We show that for any $1<p<\infty$, the space $Hank_p(\mathbb{R}_+)\subseteq B(L^p(\mathbb{R}_+))$ of all Hankel operators on $L^p(\mathbb{R}_+)$ is equal to the $w^*$-closure of the linear span of the operators $θ_u\colon L^p(\mathbb{R}_+)\to L^p(\mathbb{R}_+)$ defined by $θ_uf=f(u-\,\cdotp)$, for $u>0$. We deduce that $Hank_p(\mathbb{R}_+)$ is the dual space of$A_p(\mathbb{R}_+)$, a half-line analogue of the Figa-Talamenca-Herz algebra $A_p(\mathbb{R})$. Then we show that a function $m\colon \mathbb{R}_+^*\to \mathbb{C}$ is the symbol of a $p$-completely bounded multiplier $Hank_p(\mathbb{R}_+)\to Hank_p(\mathbb{R}_+)$ if and only if there exist $α\in L^\infty(\mathbb{R}_+;L^p(Ω))$ and $β\in L^\infty(\mathbb{R}_+;L^{p'}(Ω))$ such that $m(s+t)=\langleα(s),β(t)\rangle$ for a.e. $(s,t)\in\mathbb{R}_+^{*2}$. We also give analogues of these results in the (easier) discrete case.
We establish a connection between compactness of Hankel operators and geometry of the underlying domain through compactness multipliers for the $\overline{\partial}$-Neumann operator. In particular, we prove that any compactness multiplier induces a compact Hankel operator. We also generalize the notion of compactness multipliers to vector fields and matrices and then we use this generalization to generate compact Hankel operators.
Let H(f)(x)=\int_{(0,infty)^d} f(v) E_{x}(v) dν(v), be the multivariable Hankel transform, where E_{x}(v)=\prod_{k=1}^d (x_k v_k)^{-a_k+1/2} J_{a_k-1/2}(x_k v_k), dν(v)=v^a dv, a=(a_1,...,a_d). We give sufficient conditions on a bounded continuous function m(v) which guarantee that the operator H(m Hf) is bounded on L^p(dν) and of weak-type (1,1), or bounded on the Hardy space H^1((0,infty)^d, dν) in the sense of Coifman-Weiss.
We show that some matrices are Schur multipliers and this is applied to obtain classes of operator-valued Foguel-Hankel operators similar to contractions. This provides partial answers to a problem of K. Davidson and the second author concerning CAR-valued Foguel-Hankel operators.
We study the problem of characterizing Hankel-Schur multipliers and Toeplitz-Schur multipliers of Schatten-von Neumann class $\bS_p$ for $0<p<1$. We obtain various sharp necessary conditions and sufficient conditions for a Hankel matrix to be a Schur multiplier of $\bS_p$. We also give a characterization of the Hankel-Schur multipliers of $\bS_p$ whose symbols have lacunary power series. Then the results on Hankel-Schur multipliers are used to obtain a characterization of the Toeplitz-Schur multipliers of $\bS_p$. Finally, we return to Hankel-Schur multipliers and obtain new results in the cas…
In this article, we completely characterize absolutely norm attaining Hankel operators and absolutely minimum attaining Toeplitz operators. We also improve \cite[Theorem 2.1]{RGSSSTOE1}, by characterizing the absolutely norm attaining Toeplitz operator $T_φ$ in terms of the symbol $φ\in L^\infty$.
We give a short proof of the main result of our previous paper [2]: every Schmidt subspace of a Hankel operator is the image of a model space by an isometric multiplier. This class of subspaces is closely related to nearly $S^*$-invariant subspaces, and our proof uses Hitt's theorem on the structure of such subspaces. We also give a formula for the action of a Hankel operator on its Schmidt subspace.
For a bounded Hankel matrix $Γ$, we describe the structure of the Schmidt subspaces of $Γ$, namely the eigenspaces of $Γ^* Γ$ corresponding to non zero eigenvalues. We prove that these subspaces are in correspondence with weighted model spaces in the Hardy space on the unit circle. Here we use the term "weighted model space" to describe the range of an isometric multiplier acting on a model space. Further, we obtain similar results for Hankel operators acting in the Hardy space on the real line. Finally, we give a streamlined proof of the Adamyan-Arov-Krein theorem using the language of weight…
We completely characterize the simultaneous membership in the Schatten ideals $S_ p$, $0<p<\infty$ of the Hankel operators $H_ f$ and $H_{\bar{f}}$ on the Bergman space, in terms of the behaviour of a local mean oscillation function, proving a conjecture of Kehe Zhu from 1991.
A linear map $u\colon \ E\to F$ between operator spaces is called completely co-bounded if it is completely bounded as a map from $E$ to the opposite of $F$. We give several simple results about completely co-bounded Schur multipliers on $B(\ell_2)$ and the Schatten class $S_p$. We also consider Herz-Schur multipliers on groups.
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…
We estimate the norms of many matrix coefficients of irreducible uniformly bounded representations of SL(2, R) as completely bounded multipliers of the Fourier algebra. Our results suggest that the known inequality relating the uniformly bounded norm of a representation and the completely bounded norm of its coefficients may not be optimal.
We show that any completely positive multiplier of the convolution algebra of the dual of an operator algebraic quantum group $\G$ (either a locally compact quantum group, or a quantum group coming from a modular or manageable multiplicative unitary) is induced in a canonical fashion by a unitary corepresentation of $\G$. It follows that there is an order bijection between the completely positive multipliers of $L^1(\G)$ and the positive functionals on the universal quantum group $C_0^u(\G)$. We provide a direct link between the Junge, Neufang, Ruan representation result and the representing e…
We continue the study of multidimensional operator multipliers initiated in [arXiv:math/0701645]. We introduce the notion of the symbol of an operator multiplier. We characterise completely compact operator multipliers in terms of their symbol as well as in terms of approximation by finite rank multipliers. We give sufficient conditions for the sets of compact and completely compact multipliers to coincide and characterise the cases where an operator multiplier in the minimal tensor product of two C*-algebras is automatically compact. We give a description of multilinear modular completely com…
In the article, we find new dilatation results on non-commutative $L_p$ spaces. We prove that any selfadjoint, unital, positive measurable Schur multiplier on some $B(L^2(Σ))$ admits, for all $1\leq p<\infty$, an invertible isometric dilation on some non-commutative $L^p$-space. We obtain a similar result for selfadjoint, unital, completely positive Fourier multiplier on $VN(G)$, when $G$ is a unimodular locally compact group. Furthermore, we establish multivariable versions of these results.
Let $G$ be a discrete group. Let $λ: G \to B(\ell_2(G),\ell_2(G))$ be the left regular representation. A function $\ph : G \to \comp$ is called a completely bounded multiplier (= Herz-Schur multiplier) if the transformation defined on the linear span $K(G)$ of $\{λ(x),x \in G\}$ by $$\sum_{x \in G} f(x) λ(x) \to \sum_{x \in G} f(x) \ph(x) λ(x)$$ is completely bounded (in short c.b.) on the $C^*$-algebra $C_λ^*(G)$ which is generated by $λ$ ($C_λ^*(G)$ is the closure of $K(G)$ in $B(\ell_2(G),\ell_2(G))$.) One of our main results gives a simple characterization of the functions $\ph$ such that …
Let G be a locally compact group L^p(G) be the usual L^p-space for 1 =< p =< infty and A(G) be the Fourier algebra of G. Our goal is to study, in a new abstract context, the completely bounded multipliers of A(G), which we denote M_{cb}A(G). We show that M_{cb}A(G) can be characterised as the ``invariant part'' of the space of (completely) bounded normal L^{infty}(G)-bimodule maps on B(L^2(G)), the space of bounded operator on L^2(G). In doing this we develop a function theoretic description of the normal L^{infty}(X,mu)-bimodule maps on B(L^2(X,mu)), which we denote by V^{infty}(X,μ), and nam…
We completely describe spaces of multipliers of certain harmonic function spaces of Bergman type in R^n.This is the first sharp result of this kind for Bergman type mixed norm spaces of harmonic functions in the unit ball of R^n
Let T and C be two Hilbert space operators. We prove that if T is near, in a certain sense, to an operator completely polynomially dominated with a finite bound by C, then T is similar to an operator which is completely polynomially dominated by the direct sum of C and a suitable weighted unilateral shift. Among the applications, a refined Banach space version of Rota similarity theorem is given and partial answers to a problem of K. Davidson and V. Paulsen are obtained. The latter problem concerns CAR-valued Foguel-Hankel operators which are generalizations of the operator considered by G. Pi…
Motivated by Barr{\'ı}a-Halmos's \cite[Question 19]{barria1982asymptotic} and Halmos's \cite[Problem 237]{Halmos1978A}, we explore projections in Toeplitz algebra on the Hardy space. We show that the product of two Toeplitz (Hankel) operators is a projection if and only if it is the projection onto one of the invariant subspaces of the shift (backward shift) operator. As a consequence one obtains new proofs of criterion for Toeplitz operators and Hankel operators to be partial isometries. Furthermore, we completely characterize when the self-commutator of a Toeplitz operator is a projection. T…
The main goal of this note is to show that (not necessarily holomorphic) multipliers of a wide class of normed spaces of continuous functions over a connected Hausdorff topological space cannot attain their multiplier norms, unless they are constants. As an application, a contractive multiplication operator is either a multiplication with a constant, or is completely non-unitary. Additionally we explore possibilities for a multiplication operator to be (weakly) compact and (uniformly) mean ergodic.
A result of Gilbert shows that every completely bounded multiplier $f$ of the Fourier algebra $A(G)$ arises from a pair of bounded continuous maps $α,β:G \rightarrow K$, where $K$ is a Hilbert space, and $f(s^{-1}t) = (β(t)|α(s))$ for all $s,t\in G$. We recast this in terms of adjointable operators acting between certain Hilbert C$^*$-modules, and show that an analogous construction works for completely bounded left multipliers of a locally compact quantum group. We find various ways to deal with right multipliers: one of these involves looking at the opposite quantum group, and this leads to …
In this paper, we completely characterize when two dual truncated Toeplitz operators are essentially commuting and when the semicommutator of two dual truncated Toeplitz operators is compact. Our main idea is to study dual truncated Toeplitz operators via Hankel operators, Toeplitz operators and function algebras.
We study Fourier multipliers on free group $\mathbb{F}_\infty$ associated with the first segment of the reduced words, and prove that they are completely bounded on the noncommutative $L^p$ spaces $L^p(\hat{\mathbb{F}}_\infty)$ iff their restriction on $L^p(\hat{\mathbb{F}}_1)=L^p(\mathbb{T})$ are completely bounded. As a consequence, every classical Mikhlin multiplier extends to a $L^p$ Fourier multiplier on free groups for all $1<p<\infty$.
We investigate a new representation of general operators by means of sums of shifted Gabor multipliers. These representations arise by studying the matrix of an operator with respect to a Gabor frame. Each shifted Gabor multiplier corresponds to a side-diagonal of this matrix. This representation is especially useful for operators whose associated matrix possesses some off-diagonal decay. In this case one can completely characterize the symbol class of the operator by the size of the symbols of the Gabor multipliers. As an application we derive approximation theorems for pseudodifferential ope…
We study the operators T on the weighted space L^p commuting either with the right translations St or left translations P^+S_{-t} and we establish the existence of a symbol of T. We characterize completely the spectrum of St. We obtain a similar result for the spectrum of P^+S_{-t} and some spectral results for the bounded operators commuting with (St), t>0 or with (P^+St), t<0.
We study the dual relationship between quantum group convolution maps $L^1(\mathbb{G})\rightarrow L^{\infty}(\mathbb{G})$ and completely bounded multipliers of $\widehat{\mathbb{G}}$. For a large class of locally compact quantum groups $\mathbb{G}$ we completely isomorphically identify the mapping ideal of row Hilbert space factorizable convolution maps with $M_{cb}(L^1(\widehat{\mathbb{G}}))$, yielding a quantum Gilbert representation for completely bounded multipliers. We also identify the mapping ideals of completely integral and completely nuclear convolution maps, the latter case coincidi…
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…
Functional Gabor single-frame or multi-frame generator multipliers are the matrices of function entries that preserve Parseval Gabor single-frame or multi-frame generators. An interesting and natural question is how to characterize all such multipliers. This question has been answered for several special cases including the case of single-frame generators in two dimensions and the case of multi-frame generators in one-dimension. In this paper we completely characterize multipliers for Gabor single-frame and multi-frame generators with respect to separable time-frequency lattices in any given d…
Consider a completely bounded Fourier multiplier phi of a locally compact group G, and take 1 <= p <= infinity. One can associate to phi a Schur multiplier on the Schatten classes S_p(L^2 G), as well as a Fourier multiplier on Lp(LG), the non-commutative Lp-space of the group von Neumann algebra of G. We prove that the completely bounded norm of the Schur multiplier is not greater than the completely bounded norm of the Lp-Fourier multiplier. When G is amenable we show that equality holds, extending a result by Neuwirth and Ricard to non-discrete groups. For a discrete group G and in the speci…
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)
For a locally compact group $G$, let $A(G)$ be its Fourier algebra, let $M_{cb}A(G)$ denote the completely bounded multipliers of $A(G)$, and let $A_{Mcb}(G)$ stand for the closure of $A(G)$ in $M_{cb}A(G)$. We characterize the norm one idempotents in $M_{cb}A(G)$: the indicator function of a set $E \subset G$ is a norm one idempotent in $M_{cb}A(G)$ if and only if $E$ is a coset of an open subgroup of $G$. As applications, we describe the closed ideals of $A_{Mcb}(G)$ with an approximate identity bounded by 1, and we characterize those $G$ for which $A_{Mcb}(G)$ is 1-amenable in the sense of …
We show that the multiplier algebra of the Fourier algebra on a locally compact group $G$ can be isometrically represented on a direct sum on non-commutative $L^p$ spaces associated to the right von Neumann algebra of $G$. If these spaces are given their canonical Operator space structure, then we get a completely isometric representation of the completely bounded multiplier algebra. We make a careful study of the non-commutative $L^p$ spaces we construct, and show that they are completely isometric to those considered recently by Forrest, Lee and Samei; we improve a result about module homomo…
It was shown by Chen, Xu and Yin that completely bounded Fourier multipliers on noncommutative $L_p$-spaces of quantum tori $\mathbb T^d_θ$ do not depend on the parameter $θ$. We establish that the situation is somehow different for bounded multipliers. The arguments are based on transference from the commutative torus.
In the present paper the unconditional convergence and the invertibility of multipliers is investigated. Multipliers are operators created by (frame-like) analysis, multiplication by a fixed symbol, and resynthesis. Sufficient and/or necessary conditions for unconditional convergence and invertibility are determined depending on the properties of the analysis and synthesis sequences, as well as the symbol. Examples which show that the given assertions cover different classes of multipliers are given. If a multiplier is invertible, a formula for the inverse operator is determined. The case when…
We study the class $\mathcal{M}_p$ of Schur multipliers on the Schatten-von Neumann class $\mathcal{S}_p$ with $1 \leq p \leq \infty$ as well as the class of completely bounded Schur multipliers $\mathcal{M}_p^{cb}$. We first show that for $2 \leq p < q \leq \infty$ there exist $m \in \mathcal{M}_p^{cb}$ with $m \not \in \mathcal{M}_q$, so in particular the following inclusions that follow from interpolation are strict $\mathcal{M}_q \subsetneq \mathcal{M}_p$ and $\mathcal{M}^{cb}_q \subsetneq \mathcal{M}^{cb}_p$. In the remainder of the paper we collect computational evidence that for $p\not …
Schur Multipliers with Unequal Operator and Completely Bounded Norms on $S_p$, $1<p\ne 2<\infty $
Institute for Advanced Study in Mathematics of HIT, Harbin Institute of Technology（哈尔滨工业大学数学高等研究院）；School of Mathematics and Statistics, University of New South Wales（新南威尔士大学数学与统计学院）；Department of Mathematics, New Uzbekistan University（新乌兹别克斯坦大学数学系）；V. I. Romanovsky Institute of Mathematics of the Academy of Sciences of Uzbekistan（乌兹别克斯坦科学院V.I.罗曼诺夫斯基数学研究所）
Grothendieck's inequalities for operators and bilinear forms imply some factorization results for complex m x n matrices. Based on the theory of operator spaces and completely bounded mappings we present norm optimal versions of these results and two norm optimal factorization results related to the Schur product. We show that the spaces of respectively bilinear forms and Schur multipliers are conjugate duals to each other with respect to their completely bounded norms.
We study the space of functions $ϕ\colon \NN\to \CC$ such that there is a Hilbert space $H$, a power bounded operator $T$ in $B(H)$ and vectors $ξ,η$ in $H$ such that $$ϕ(n) = < T^nξ,η>.$$ This implies that the matrix $(ϕ(i+j))_{i,j\ge 0}$ is a Schur multiplier of $B(\ell_2)$ or equivalently is in the space $(\ell_1 \buildrel {\vee}\over {\otimes} \ell_1)^*$. We show that the converse does not hold, which answers a question raised by Peller [Pe]. Our approach makes use of a new class of Fourier multipliers of $H^1$ which we call ``shift-bounded''. We show that there is a $ϕ$ which is a ``compl…
Let $\mathscr{H}^2$ denote the space of ordinary Dirichlet series with square summable coefficients, and let $\mathscr{H}^2_0$ denote its subspace consisting of series vanishing at $+\infty$. We investigate the weak product spaces $\mathscr{H}^2\odot\mathscr{H}^2$ and $\mathscr{H}^2_0\odot\mathscr{H}^2_0$, finding that several pertinent problems are more tractable for the latter space. This surprising phenomenon is related to the fact that $\mathscr{H}^2_0\odot\mathscr{H}^2_0$ does not contain the infinite-dimensional subspace of $\mathscr{H}^2$ of series which lift to linear functions on the …
Let $G$ be a locally compact group, and let $A_\cb(G)$ denote the closure of $A(G)$, the Fourier algebra of $G$, in the space of completely bounded multipliers of $A(G)$. If $G$ is a weakly amenable, discrete group such that $\cstar(G)$ is residually finite-dimensional, we show that $A_\cb(G)$ is operator amenable. In particular, $A_\cb(F_2)$ is operator amenable even though $F_2$, the free group in two generators, is not an amenable group. Moreover, we show that, if $G$ is a discrete group such that $A_\cb(G)$ is operator amenable, a closed ideal of $A(G)$ is weakly completely complemented in…
We introduce a noncommutative analogue of the Figá-Talamanca-Herz algebra $A_p(G)$ on the natural predual of the operator space $\frak{M}_{p,cb}$ of completely bounded Schur multipliers on Schatten space $S_p$. We determine the isometric Schur multipliers and prove that the space $\frak{M}_{p}$ of bounded Schur multipliers on Schatten space $S_p$ is the closure in the weak operator topology of the span of isometric multipliers.
We obtain descriptions of central operator-valued Schur and Herz-Schur multipliers, akin to a classical characterisation due to Grothendieck, that reveals a close link between central (linear) multipliers and bilinear multipliers into the trace class. Restricting to dynamical systems where a locally compact group acts on itself by translation, we identify their convolution multipliers as the right completely bounded multipliers, in the sense of Junge-Neufang-Ruan, of a canonical quantum group associated with the underlying group. We provide characterisations of contractive idempotent operator-…
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…
We establish a platform to transfer $L_p$-completely bounded maps on tensor products of von Neumann algebras to $L_p$-completely bounded maps on the corresponding amalgamated free products. As a consequence, we obtain a Hörmander-Mikhlin multiplier theory for free products of groups. Let $\mathbb{F}_\infty$ be a free group on infinite generators $\{g_1, g_2,\cdots\}$. Given $d\ge1$ and a bounded symbol $m$ on $\mathbb{Z}^d$ satisfying the classical Hörmander-Mikhlin condition, the linear map $M_m:\mathbb{C}[\mathbb{F}_\infty]\to \mathbb{C}[\mathbb{F}_\infty]$ defined by $λ(g)\mapsto m(k_1,\cdo…
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…
