征描述。
Invariant subspaces of the direct sum of forward and backward shifts on vector-valued Hardy spaces
研究向量值Hardy空间上前向与后向移位直和算子的不变子空间，建立其与双边移位一类不变子空间的一一对应，并将其表示为混合Toeplitz与Hankel算子的核或值域，推广了一维情形下的已有结果。
Similarity of Cowen-Douglas operators to the backward Dirichlet shift
我们证明了在具有解析核的再生核希尔伯特空间上Cowen-Douglas算子与后向移位算子的相似性刻画可以用来描述Dirichlet空间中的相似性.在前面的证明中，一个模型定理，允许一个得到本征向量丛结构的运营商起着至关重要的作用。
In this article, we characterize nearly invariant subspaces of finite defect for the backward shift operator acting on the vector-valued Hardy space which is a vectorial generalization of a result of Chalendar-Gallardo-Partington (C-G-P). Using this characterization of nearly invariant subspace under the backward shift we completely describe the almost invariant subspaces for the shift and its adjoint acting on the vector valued Hardy space.
Invariant subspaces of perturbed backward shift
本文研究 Hardy 空间中后向移位有限秩扰动的不变闭子空间，并据此分类几乎不变子空间、细化近不变子空间概念。
It is well known that the kernel of a Toeplitz operator is nearly invariant under the backward shift $S^*$. This paper shows that kernels of finite-rank perturbations of Toeplitz operators are nearly $S^*$-invariant with finite defect. This enables us to apply a recent theorem by Chalendar--Gallardo--Partington to represent the kernel in terms of backward shift-invariant subspaces, which we identify in several important cases.
We study the structure of the backward shift invariant and nearly invariant subspaces in weighted Fock-type spaces $\mathcal{F}_W^p$, whose weight $W$ is not necessarily radial. We show that in the spaces $\mathcal{F}_W^p$ which contain the polynomials as a dense subspace (in particular, in the radial case) all nontrivial backward shift invariant subspaces are of the form $\mathcal{P}_n$, i.e., finite dimensional subspaces consisting of polynomials of degree at most $n$. In general, the structure of the nearly invariant subspaces is more complicated. In the case of spaces of slow growth (up to…
Solution of Invariant Subspace Problem in the Hilbert space
利用Duhamel代数和再生核的方法，证明了Hardy-希尔伯特空间H^{2}（D）上的每个线性有界算子都有非平凡的不变子空间.这肯定地解决了希尔伯特空间中的不变子空间问题。
We present two alternative proofs of Mandrekar's theorem, which states that an invariant subspaces of the Hardy space on the bidisc is of Beurling type precisely when the shifts satisfy a doubly commuting condition. The first proof uses properties of Toeplitz operators to derive a formula for the reproducing kernel of certain shift invariant subspaces, which can then be used to characterize them. The second proof relies on the reproducing property in order to show that the reproducing kernel at the origin must generate the entire shift invariant subspace.
Bounded symbols and reproducing kernel thesis for truncated Toeplitz operators
Toeplitz算子到$H^2$的余不变子空间的可积称为截断Toeplitz算子。我们研究了与这些算子相关的两个问题。第一，提出了Sarason，是是否有界的运营商意味着存在一个有界的符号;第二是再生核论文。我们表明，在一般情况下，第一个问题的答案是否定的，我们展示了一些类的空间，这两个问题的答案是肯定的。
Reproducing kernel for a class of weighted Bergman spaces on the symmetrized polydisc
对称多圆盘上的一类自然的加权Bergman空间被等距嵌入到相应的加权Bergman空间中。我们找到了这个子空间的一个规范正交基。它使我们能够计算核函数的加权Bergman空间的对称化多圆盘使用显式性质，我们的嵌入。这个核函数家族包括对称化多圆盘上的Szegö核和Bergman核。
An Invariant Subspace Theorem and Invariant Subspaces of Analytic Reproducing Kernel Hilbert Spaces - II
本文是对我们工作[20]的后续贡献，在[20]中我们讨论了希尔伯特空间上压缩的一些不变子空间结果。本文将[20]的结果推广到希尔伯特空间上有界线性算子的n元组的情形。设T =（T_1，\ldots，T_n）是Hilbert空间\mathcal{H}上的一个纯交换同球压缩n元组，\mathcal{S}是\mathcal{H}的一个非平凡闭子空间.我们的一个主要结果表明：\mathcal{S}是一个联合T不变子空间当且仅当存在一个部分等距算子\in \mathcal{B}（H^2_n（\mathcal{E}），\mathcal{H}）$ such that $ \mathcal{S} = H^2_n（\mathcal{E}）$，其中H^2_n是Drury-Arveson空间，\mathcal{E}是系数Hilbert空间，T_i = H \mathcal{E} MT，i = 1，\ldots，n.特别地，我们的工作解决的情况下，联合移位不变子空间的Hardy空间和加权Bergman空间的单位球在\mathbb{C}^n。
In this paper we study subspaces which are invariant under squares and cubes (separately as well as jointly) of unicellular backward weighted shift operators on a separable Hilbert space. The finite-dimensional subspaces are characterized for all weights and the infinite-dimensional subspaces are characterized for two classes of weights.
HCIZ integral formula as unitarity of a canonical map between reproducing kernel spaces
该研究证明HCIZ积分公式等价于Segal-Bargmann空间不变子空间间典范映射的酉性，由此得到HCIZ积分公式的两种新证法及相关结果的替代证明。
Systems of reproducing kernels and their biorthogonal: completeness or incompleteness?
设$\{v_n\}$是希尔伯特空间$\mathcal{H}$中的完备极小系，$\{w_m\}$是它的双正交系.众所周知，$\{w_m\}$不一定是完备的。然而，如果我们考虑解析函数的再生核希尔伯特空间$\mathcal{H}$中的再生核系统，情况可能会改变。研究一类具有再生核Riesz基的空间和Hardy空间的模型子空间$K_Θ$的完备性问题.我们发现一类空间的系统双正交完整系统的再生核总是完整的，并表明，在一般情况下，这是不正确的。特别是，我们回答了N.K.提出的问题。Nikolski的方法，构造了一个非完全双正交系的模型子空间。
Nearly invariant brangesian subspaces
本文将Hitt关于$H^2(\bb D)$上后移算子近不变子空间的定理及其推广扩展到布兰奇空间框架，研究开单位圆盘上解析函数再生核希尔伯特空间中被内函数除时近不变的压缩包含希尔伯特空间。
We consider Hilbert spaces of analytic functions in the disk with a normalized reproducing kernel and such that the backward shift $f(z) \mapsto \frac{f(z)-f(0)}{z}$ is a contraction on the space. We present a model for this operator and use it to prove the surprising result that functions which extend continuously to the closure of the disk are dense in the space. This has several applications, for example we can answer a question regarding reverse Carleson embeddings for these spaces. We also identify a large class of spaces which are similar to the de Branges-Rovnyak spaces and prove some r…
Invariant Krein subspaces, regular irreducibility and integral representations
我们研究了Krein空间中群的酉表示、不可约准则和积分分解。我们的主要工具是克莱因子空间及其（再生）内核和Choquet定理的变体理论。
Recently, Liang and Partington \cite{YP} show that kernels of finite-rank perturbations of Toeplitz operators are nearly invariant with finite defect under the backward shift operator acting on the scalar-valued Hardy space. In this article we provide a vectorial generalization of a result of Liang and Partington. As an immediate application we identify the kernel of perturbed Toeplitz operator in terms of backward shift-invariant subspaces in various important cases by applying the recent theorem (\cite{CDP, OR}) in connection with nearly invariant subspaces of finite defect for the backward …
Let $\mathcal{H}$ be a reproducing kernel Hilbert space of functions on a set $X$. We study the problem of finding a minimal geodesic of the Grassmann manifold of $\mathcal{H}$ that joins two subspaces consisting of functions which vanish on given finite subsets of $X$. We establish a necessary and sufficient condition for existence and uniqueness of geodesics, and we then analyze it in examples. We discuss the relation of the geodesic distance with other known metrics when the mentioned finite subsets are singletons. We find estimates on the upper and lower eigenvalues of the unique self-adjo…
An algebra structure for reproducing kernel Hilbert spaces
本文在再生核希尔伯特空间上定义代数结构，等价于权的次卷积性，证明该类在张量积和拉回下封闭，且谱函子像包含所有紧子空间。
Generalized Reproducing Kernel Banach Spaces: A Functional Analytic Framework for Abstract Neural Networks
本文提出广义再生核巴拿赫空间框架，用于构建抽象神经网络，扩展了传统RKBS在处理巴拿赫值余域问题中的应用，推动了神经架构设计与数学建模的结合。
In this article, we briefly describe nearly $T^{-1}$ invariant subspaces with finite defect for a shift operator $T$ having finite multiplicity acting on a separable Hilbert space $\mathcal{H}$ as a generalization of nearly $T^{-1}$ invariant subspaces introduced by Liang and Partington in \cite{YP}. In other words we characterize nearly $T^{-1}$ invariant subspaces with finite defect in terms of backward shift invariant subspaces in vector-valued Hardy spaces by using Theorem 3.5 in \cite{CDP}. Furthermore, we also provide a concrete representation of the nearly $T_B^{-1}$ invariant subspaces…
In this note we prove that the reproducing kernel of a Hilbert space satisfying the division property has integrable form, is locally of trace class, and the Hilbert space itself is a Hilbert space of holomorphic functions.
Refinement of Operator-valued Reproducing Kernels
研究了一个算子值再生核的加细核的构造，使得加细核的向量值再生核Hilbert空间包含给定算子值再生核的向量值再生核Hilbert空间作为子空间.该研究的动机是在多任务学习中，当欠拟合或过拟合发生时，需要更新当前的算子值再生核。数值模拟表明，所建立的精化核方法能够满足这一要求。基于特征映射和算子值再生核的向量值积分表示，提供了各种表征。具体的例子精炼平移不变和有限的希尔伯特-施密特算子值再生核。其他示例包括标量值的变换不变核和变换核的Hessian的细化。在精化过程中保持的算子值再生核的存在性和性质也进行了研究。
For $b\in H^\infty_1$, the closed unit ball of $H^\infty$, the de Branges-Rovnyak spaces $\mathcal{H}(b)$ is a Hilbert space contractively contained in the Hardy space $H^2$ that is invariant by the backward shift operator $S^*$. We consider the reducing subspaces of the operator $S^{*2}|_{\mathcal{H}(b)}$. When $b$ is an inner function, $S^{*2}|_{\mathcal{H}(b)}$ is a truncated Toepltiz operator and its reducibility was characterized by Douglas and Foias using model theory. We use another approach to extend their result to the case where $b$ is extreme. We prove that if $b$ is extreme but not…
Generalized Gramians: Creating frame vectors in maximal subspaces
框架是希尔伯特空间$\mathscr{H}$中的向量$S$的系统，其属性允许人们针对$\mathscr{H}$中的所有向量编写相对于$S$的两个操作（分析和综合）的算法;以范数-收敛级数表示。传统上，框架性质用$S$ -Gramian表示，即$G_{S}$（元素等于$S$中的向量对的内积的无限矩阵）;但为了保证框架边界，仍然对$S$中的给定向量系统有很强的限制。在本文中，我们删除这些限制$G_{S}$，我们得到的直接积分分析/综合公式。证明了在正半直线上的每个有限区间$J$的谱子空间中，存在相应的标准标架，标架界等于$J$的端点.应用给出再生核希尔伯特空间，和随机场。
A complete characterization of nearly-invariant subspaces of finite defect for the backward shift operator acting on the Hardy space is provided in the spirit of Hitt and Sarason's theorem. As a corollary we describe the almost-invariant subspaces for the shift and its adjoint.
For a shift operator $T$ with finite multiplicity acting on a separable infinite dimensional Hilbert space we represent its nearly $T^{-1}$ invariant subspaces in terms of invariant subspaces under the backward shift. Going further, given any finite Blaschke product $B$, we give a description of the nearly $T_{B}^{-1}$ invariant subspaces for the operator $T_B$ of multiplication by $B$ in a scale of Dirichlet-type spaces.
This paper focuses on representations of contractively embedded invariant subspaces in several variables. We present a version of the de Branges theorem for $n$-tuples of multiplication operators by the coordinate functions on analytic reproducing kernel Hilbert spaces over the unit ball $\mathbb{B}^n$ and the Hardy space over the unit polydics $\mathbb{D}^n$ in $\mathbb{C}^n$.
It is known that (i) a subspace ${\mathcal N}$ of the Hardy space $H^2$ which is invariant under the backward shift operator can be represented as the range of the observability operator of a conservative discrete-time linear system, (ii) the transfer-function of this conservative linear system in turn is the inner Beurling-Lax representer for the forward-shift invariant subspace ${\mathcal M} : = {\mathcal N}^\perp$, and (iii) this transfer function also serves as the Sz.-Nagy-Foias characteristic function of the pure contraction operator $T$ given by $T = P_{\mathcal N} M_z |_{\mathcal N}$. …
In this paper we first study the structure of the scalar and vector-valued nearly invariant subspaces with a finite defect. We then subsequently produce some fruitful applications of our new results. We produce a decomposition theorem for the vector-valued nearly invariant subspaces with a finite defect. More specifically, we show every vector-valued nearly invariant subspace with a finite defect can be written as the isometric image of a backwards shift invariant subspace. We also show that there is a link between the vector-valued nearly invariant subspaces and the scalar-valued nearly invar…
Invariant subspaces of compressions of the Hardy shift on some parametric spaces
本文研究 Hardy 移位在参数空间上的压缩算子 $S_{\alpha,\beta}$，证明其为再生核 Hilbert 空间上的移位，并完整分类其不变子空间，指出它们虽循环但可能不由游荡子空间生成，并给出充要条件。
The Beurling-Lax-Halmos Theorem for Infinite Multiplicity
本文探讨了贝鲁林-拉克斯-哈洛姆定理的扩展问题，研究了移不变子空间的结构及与汉克尔算子核的关系，提出了贝鲁林度数的概念，并探讨了模型算子的谱乘积性。
In this paper we consider compressions of $k^{th}$--order slant Toeplitz operators to the backward shift invariant subspaces of the classical Hardy space $H^2$. In particular, we characterize these operators using compressed shifts and finite rank operators of special kind.
Convergent series of coherent states
针对齐性空间离散子空间上的部分惠特尼函数级数，本文在正定再生核希尔伯特空间框架下完成了其收敛性的证明。
By analytic perturbations, we refer to shifts that are finite rank perturbations of the form $M_z + F$, where $M_z$ is the unilateral shift and $F$ is a finite rank operator on the Hardy space over the open unit disc. Here shift refers to the multiplication operator $M_z$ on some analytic reproducing kernel Hilbert space. In this paper, we first isolate a natural class of finite rank operators for which the corresponding perturbations are analytic, and then we present a complete classification of invariant subspaces of those analytic perturbations. We also exhibit some instructive examples and…
Indices of quadratic programs over reproducing kernel Hilbert spaces for fun and profit
本文从抽象视角研究再生核Hilbert空间上的二次规划，通过最大值原理解释投资组合理论中的稀疏性，发现最优分布的支撑集位于特定边界交集内，并利用该方法解决迷宫问题。
Cyclic nearly invariant subspaces for semigroups of isometries
本文研究了由单位圆盘自同构生成的离散半群及等距的强连续乘法半群的几乎不变子空间结构，利用Toeplitz核与再生核理论，在特定外因子条件下通过模型空间刻画了循环子空间。
Motivated by the theory of weighted shifts on directed trees and its multivariable counterpart, we address the question of identifying commutant and reflexivity of the multiplication $d$-tuple $\mathscr M_z$ on a reproducing kernel Hilbert space $\mathscr H$ of $E$-valued holomorphic functions on $Ω$, where $E$ is a separable Hilbert space and $Ω$ is a bounded domain in $\mathbb C^d$ admitting bounded approximation by polynomials. In case $E$ is a finite dimensional cyclic subspace for $\mathscr M_z$, under some natural conditions on the $B(E)$-valued kernel associated with $\mathscr H$, the c…
We consider Banach spaces E of functions holomorphic on the open unit disc D such that the unilateral shift S and the backward shift T are bounded on E. Assuming that the spectra of S and T are equal to the closed unit disc we discuss the existence of closed z-invariant of N of E having the "division property", which means that the function f $λ$ : z $\rightarrow$ f (z)/ z--$λ$ belongs to N for every $λ$ $\in$ D and for every f $\in$ N such that f ($λ$) = 0. This question is related to the existence of nontrivial bi-invariant subspaces of Banach spaces of hyperfunctions on the unit circle T.
Truncated Toeplitz operators are compressions of multiplication operators on $L^2$ to model spaces (that is, subspaces of $H^2$ which are invariant with respect to the backward shift). For this class of operators we prove certain Szegö type theorems concerning the asymptotics of their compressions to an increasing chain of finite dimensional model spaces.
Approximation processes in the reproducing kernel Hilbert space associated to a continuous kernel on the unit sphere $S^m$ in the Euclidean space $\mathbb{R}^{m+1}$ are known to depend upon the Mercer's expansion of the compact and self-adjoint $L^2(S^m)$-operator associated to the kernel. The estimation of the Kolmogorov $n$-th width of the unit ball of the reproducing kernel Hilbert space in $L^2(S^m)$ and the identification of the so-called optimal subspace usually suffice. These Kolmogorov widths can be computed through the eigenvalues of the integral operator associated to the kernel. Thi…
Let $μ$ be a probability measure on $\mathbb{T}$ that is singular with respect to the Haar measure. In this paper we study Fourier expansions in $L^2(\mathbb{T},μ)$ using techniques from the theory of model subspaces of the Hardy space. Since the sequence of monomials $\{z^n\}_{n\in \mathbb{N}}$ is effective in $L^2(\mathbb{T},μ)$, it has a Parseval frame associated via the Kaczmarz algorithm. Our first main goal is to identify the aforementioned frame with boundary values of the frame $P_φ(z^n)$ for the model subspace $\mathcal{H}(φ)= H^2 \ominus φH^2$, where $P_φ$ is the orthogonal projectio…
Let $G$ be a finite pseudoreflection group, $Ω\subseteq \mathbb C^n$ be a bounded domain which is a $G$-space and $\mathcal H\subseteq\mathcal O(Ω)$ be an analytic Hilbert module possessing a $G$-invariant reproducing kernel. We study the structure of joint reducing subspaces of the multiplication operator $\mathbf M_{\boldsymbolθ}$ on $\mathcal H,$ where $\{θ_i\}_{i=1}^n$ is a homogeneous system of parameters associated to $G$ and $\boldsymbolθ= (θ_1, \ldots, θ_n)$ is a polynomial map of $\mathbb C^n$. We show that it admits a family $\{\mathbb P_\varrho\mathcal H:\varrho\in\widehat G\}$ of n…
Dynamics of weighted backward shifts on certain analytic function spaces
本文引入两类解析函数空间，研究加权后移算子的有界性、相似性及动力学性质，证明其超循环子空间存在，并推广至算子的函数。
Noncommutative Shift-Invariant Spaces
在离散群的酉表示下不变的希尔伯特空间的子空间的结构与具有在无界算子空间中取值的内积的希尔伯特模的概念有关。再生系统的理论，在这样的模块化结构的发展，提供了一个一般的框架，其中包括移位不变空间的基本结果。特别是，提供了Riesz和与群表示相关的框架序列的一般特征，扩展了以前关于阿贝尔群和非交换离散群的酉表示的循环子空间的结果。
Generalized $q$-Fock spaces and structural identities
本文利用 $q$-calculus 研究了一族在 Hardy 空间与 Fock 空间间插值的再生核 Hilbert 空间，通过经典算子及其 $q$-calculus 对应算子对其进行刻画，并借此研究了经典后向移位算子与 q-Jackson 导数间的交织算子。
Beurling Criteria for Reproducing Kernels
研究再生核的贝林准则，通过多种方式刻画贝林 - 拉克斯 - 哈尔莫斯（BLH）再生核对类别，统一扩展相关结果，举例说明理论并与完全皮卡对研究对比，指出完全皮卡核类别与BLH核类别反向包含关系不成立。
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