Scott McCullough, Tavan T. Trent
For a collection of reproducing kernels k which includes those for the Hardy space of the polydisk and ball and for the Bergman space, k is a complete Pick kernel if and only if the multiplier algebra of the Hilbert space H^2(k) associated to k has the Douglas property. Consequences for solving the operator equation AX=Y are examined.
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On the metric of the jet bundle and similarity of Cowen-Douglas operators
本文利用度量矩阵之比和射流丛条件，给出了Cowen-Douglas算子与加权Dirichlet空间上后向移位算子相似的充分条件，并借助Müller及Kidane-Trent的结果证明了该相似性。
We consider bijective maps $ϕ$ on the full operator algebra $\mathcal{B}(\mathcal{H})$ of an infinite dimensional Hilbert space with the property that, for every $A,B,X\in \mathcal{B}(\mathcal{H})$, $X$ is the Douglas solution of the equation $A=BX$ if and only if $Y=ϕ(X)$ is the Douglas solution of the equation $ϕ(A)=ϕ(B)Y$. We prove that those maps are implemented by a unitary or anti-unitary map $U$, i.e., $ϕ(A)=UAU^*$.
In resonance to a recent geometric framework proposed by Douglas and Yang, a functional model for certain linear bounded operators with rank-one self-commutator acting on a Hilbert space is developed. By taking advantage of the refined existing theory of the principal function of a hyponormal operator we transfer the whole action outside the spectrum, on the resolvent of the underlying operator, localized at a distinguished vector. The whole construction turns out to rely on an elementary algebra body involving analytic multipliers and Cauchy transforms. A natural field theory interpretation o…
The convex-transitivity property can be seen as a convex generalization of the almost transitive (or quasi-isotropic) group action of the isometry group of a Banach space on its unit sphere. We will show that certain Banach algebras, including conformal invariant Douglas algebras, are weak-star convex-transitive. Geometrically speaking, this means that the investigated spaces are highly symmetric. Moreover, it turns out that the symmetry property is satisfied by using only `inner' isometries, i.e. a subgroup consisting of isometries which are homomorphisms on the algebra. In fact, weighted com…
The well known Douglas Lemma says that for operators $A,B$ on Hilbert space that $AA^*-BB^*\succeq 0$ implies $B=AC$ for some contraction operator $C$. The result carries over directly to classical operator-valued Toeplitz operators by simply replacing operator by Toeplitz operator. Free functions generalize the notion of free polynomials and formal power series and trace back to the work of J. Taylor in the 1970s. They are of current interest, in part because of their connections with free probability and engineering systems theory. For free functions $a$ and $b$ on a free domain $\cK$ define…
We prove a local Douglas formula for higher order weighted Dirichlet-type integrals. With the help of this formula, we study the multiplier algebra of the associated higher order weighted Dirichlet-type spaces $\mathcal H_{\pmbμ},$ induced by an $m$-tuple $\pmb μ=(μ_1,\ldots,μ_{m})$ of finite non-negative Borel measures on the unit circle. In particular, it is shown that any weighted Dirichlet-type space of order $m,$ for $m\geqslant 3,$ forms an algebra under pointwise product. We also prove that every non-zero closed $M_z$-invariant subspace of $\mathcal H_{\pmbμ},$ has codimension $1$ prope…
The Lifting Property for Frame Multipliers and Toeplitz Operators
本文证明正符号框架乘子是coorbit空间间的Banach同构，推广提升定理至Gabor框架和Fock空间，得到调制空间与加权Fock空间的新同构结果。
We prove that if $G$ is a discrete group and $(A,G,α)$ is a C*-dynamical system such that the reduced crossed product $A\rtimes_{r,α} G$ possesses property (SOAP) then every completely compact Herz-Schur $(A,G,α)$-multiplier can be approximated in the completely bounded norm by Herz-Schur $(A,G,α)$-multipliers of finite rank. As a consequence, if $G$ has the approximation property (AP) then the completely compact Herz-Schur multipliers of $A(G)$ coincide with the closure of $A(G)$ in the completely bounded multiplier norm. We study the class of invariant completely compact Herz-Schur multiplie…
A committee space is a Hilbert space of power series, perhaps in several or noncommuting variables, such that $\|z^α\|\|z^β\| \geq \|z^{α+β}\|.$ Such a space satisfies the true column-row property when ever the map transposing a column multiplier to a row multiplier is contractive. We describe a model for random multipliers and show that such random multipliers satisfy the true column-row property. We also show that the column-row property holds asymptotically for large random Nevanlinna-Pick interpolation problems where the nodes are chosen identically and independently. These results suggest…
Let $G$ be an infinite locally compact abelian group. If $X$ is Banach space, we show that if every bounded Fourier multiplier $T$ on $L^2(G)$ has the property that $T\ot Id_X$ is bounded on $L^2(G,X)$ then the Banach space $X$ is isomorphic to a Hilbert space. Moreover, if $1<p<\infty$, $p\not=2$, we prove that there exists a bounded Fourier multiplier on $L^p(G)$ which is not completely bounded. Finally, we examine unconditionality from the point of view of Schur multipliers. More precisely, we give several necessary and sufficient conditions to determine if an operator space is completely i…
In order to investigate the relationship between weak amenability and the Haagerup property for groups, we introduce the weak Haagerup property, and we prove that having this approximation property is equivalent to the existence of a semigroup of Herz-Schur multipliers generated by a proper function. It is then shown that a (not necessarily proper) generator of a semigroup of Herz-Schur multipliers splits into a positive definite kernel and a conditionally negative definite kernel. We also show that the generator has a particularly pleasant form if and only if the group is amenable. In the sec…
In the theory of complete Pick spaces, the column-row property has appeared in a variety of contexts. We show that it is satisfied by every complete Pick space in the following strong form: each sequence of multipliers that induces a contractive column multiplication operator also induces a contractive row multiplication operator. In combination with known results, this yields a number of consequences. Firstly, we obtain multiple applications to the theory of weak product spaces, including factorization, multipliers and invariant subspaces. Secondly, there is a short proof of the characterizat…
If an invertible linear dynamical systems is Li-York chaotic or other chaotic, what's about it's inverse dynamics? what's about it's adjoint dynamics? With this unresolved but basic problems, this paper will give a criterion for Lebesgue operator on separable Hilbert space. Also we give a criterion for the adjoint multiplier of Cowen-Douglas functions on $2$-th Hardy space. Last we give some chaos about scalars perturbation of operator and some examples of invertible bounded linear operator such that $T$ is chaotic but $T^{-1}$ is not.
We prove an extension property for $M_d$-multipliers from a subgroup to the ambient group, showing that $M_{d+1}(G)$ is strictly contained in $M_d(G)$ whenever $G$ contains a free subgroup. Another consequence of this result is the stability of the $M_d$-approximation property under group extensions. We also show that Baumslag-Solitar groups are $M_d$-weakly amenable with $\boldsymbolΛ(\operatorname{BS}(m,n),d)=1$ for all $d\geq 2$. Finally, we show that, for simple Lie groups with finite centre, $M_d$-weak amenability is equivalent to weak amenability, and we provide some estimates on the con…
The $p$-Operator Approximation Property
本文在算子空间非交换框架下引入p-算子逼近性质（p-OAP），建立其与算子p-紧映射理想及切片映射性质的等价刻画，研究其对偶传递性，并应用于离散群约化C*-代数中Herz-Schur乘子的cb-范数逼近。
Multiplier varieties and multiplier algebras of CNP Dirichlet series kernels
研究CNP狄利克雷级数核的乘子代数等距和代数同构问题，通过多项式方程确定乘子品种，从而分类其同构性，并解决McCarthy和Shalit提出的问题。
In 2010, Lafforgue and de la Salle gave examples of noncommutative Lp-spaces without the operator space approximation property (OAP) and, hence, without the completely bounded approximation property (CBAP). To this purpose, they introduced the property of completely bounded approximation by Schur multipliers on Sp and proved that for p < 4/3 and p > 4 the groups SL(n,Z), with n \geq 3, do not have it. Since for 1 < p < \infty the property of completely bounded approximation by Schur multipliers on Sp is weaker than the approximation property of Haagerup and Kraus (AP), these groups were also t…
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…
Representations of the Möbius group and pairs of homogeneous operators in the Cowen-Douglas class
本文研究了Möbius群在再生核Hilbert空间上的乘子表示，证明了其酉等价于全纯离散序列表示的张量积，并利用该结果刻画了Cowen-Douglas类中秩为1的Möb-齐次算子对的上三角结构，同时给出了对称与反对称情形下的表示分解。
We continue the investigation of the isomorphism problem for multiplier algebras of reproducing kernel Hilbert spaces with the complete Nevanlinna-Pick property. In contrast to previous work in this area, we do not study these spaces by identifying them with restrictions of a universal space, namely the Drury-Arveson space. Instead, we work directly with the Hilbert spaces and their reproducing kernels. In particular, we show that two multiplier algebras of Nevanlinna-Pick spaces on the same set are equal if and only if the Hilbert spaces are equal. Most of the article is devoted to the study …
$K_{0}$-groups and strongly irreducible decompositions of operator tuples
本文利用算子代数的$K$-理论，研究算子组强不可约分解的唯一性，给出Cowen-Douglas组的代数相似不变量，并计算乘子代数的$K_{0}$-群及描述Drury-Arveson空间上向后多移位的相似性。
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…
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…
By adapting an ultraproduct technique of Junge and Zeng, we prove that radial completely bounded multipliers on $q$-Gaussian algebras transfer to $q$-Araki-Woods algebras. As a consequence, we establish the $w^{\ast}$-complete metric approximation property for all $q$-Araki-Woods algebras. We apply the latter result to show that the canonical ultraweakly dense C$^\ast$-subalgebras of $q$-Araki-Woods algebras are always QWEP.
We continue our investigations on pointwise multipliers for Besov spaces of dominating mixed smoothness. This time we study the algebra property of the classes $S^r_{p,q}B(\mathbb{R}^d)$ with respect to pointwise multiplication. In addition if $p\leq q$, we are able to describe the space of all pointwise multipliers for $S^r_{p,q}B(\mathbb{R}^d)$.
Spaces of Dirichlet series with the complete Pick property
本文刻画了核为$k(s,u) = \sum a_n n^{-s-\bar u}$形式的狄利克雷级数再生核希尔伯特空间何时为完全Pick空间，引入弱同构概念并证明多类此类空间与Drury-Arveson空间$H^2_d$弱同构，揭示其乘子代数性质。
In this paper we prove a randomized difference norm characterization for Bessel potential spaces with values in UMD Banach spaces. The main ingredients are $\mathcal{R}$-boundedness results for Fourier multiplier operators, which are of independent interest. As an application we characterize the pointwise multiplier property of the indicator function of the half-space on these spaces. All results are proved in the setting of weighted spaces.
We present a class of weight functions $ w$ on the circle $ \mathbb{T}$, called Lévy-Khinchin-Schoenberg (LKS) weights, for which we are able to completely characterize (in terms of a capacitary inequality) all Fourier multipliers for the weighted space $ L^{2}(\mathbb{T},w)$. We show that the multiplier algebra is nontrivial if and only if $ 1/w\in L^{1}(\mathbb{T})$, and in this case multipliers satisfy the Spectral Localization Property (no "hidden spectrum"). On the other hand, the Muckenhoupt $ (A_{2})$ condition responsible for the basis property of exponentials $ (e^{ikx})$ is more or l…
We study various operator homological properties of the Fourier algebra $A(G)$ of a locally compact group $G$. Establishing the converse of two results of Ruan and Xu, we show that $A(G)$ is relatively operator 1-projective if and only if $G$ is IN, and that $A(G)$ is relatively operator 1-flat if and only if $G$ is inner amenable. We also exhibit the first known class of groups for which $A(G)$ is not relatively operator $C$-flat for any $C\geq1$. As applications of our techniques, we establish a hereditary property of inner amenability, answer an open question of Lau and Paterson, and answer…
We show that every function in a reproducing kernel Hilbert space with a normalized complete Pick kernel is the quotient of a multiplier and a cyclic multiplier. This extends a theorem of Alpay, Bolotnikov and Kaptanoğlu. We explore various consequences of this result regarding zero sets, spaces on compact sets and Gleason parts. In particular, using a construction of Salas, we exhibit a rotationally invariant complete Pick space of analytic functions on the unit disc for which the corona theorem fails.
Discrete Hilbert Transform And Discrete Mikhlin Multiplier On Discrete Variable Lebesgue Space
本文在变量离散Lebesgue空间上证明了离散Hilbert变换的有界性，并给出了离散Mikhlin乘子在该空间上有界的充分条件。
Weighted function spaces: convolutors, multipliers, and mollifiers
本文研究Gelfand-Shilov型加权光滑函数空间，引入紧支集光滑函数对的矩向分解性质，通过磨光化方法完整刻画了该类空间及对应的卷积子、乘子空间，明确了相关算子与权函数系统的作用关系。
We consider the classical Besov and Triebel-Lizorkin spaces defined via differences and prove a homogeneity property for functions with bounded support in the frame of these spaces. As the proof is based on compact embeddings between the studied function spaces we present also some results on the entropy numbers of these embeddings. Moreover, we derive some applications in terms of pointwise multipliers.
Suppose $H$ is a finite dimensional reproducing kernel Hilbert space of functions on $X.$ If $H$ has the complete Pick property then there is an isometric map, $Φ,$ from $X,$ with the metric induced by $H,$ into complex hyperbolic space, $\mathbb{CH}^{n},$ with its pseudohyperbolic metric. We investigate the relationships between the geometry of $Φ(X)$ and the function theory of $H$ and its multiplier algebra.
Nonexistence of Henkin type projections via a Wiener theorem for multipliers
本文通过建立新的傅里叶乘法性质，证明在特定函数空间中，适当的矩阵算子A(D)下的自由子空间非补集，扩展了Henkin(1967)的结果。
In this paper, we characterize the (left) multiplier algebra of a semidirect product algebra ${\mathcal A}={\mathcal B}\oplus {\mathcal I}$, where ${\mathcal I}$ and ${\mathcal B}$ are closed two-sided ideal and closed subalgebra of ${\mathcal A}$, respectively. As an application of this result we investigate the BSE-property of this class of Banach algebras. We then for two commutative semisimple Banach algebras ${\mathcal A}$ and ${\mathcal B}$ characterize the BSE-functions on the carrier space of ${\mathcal A}\times_ϕ{\mathcal B}$, the $ϕ$-Lau product of ${\mathcal A}$ and ${\mathcal B}$, …
We characterize interpolating sequences for multiplier algebras of spaces with the complete Pick property. Specifically, we show that a sequence is interpolating if and only if it is separated and generates a Carleson measure. This generalizes results of Carleson for the Hardy space and of Bishop, Marshall and Sundberg for the Dirichlet space. Furthermore, we investigate interpolating sequences for pairs of Hilbert function spaces.
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。
In this paper we introduce a new property for normed algebras. This property which we call it stability, plays a key role in the studying of the theory of almost multiplier maps. In this note we study some of the basic properties of this notion and characterize stable algebras among similar algebras.
We consider abstract non-negative self-adjoint operators on $L^2(X)$ which satisfy the finite speed propagation property for the corresponding wave equation. For such operators we introduce a restriction type condition which in the case of the standard Laplace operator is equivalent to $(p,2)$ restriction estimate of Stein and Tomas. Next we show that in the considered abstract setting our restriction type condition implies sharp spectral multipliers and endpoint estimates for the Bochner-Riesz summability. We also observe that this restriction estimate holds for operators satisfying dispersiv…
Continuous frames and tensor products are important topics in theoretical physics. This paper combines those concepts. We derive fundamental properties of continuous frames for tensor product of Hilbert spaces. This includes, for example, the consistency property, i.e. preservation of the frame property under the tensor product, and the description of the canonical dual tensors by those on the Hilbert space level. We show the full characterization of all dual systems for a given continuous frame, a result interesting by itself, and apply this to dual tensor frames. Furthermore, we discuss the …
Singular multipliers on multiscale Zygmund sets
本文刻画了使 Hörmander-Mihlin 或 Marcinkiewicz 乘子满足弱型端点模不等式的闭零集，引入多尺度 Zygmund 性质，并给出稀疏加权估计，解决了 Lerner 猜想。
Characterization of multipliers on vector-valued Hardy spaces
本文研究了向量值 Hardy 空间上乘法子的特征，揭示了无限多边形盘与无限多边形 тор 上的乘法子空间差异，并将结果扩展到狄利克雷级数的 Hardy 空间。
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