arXiv subjects
Publications and source records attributed to Christian Le Merdy.
Let $1<p<\infty$ and let $n\geq 1$. It is proved that a function $f:{\mathbb R}\to {\mathbb C}$ is $n$-times Fréchet differentiable on ${\mathcal S}^p$ at every self-adjoint operator if and only if $f$ is $n$-times differentiable, $f',f'',\ldots,f^{(n)}$ are bounded and $f^{(n)}$ is uniformly continuous.
Consider three normal operators $A,B,C$ on separable Hilbert space $\H$ as well as scalar-valued spectral measures $λ_A$ on $σ(A)$, $λ_B$ on $σ(B)$ and $λ_C$ on $σ(C)$. For any $ϕ\in L^\infty(λ_A\times λ_B\times λ_C)$ and any $X,Y\in S^2(\H)$, the space of Hilbert-Schmidt operators on $\H$, we provide a general definition of a triple operator integral $Γ^{A,B,C}(ϕ)(X,Y)$ belonging to $S^2(\H)$ in such a way that $Γ^{A,B,C}(ϕ)$ belongs to the space $B_2(S^2(\H)\times S^2(\H), S^2(\H))$ of bounded bilinear operators on $S^2(\H)$, and the resulting mapping $Γ^{A,B,C}\colon L^\infty(λ_A\times λ_B\times λ_C) \to B_2(S^2(\H)\times S^2(\H), S^2(\H))$ is a $w^*$-continuous isometry. Then we show that a function $ϕ\in L^\infty(λ_A\times λ_B\times λ_C)$ has the property that $Γ^{A,B,C}(ϕ)$ maps $S^2(\H)\times S^2(\H)$ into $S^1(\H)$, the space of trace class operators on $\H$, if and only if it has the following factorization property: there exist a Hilbert space $H$ and two functions $a\in L^{\infty}(λ_A \times λ_B ; H)$ and $b\in L^{\infty}(λ_B\times λ_C ; H)$ such that $ϕ(t_1,t_2,t_3)= \left\langle a(t_1,t_2),b(t_2,t_3) \right\rangle$ for a.e. $(t_1,t_2,t_3) \in σ(A) \times σ(B) \times σ(C).$ This is a bilinear version of Peller's Theorem characterizing double operator integral mappings $S^1(\H)\to S^1(\H)$. In passing we show that for any separable Banach spaces $E,F$, any $w^*$-measurable esssentially bounded function valued in the Banach space $Γ_2(E,F^*)$ of operators from $E$ into $F^*$ factoring through Hilbert space admits a $w^*$-measurable Hilbert space factorization.
Given Hilbert spaces $H_1,H_2,H_3$, we consider bilinear maps defined on the cartesian product $S^2(H_2,H_3)\times S^2(H_1,H_2)$ of spaces of Hilbert-Schmidt operators and valued in either the space $B(H_1,H_3)$ of bounded operators, or in the space $S^1(H_1,H_3)$ of trace class operators. We introduce modular properties of such maps with respect to the commutants of von Neumann algebras $M_i\subset B(H_i)$, $i=1,2,3$, as well as an appropriate notion of complete boundedness for such maps. We characterize completely bounded module maps $u\colon S^2(H_2,H_3)\times S^2(H_1,H_2)\to B(H_1,H_3)$ by the membership of a natural symbol of $u$ to the von Neumann algebra tensor product $M_1\overline{\otimes} M_2^{op}\overline{\otimes} M_3$. In the case when $M_2$ is injective, we characterize completely bounded module maps $u\colon S^2(H_2,H_3)\times S^2(H_1,H_2)\to S^1(H_1,H_3)$ by a weak factorization property, which extends to the bilinear setting a famous description of bimodule linear mappings going back to Haagerup, Effros-Kishimoto, Smith and Blecher-Smith. We make crucial use of a theorem of Sinclair-Smith on completely bounded bilinear maps valued in an injective von Neumann algebra, and provide a new proof of it, based on Hilbert $C^*$-modules.
Let $T\colon L^p({\mathcal M})\to L^p({\mathcal N})$ be a bounded operator between two noncommutative $L^p$-spaces, $1\leq p<\infty$. We say that $T$ is $\ell^1$-bounded (resp. $\ell^1$-contractive) if $T\otimes I_{\ell^1}$ extends to a bounded (resp. contractive) map from $L^p({\mathcal M};\ell^1)$ into $L^p({\mathcal N};\ell^1)$. We show that Yeadon's factorization theorem for $L^p$-isometries, $1\leq p\not=2 <\infty$, applies to an isometry $T\colon L^2({\mathcal M})\to L^2({\mathcal N})$ if and only if $T$ is $\ell^1$-contractive. We also show that a contractive operator $T\colon L^p({\mathcal M})\to L^p({\mathcal N})$ is automatically $\ell^1$-contractive if it satisfies one of the following two conditions: either $T$ is $2$-positive; or $T$ is separating, that is, for any disjoint $a,b\in L^p({\mathcal M})$ (i.e. $a^*b=ab^*=0)$, the images $T(a),T(b)$ are disjoint as well.
We introduce and investigate $H^\infty$-functional calculus for commuting finite families of Ritt operators on Banach space $X$. We show that if either $X$ is a Banach lattice or $X$ or $X^*$ has property $(α)$, then a commuting $d$-tuple $(T_1,\ldots, T_d)$ of Ritt operators on $X$ has an $H^\infty$ joint functional calculus if and only if each $T_k$ admits an $H^\infty$ functional calculus. Next for $p\in(1,\infty)$, we characterize commuting $d$-tuple of Ritt operators on $L^p(Ω)$ which admit an $H^\infty$ joint functional calculus, by a joint dilation property. We also obtain a similar characterisation for operators acting on a UMD Banach space with property $(α)$. Then we study commuting $d$-tuples $(T_1,\ldots, T_d)$ of Ritt operators on Hilbert space. In particular we show that if $\Vert T_k\Vert\leq 1$ for every $k=1,\ldots,d$, then $(T_1,\ldots, T_d)$ satisfies a multivariable analogue of von Neumann's inequality. Further we show analogues of most of the above results for commuting finite families of sectorial operators.
We establish the following results on higher order $\mathcal{S}^p$-differentiability, $1<p<\infty$, of the operator function arising from a continuous scalar function $f$ and self-adjoint operators defined on a fixed separable Hilbert space: (i) $f$ is $n$ times continuously Fréchet $\mathcal{S}^p$-differentiable at every bounded self-adjoint operator if and only if $f\in C^n(\mathbb{R})$; (ii) if $f',\ldots,f^{(n-1)}\in C_b(\mathbb{R})$ and $f^{(n)}\in C_0(\mathbb{R})$, then $f$ is $n$ times continuously Fréchet $\mathcal{S}^p$-differentiable at every self-adjoint operator; (iii) if $f',\ldots,f^{(n)}\in C_b(\mathbb{R})$, then $f$ is $n-1$ times continuously Fréchet $\mathcal{S}^p$-differentiable and $n$ times Gâteaux $\mathcal{S}^p$-differentiable at every self-adjoint operator. We also prove that if $f\in B_{\infty1}^n(\mathbb{R})\cap B_{\infty1}^1(\mathbb{R})$, then $f$ is $n$ times continuously Fréchet $\mathcal{S}^q$-differentiable, $1\le q<\infty$, at every self-adjoint operator. These results generalize and extend analogous results of [10] to arbitrary $n$ and unbounded operators as well as substantially extend the results of [2,4,19] on higher order $\mathcal{S}^p$-differentiability of $f$ in a certain Wiener class, Gâteaux $\mathcal{S}^2$-differentiability of $f\in C^n(\mathbb{R})$ with $f',\ldots,f^{(n)}\in C_b(\mathbb{R})$, and Gâteaux $\mathcal{S}^q$-differentiability of $f$ in the intersection of the Besov classes $B_{\infty1}^n(\mathbb{R})\cap B_{\infty1}^1(\mathbb{R})$. As an application, we extend $\mathcal{S}^p$-estimates for operator Taylor remainders to a broad set of symbols. Finally, we establish explicit formulas for Fréchet differentials and Gâteaux derivatives.
Let $\mathcal D$ be a Schauder decomposition on some Banach space $X$. We prove that if $\mathcal D$ is not $R$-Schauder, then there exists a Ritt operator $T\in B(X)$ which is a multiplier with respect to $\mathcal D$, such that the set $\{T^n\, :\, n\geq 0\}$ is not $R$-bounded. Likewise we prove that there exists a bounded sectorial operator $A$ of type $0$ on $X$ which is a multiplier with respect to $\mathcal D$, such that the set $\{e^{-tA}\, : \, t\geq 0\}$ is not $R$-bounded.
Let $A$ be a selfadjoint operator in a separable Hilbert space, $K$ a selfadjoint Hilbert-Schmidt operator, and $f\in C^n(\mathbb{R})$. We establish that $φ(t)=f(A+tK)-f(A)$ is $n$-times continuously differentiable on $\mathbb{R}$ in the Hilbert-Schmidt norm, provided either $A$ is bounded or the derivatives $f^{(i)}$, $i=1,\ldots,n$, are bounded. As an application of the second order $\Sc^2$-differentiability, we extend the Koplienko trace formula from the Besov class $B_{\infty1}^2(\R)$ to functions $f$ for which the divided difference $f^{[2]}$ admits a certain Hilbert space factorization.
Let $G$ be a locally compact abelian group, let $ν$ be a regular probability measure on $G$, let $X$ be a Banach space, let $π\colon G\to B(X)$ be a bounded strongly continuous representation. Consider the average (or subordinated) operator $S(π,ν) = \int_{G} π(t)\,dν(t)\,\colon X\to X$. We show that if $X$ is a UMD Banach lattice and $ν$ has bounded angular ratio, then $S(π,ν)$ is a Ritt operator with a bounded $H^\infty$ functional calculus. Next we show that if $ν$ is the square of a symmetric probability measure and $X$ is $K$-convex, then $S(π,ν)$ is a Ritt operator. We further show that this assertion is false on any non $K$-convex space $X$.
We show that any bounded analytic semigroup on $L^p$ (with $1<p<\infty$) whose negative generator admits a bounded $H^{\infty}$ functional calculus with respect to some angle $< π/2$ can be dilated into a bounded analytic semigroup $(R_t)_{t\geq 0}$ on a bigger $L^p$-space in such a way that $R_t$ is a positive contraction for any $t$. We also establish a discrete analogue for Ritt operators and consider the case when $L^p$-spaces are replaced by more general Banach spaces. In connection with these functional calculus issues, we study isometric dilations of bounded continuous representations of amenable groups on Banach spaces and establish various generalizations of Dixmier's unitarization theorem.
For any Ritt operator T:L^{p}(Ω) --> L^{p}(Ω), for any positive real number α, and for any x in L^{p}, we consider the square functions |x |_{T,α} = \Bigl| \Bigl(\sum_{k=1}^{\infty} k^{2α-1}\bigl |T^{k-1}(I-T)^αx \bigr|^2 \Bigr)^{1/2}_{L^{p}}. We show that if T is actually an R-Ritt operator, then these square functions are pairwise equivalent. Then we show that T and its adjoint T* acting on L^{p'} both satisfy uniform estimates |x|_{T,1} \lesssim |x|_{L^{p}} and |y|_{T*,1} \lesssim |y|_{L^{p'}} for x in L^{p} and y in L^{p'} if and only if T is R-Ritt and admits a dilation in the following sense: there exist a measure space \widetildeΩ, an isomorphism U of L^{p}(\widetildeΩ) such that the sequence of all U^{n} for n varying in Z is bounded, as well as two bounded maps J : L^{p}(Ω) --> L^{p}(\widetildeΩ) and Q : L^{p}(\widetildeΩ) --> L^p(Ω) such that T^{n}=QU^{n}J for any nonnegative integer n. We also investigate functional calculus properties of Ritt operators and analogs of the above results on noncommutative L^{p}-spaces.
We prove the existence of a complex valued $C^2$-function on the unit circle, a unitary operator U and a self-adjoint operator Z in the Hilbert-Schmidt class $S^2$, such that the perturbated operator $$ f(e^{iZ}U)-f(U) -\frac{d}{dt}\bigl(f(e^{itZ}U)\bigr)_{\vert t=0} $$ does not belong to the space $S^1$ of trace class operators. This resolves a problem of Peller concerning the validity of the Koplienko-Neidhardt trace formula for unitaries.
A formula for the norm of a bilinear Schur multiplier acting from the Cartesian product $\mathcal S^2\times \mathcal S^2$ of two copies of the Hilbert-Schmidt classes into the trace class $\mathcal S^1$ is established in terms of linear Schur multipliers acting on the space $\mathcal S^\infty$ of all compact operators. Using this formula, we resolve Peller's problem on Koplienko-Neidhardt trace formulae. Namely, we prove that there exist a twice continuously differentiable function $f$ with a bounded second derivative, a self-adjoint (unbounded) operator $A$ and a self-adjoint operator $B\in \mathcal S^2$ such that $$ f(A+B)-f(A)-\frac{d}{dt}(f(A+tB))\big\vert_{t=0}\notin \mathcal S^1. $$
Consider a unital C*-algebra A, a von Neumann algebra M, a unital sub-C*-algebra C of A and a unital *-homomorphism $π$ from C to M. Let u: A --> M be a decomposable map (i.e. a linear combination of completely positive maps) which is a C-bimodule map with respect to $π$. We show that u is a linear combination of C-bimodule completely positive maps if and only if there exists a projection e in the commutant of $π(C)$ such that u is valued in eMe and $eπ(.)e$ has a completely positive extension A --> eMe.
We compare various functional calculus properties of Ritt operators. We show the existence of a Ritt operator T : X --> X on some Banach space X with the following property: T has a bounded $\H^\infty$ functional calculus with respect to the unit disc $\D$ (that is, T is polynomially bounded) but T does not have any bounded $\H^\infty$ functional calculus with respect to a Stolz domain of $\D$ with vertex at 1. Also we show that for an R-Ritt operator, the unconditional Ritt condition of Kalton-Portal is equivalent to the existence of a bounded $\H^\infty$ functional calculus with respect to such a Stolz domain.
Let T : X --> X$ be a power bounded operator on Banach space. An operator C : X --> Y$ is called admissible for T if it satisfies an estimate $\sum_k\norm{CT^k(x)}^2\,\leq M^2\norm{x}^2$. Following Harper and Wynn, we study the validity of a certain Weiss conjecture in this discrete setting. We show that when X is reflexive and T is a Ritt operator satisfying a appropriate square function estimate, C is admissible for T if and only if it satisfies a uniform estimate $(1-| ω|^2)^{1/2}\norm{C(I-ωT)^{-1}}\,\leq K$ for $ω\in \Cdb$, $|ω|<1$. We extend this result to the more general setting of alpha-admissibility. Then we investigate a natural variant of admissibility involving R-boundedness and provide examples to which our general results apply.
A Ritt operator T : X --> X on Banach space is a power bounded operator such that the sequence of all n(T^{n} -T^{n-1}) is bounded. When X=Lp for some 1<p<\infty, we study the validity of square functions estimates Norm{(\sum_k k |T^{k}(x) - T^{k-1}(x)|^2)^{1/2}}_{Lp} \leq K Norm{x}_{Lp} for such operators. We show that T and its adjoint T^* both satisfy such estimates if and only if T admits a bounded functional calculus with respect to a Stolz domain. This is a single operator analog of the famous Cowling-Doust-McIntosh-Yagi characterization of bounded H^\infty-calculus on $Lp$-spaces by the boundedness of certain Littlewood-Paley-Stein square functions. We also prove a similar result on Hilbert space. Then we extend the above to more general Banach spaces, where square functions have to be defined in terms of certain Rademacher averages. We focus on noncommutative Lp-spaces, where square functions are quite explicit, and we give applications, examples and illustrations on those spaces, as well as on classical Lp.
Let T_t = e^{-tA} be a bounded analytic semigroup on Lp, with 1 \fracπ{2}. We show that this actually holds true for some θ<\fracπ{2}.