On the multiplier problem for the ball on graded Lie groups

Riemann-Stieltjes operators and multipliers on $Q_p$ spaces in the unit ball of $C^n$

This paper is devoted to characterizing the Riemann-Stieltjes operators and pointwise multipliers acting on M${\rm \ddot{o}}$bius invariant spaces $Q_p$, which unify BMOA and Bloch space in the scale of $p$. The boundedness and compactness of these operators on $Q_p$ spaces are determined by means of an embedding theorem, i.e. $Q_p$ spaces boundedly embedded in the non-isotropic tent type spaces $T_q^\infty$.

Interpolation and duality in algebras of multipliers on the ball

We study the multiplier algebras $A(\mathcal{H})$ obtained as the closure of the polynomials on certain reproducing kernel Hilbert spaces $\mathcal{H}$ on the ball $\mathbb{B}_d$ of $\mathbb{C}^d$. Our results apply, in particular, to the Drury-Arveson space, the Dirichlet space and the Hardy space on the ball. We first obtain a complete description of the dual and second dual spaces of $A(\mathcal H)$ in terms of the complementary bands of Henkin and totally singular measures for $\operatorname{Mult}(\mathcal{H})$. This is applied to obtain several definitive results in interpolation. In part…

多乘子在$\mathbb{C}^n$单位球上的循环性：基于圆冠的方法

Cyclicity of Multipliers on the Unit Ball of $\mathbb{C}^n$: A Corona-Based Approach

本文基于圆冠方法，研究了Dirichlet型空间中乘子的循环性条件，得出了乘子在该空间中为循环的充要条件。

单位球齐次子流形上再生核希尔伯特空间的变形

Deformations of Reproducing Kernel Hilbert Spaces on Homogeneous Subvarieties of the Unit Ball

研究单位球齐次子流形、其再生核希尔伯特空间及乘子代数关系，证明RKHS等距同构时乘子代数也同构，对于易处理齐次流形，乘子代数同构或有酉变换关系时，相关RKHS也等距同构。

Multipliers of embedded discs

We consider a number of examples of multiplier algebras on Hilbert spaces associated to discs embedded into a complex ball in order to examine the isomorphism problem for multiplier algebras on complete Nevanlinna-Pick reproducing kernel Hilbert spaces. In particular, we exhibit uncountably many discs in the ball of $\ell^2$ which are multiplier biholomorphic but have non-isomorphic multiplier algebras. We also show that there are closed discs in the ball of $\ell^2$ which are varieties, and examine their multiplier algebras. In finite balls, we provide a counterpoint to a result of Alpay, Put…

Unified approach to spectral properties of multipliers

Let $\mathbb B_n$ be the open unit ball in $\mathbb C^n$. We characterize the spectra of pointwise multipliers $M_u$ acting on Banach spaces of analytic functions on $\mathbb B_n$ satisfying some general conditions. These spaces include Bergman-Sobolev spaces $A^p_{α,β}$, Bloch-type spaces $\mathcal B_α$, weighted Hardy spaces $H^p_w$ with Muckenhoupt weights and Hardy-Sobolev Hilbert spaces $H^2_β$. Moreover, we describe the essential spectra of multipliers in most of the aforementioned spaces, in particular, in those spaces for which the set of multipliers is a subset of the ball algebra.

Canonical transfer-function realization for Schur multipliers on the Drury-Arveson space and models for commuting row contractions

We develop a $d$-variable analog of the two-component de Bran-ges-Rovnyak reproducing kernel Hilbert space associated with a Schur-class function on the unit disk. In this generalization, the unit disk is replaced by the unit ball in $d$-dimensional complex Euclidean space, and the Schur class becomes the class of contractive multipliers on the Drury-Arveson space over the ball. We also develop some results on a model theory for commutative row contractions which are not necessarily completely noncoisometric (the case considered in earlier work of Bhattacharyya, Eschmeier and Sarkar)

Extreme points of the unit ball of an operator space

This paper is a revision and an enlargement of the previous version titled "Extreme points of the unit ball of a quasi-multiplier space" which had been circulated since 2004. We study extreme points of the unit ball of an operator space by introducing the new notion (approximate) "quasi-identities". Then we characterize an operator algebra with a contractive approximate quasi- (respectively, left, right, two-sided) identity in terms of quasi-multipliers and extreme points. Furthermore, we give a very neat necessary and sufficient condition for a given operator space to become a $C^*$-algebra o…

Hypercontractions and factorizations of multipliers in one and several variables

We introduce the notion of characteristic functions for commuting tuples of hypercontractions on Hilbert spaces, as a generalization of the notion of Sz.-Nagy and Foias characteristic functions of contractions. We present an explicit method to compute characteristic functions of hypercontractions and relate characteristic functions by means of the factors of Schur-Agler class of functions and universal multipliers on the unit ball in $\mathbb{C}^n$. We also offer some factorization properties of multipliers. Characteristic functions of hypercontrctions are complete unitary invariant. The Drury…

Noncoherence of the multiplier algebra of the Drury-Arveson space H^2_n for n>=3

Let H_n^2 denote the Drury-Arveson Hilbert space on the unit ball B_n in C^n, and let M(H_n^2) be its multiplier algebra. We show that for n>=3, the ring M(H_n^2) is not coherent.

Localizable points in the support of a multiplier ideal and spectra of constrained operators

A unitarily invariant, complete Nevanlinna--Pick kernel $K$ on the unit ball determines a class of operators on Hilbert space called $K$-contractions. We study those $K$-contractions that are constrained, in the sense that they are annihilated by an ideal of multipliers. Our overarching goal is to identify various joint spectra of these constrained $K$-contractions through the vanishing locus of their annihilators. Our methods are based around a careful analysis of a subset of the ball associated to the annihilator, which we call its support. For the functional models, we show how this support…

Continuous Deformations of Algebras of Holomorphic Functions on Subvarieties of a Noncommutative Ball

We propose a general method for constructing continuous Banach bundles whose fibers are algebras of holomorphic functions on subvarieties of a closed noncommutative ball. These algebras are of the form $\mathcal{A}_d/I_x$, where $\mathcal{A}_d$ is the noncommutative disk algebra introduced by G. Popescu, and $I_x$ is a graded ideal in $\mathcal{ A}_d$, which depends continuously on the point $x$ of the topological space $X$. Similarly, we construct bundles with fibers isomorphic to the algebras $\mathcal{F}_d/I_x$ of holomorphic functions on subvarieties of an open noncommutative ball. Here $\…

Trace theorems in harmonic function spaces on (R^n+1)^m,multiplier theorems and related problems

We introduce and study properties of certain new harmonic function spaces on products of upper half-spaces.Norm estimates for the so-called expanded Bergman projections are obtained.Sharp theorems on multipliers acting on certain Sobolev type spaces of harmonic functions on the unit ball are also obtained

单位谱球上的Schur分析

Schur analysis over the unit spectral ball

本文在单位谱球上定义了Hardy空间、Schur乘子及其实现与插值，并指出其在四元数矩阵及超复数代数等领域的应用前景。

Multiplier algebras of complete Nevanlinna-Pick spaces: dilations, boundary representations and hyperrigidity

We study reproducing kernel Hilbert spaces on the unit ball with the complete Nevanlinna-Pick property through the representation theory of their algebras of multipliers. We give a complete description of the representations in terms of the reproducing kernels. While representations always dilate to $*$-representations of the ambient $C^*$-algebra, we show that in our setting we automatically obtain coextensions. In fact, we show that in many cases, this phenomenon characterizes the complete Nevanlinna-Pick property. We also deduce operator theoretic dilation results which are in the spirit of…

A multiplier algebra functional calculus

This paper generalizes the classical Sz.-Nagy--Foias $H^{\infty}(\mathbb{D})$ functional calculus for Hilbert space contractions. In particular, we replace the single contraction $T$ with a tuple $T=(T_1, \dots, T_d)$ of commuting bounded operators on a Hilbert space and replace $H^{\infty}(\mathbb{D})$ with a large class of multiplier algebras of Hilbert function spaces on the unit ball in $\mathbb C^d$.

完全 Nevanlinna-Pick 核、Schwarz 引理与 Schur 算法

Complete Nevanlinna-Pick kernels, the Schwarz lemma and the Schur algorithm

本文研究满足完全 Nevanlinna-Pick 性质的再生核 Hilbert 空间中乘子代数单位球元素的 Schwarz 引理、Schur 算法及 Poincaré 压缩性。

再生核希尔伯特空间之间的距离与单位球中有限集的几何

Distance between reproducing kernel Hilbert spaces and geometry of finite sets in the unit ball

本文研究再生核希尔伯特空间、乘子代数与点集几何的关系，引入适用于两者的距离变体，证明在Drury-Arveson空间有限维商空间中，空间接近等价于乘子代数接近及点集几乎全等。

Duality, convexity and peak interpolation in the Drury-Arveson space

We consider the closed algebra $\mathcal{A}_d$ generated by the polynomial multipliers on the Drury-Arveson space. We identify $\mathcal{A}_d^*$ as a direct sum of the preduals of the full multiplier algebra and of a commutative von Neumann algebra, and establish analogues of many classical results concerning the dual space of the ball algebra. These developments are deeply intertwined with the problem of peak interpolation for multipliers, and we generalize a theorem of Bishop-Carleson-Rudin to this setting by means of Choquet type integral representations. As a byproduct we shed some light o…

Multipliers and operator space structure of weak product spaces

In the theory of reproducing kernel Hilbert spaces, weak product spaces generalize the notion of the Hardy space $H^1$. For complete Nevanlinna-Pick spaces $\mathcal H$, we characterize all multipliers of the weak product space $\mathcal H \odot \mathcal H$. In particular, we show that if $\mathcal H$ has the so-called column-row property, then the multipliers of $\mathcal H$ and of $\mathcal H \odot \mathcal H$ coincide. This result applies in particular to the classical Dirichlet space and to the Drury-Arveson space on a finite dimensional ball. As a key device, we exhibit a natural operator…

$N(p, q, s)$-type spaces in the unit ball of $C^n$

In this paper, we consider a new class of space, called $N(p, q, s)$-type spaces, in the unit ball $B$ of $C^n$. We study some basic properties, Hadamard gaps, Hadamard products, Random power series, Korenblum's inequality, Gleason's problem, atomic decomposition of $N(p, q, s)$-type spaces. Moreover, we also establish several equivalent characterizations, including Carleson measure characterization and various derivative characterizations. Finally, we also characterize the distance between Bergman-type spaces and $N(p, q, s)$-type spaces, Riemann-Stieltjes operators and multipliers on $N(p, q…

The Toeplitz corona problem for algebras of multipliers on a Nevanlinna-Pick space

Suppose $\fA$ is an algebra of operators on a Hilbert space $H$ and $A_1,..., A_n \in \fA$. If the row operator $[A_1,..., A_n] \in B(H^{(n)},H)$ has a right inverse in $B(H, H^{(n)})$, the Toeplitz corona problem for $\fA$ asks if a right inverse can be found with entries in $\fA$. When $H$ is a complete Nevanlinna-Pick space and $\fA$ is a weakly-closed algebra of multiplication operators on $H$, we show that under a stronger hypothesis, the corona problem for $\fA$ has a solution. When $\fA$ is the full multiplier algebra of $H$, the Toeplitz corona theorems of Arveson, Schubert and Ball-Tr…

Column extreme multipliers of the Free Hardy space

The full Fock space over $\mathbb C ^d$ can be identified with the free Hardy space, $H^2 (\mathbb B ^d _\mathbb N)$ - the unique non-commutative reproducing kernel Hilbert space corresponding to a non-commutative Szegö kernel on the non-commutative, multi-variable open unit ball $\mathbb B ^d _\mathbb N := \bigsqcup _{n=1} ^\infty \left( \mathbb C^{n\times n} \otimes \mathbb C ^d \right) _1$. Elements of this space are free or non-commutative functions on $\mathbb B ^d _\mathbb N$. Under this identification, the full Fock space is the canonical non-commutative and several-variable analogue of…

Asymptotics and zeta functions on compact nilmanifolds

In this paper, we obtain asymptotic formulae on nilmanifolds $Γ\backslash G$, wher $G$ is any stratified (or even graded) nilpotent Lie group equipped with a co-compact discrete subgroup $Γ$. We study especially the asymptotics related to the sub-Laplacians naturally coming from the stratified structure of the group $G$ (and more generally any positive Rockland operators when $G$ is graded). We show that the short-time asymptotic on the diagonal of the kernels of spectral multipliers contains only a single non-trivial term. We also study the associated zeta functions.

Spectral Multipliers on $2$-Step Stratified Groups, II

Given a graded group $G$ and commuting, formally self-adjoint, left-invariant, homogeneous differential operators $\mathcal{L}_1,\dots, \mathcal{L}_n$ on $G$, one of which is Rockland, we study the convolution operators $m(\mathcal{L}_1,\dots, \mathcal{L}_n)$ and their convolution kernels, with particular reference to the case in which $G$ is abelian and $n=1$, and the case in which $G$ is a $2$-step stratified group which satisfies a slight strengthening of the Moore-Wolf condition and $\mathcal{L}_1,\dots,\mathcal{L}_n$ are either sub-Laplacians or central elements of the Lie algebra of $G$.…

子哈代希尔伯特空间的通用乘子

Universal multipliers for Sub-Hardy Hilbert spaces

本文研究 de Branges-Rovnyak 空间的通用乘子，建立一般框架，重新证明 Davis-McCarthy 定理，并用商 b/a 所属的函数类刻画 Lipschitz 类与 Gevrey 类乘子。

Pure contractive multipliers of some reproducing kernel Hilbert spaces and applications

A contraction $T$ on a Hilbert space $\mathcal{H}$ is said to be pure if the sequence $\lbrace T^{*n} \rbrace_{n}$ converges to $0$ in the strong operator topology. In this article, we prove that for contractions $T$, which commute with certain tractable tuples of commuting operators $X = (X_1,\ldots,X_n)$ on $\mathcal{H}$, the following statements are equivalent: (i) $T$ is a pure contraction on $\mathcal{H}$, (ii) the compression $P_{\mathcal{W}(X)}T|_{\mathcal{W}(X)}$ is a pure contraction, where $\mathcal{W}(X)$ is the wandering subspace corresponding to the tuple $X$. An operator-valued m…

Heisenberg群上的调和与非调和振子

Harmonic and Anharmonic Oscillators on the Heisenberg Group

本文利用Dynin–Folland群表示定义Heisenberg群上的调和振子，并推广到非调和振子，获得谱估计、谱乘子 $L^p$–$L^q$ 估计及热半群估计。

Weak products of complete Pick spaces

Let $\mathcal H$ be the Drury-Arveson or Dirichlet space of the unit ball of $\mathbb C^d$. The weak product $\mathcal H\odot\mathcal H$ of $\mathcal H$ is the collection of all functions $h$ that can be written as $h=\sum_{n=1}^\infty f_n g_n$, where $\sum_{n=1}^\infty \|f_n\|\|g_n\|<\infty$. We show that $\mathcal H\odot\mathcal H$ is contained in the Smirnov class of $\mathcal H$, i.e. every function in $\mathcal H\odot\mathcal H$ is a quotient of two multipliers of $\mathcal H$, where the function in the denominator can be chosen to be cyclic in $\mathcal H$. As a consequence we show that …

Multipliers of Dirichlet series and monomial series expansions of holomorphic functions in infinitely many variables

Let $\mathcal{H}_\infty$ be the set of all ordinary Dirichlet series $D=\sum_n a_n n^{-s}$ representing bounded holomorphic functions on the right half plane. A multiplicative sequence $(b_n)$ of complex numbers is said to be an $\ell_1$-multiplier for $\mathcal{H}_\infty$ whenever $\sum_n |a_n b_n| < \infty$ for every $D \in \mathcal{H}_\infty$. We study the problem of describing such sequences $(b_n)$ in terms of the asymptotic decay of the subsequence $(b_{p_j})$, where $p_j$ denotes the $j$th prime number. Given a multiplicative sequence $b=(b_n)$ we prove (among other results): $b$ is an …

非交换德布兰-罗文纳克模型用于行收缩

A non-commutative de Branges-Rovnyak model for row contractions

本文提出非交换德布兰-罗文纳克模型，用于描述CNC行收缩的特征函数及其单位不变性。

Local and multilinear noncommutative de Leeuw theorems

Let $Γ< G$ be a discrete subgroup of a locally compact unimodular group $G$. Let $m\in C_b(G)$ be a $p$-multiplier on $G$ with $1 \leq p < \infty$ and let $T_{m}: L_p(\widehat{G}) \rightarrow L_p(\widehat{G})$ be the corresponding Fourier multiplier. Similarly, let $T_{m \vert_Γ}: L_p(\widehatΓ) \rightarrow L_p(\widehatΓ)$ be the Fourier multiplier associated to the restriction $m|_Γ$ of $m$ to $Γ$. We show that \[ c( {\rm supp}( m\vert_Γ ) ) \Vert T_{m \vert_Γ}: L_p(\widehatΓ) \rightarrow L_p(\widehatΓ) \Vert \leq \Vert T_{m }: L_p(\widehat{G}) \rightarrow L_p(\widehat{G}) \Vert, \] for a spe…

完全Pick代数的同构问题：综述

The isomorphism problem for complete Pick algebras: a survey

本综述围绕完全Pick代数的同构问题展开，阐述其作为满足Pick插值定理的乘子代数的研究背景，介绍其通过Drury-Arveson空间乘子限制到单位球子簇的具体实现形式，总结基于复几何分类该代数的相关研究，描述所用工具方法并澄清修正文献内容。

Operators affiliated to the free shift on the free Hardy space

The Smirnov class for the classical Hardy space is the set of ratios of bounded analytic functions on the open complex unit disk with outer denominators. This definition extends naturally to the commutative and non-commutative multi-variable settings of the Drury-Arveson space and the full Fock space over $\mathbb C ^d$. Identifying the Fock space with the free multi-variable Hardy space of non-commutative or free holomorphic functions on the non-commutative open unit ball, we prove that any closed, densely-defined operator affiliated to the right free multiplier algebra of the full Fock space…

齐型空间上与球拟巴拿赫函数空间和非负自伴算子相关的局部Hardy空间及其应用

Local Hardy Spaces Associated with Ball Quasi-Banach Function Spaces and Non-negative Self-adjoint Operators on Spaces of Homogeneous Type and Their Applications

本文在齐型空间框架下引入与球拟巴拿赫函数空间和非负自伴算子相关的局部Hardy空间，研究其性质并建立局部Riesz变换有界性、谱乘子定理等应用，推广了经典局部Hardy空间理论。

Gelfand transforms and boundary representations of complete Nevanlinna--Pick quotients

The main objects under study are quotients of multiplier algebras of certain complete Nevanlinna--Pick spaces, examples of which include the Drury--Arveson space on the ball and the Dirichlet space on the disc. We are particularly interested in the non-commutative Choquet boundaries for these quotients. Arveson's notion of hyperrigidity is shown to be detectable through the essential normality of some natural multiplication operators, thus extending previously known results on the Arveson--Douglas conjecture. We also highlight how the non-commutative Choquet boundaries of these quotients are i…

解析圆盘的 Hilbert 函数空间与乘子代数

Hilbert function spaces and multiplier algebras of analytic discs

本文研究解析圆盘同构问题与Complete Pick空间嵌入维数问题，证明了边界自相交圆盘乘子代数同构的映射复合条件，并揭示了旋转不变空间嵌入维数与其核的关系。