Let $0<p,q\leq \infty$ and denote by $\mathcal{S}_p^N$ and $\mathcal{S}_q^N$ the corresponding Schatten classes of real $N\times N$ matrices. We study the Gelfand numbers of natural identities $\mathcal{S}_p^N\hookrightarrow \mathcal{S}_q^N$ between Schatten classes and prove asymptotically sharp bounds up to constants only depending on $p$ and $q$. This extends classical results for finite-dimensional $\ell_p$ sequence spaces by E. Gluskin to the non-commutative setting and complements bounds previously obtained by B. Carl and A. Defant, A. Hinrichs and C. Michels, and J. Chávez-Domínguez and D. Kutzarova.
Let $0<p,q\leq \infty$ and denote by $\mathcal S_p^N$ and $\mathcal S_q^N$ the corresponding Schatten classes of real $N\times N$ matrices. We study approximation quantities of natural identities $\mathcal S_p^N\hookrightarrow \mathcal S_q^N$ between Schatten classes and prove asymptotically sharp bounds up to constants only depending on $p$ and $q$, showing how approximation numbers are intimately related to the Gelfand numbers and their duals, the Kolmogorov numbers. In particular, we obtain new bounds for those sequences of $s$-numbers. Our results improve and complement bounds previously o…
In this paper, we establish the Schatten class and endpoint weak Schatten class estimates for the commutator of Riesz transforms on weighted $L^2$ spaces. It provides a weighted version for the estimate of the quantised derivative introduced by Alain Connes and studied recently by Lord--McDonald--Sukochev--Zanin.
This article provides a deeper study of the Riesz transform commutators associated with the Neumann Laplacian operator $Δ_N$ on $\mathbb R^n$. Along the line of singular value estimates for Riesz transform commutators established by Janson--Wolff and Rochberg--Semmes, we establish a full range of Schatten-$p$ class characterization for these commutators.
The first our aim is to clarify the results obtained by Lidskii devoted to the decomposition on the root vector system of the non-selfadjoint operator. We use a technique of the entire function theory and introduce a so-called Schatten-von Neumann class of the convergence exponent. Considering strictly accretive operators satisfying special conditions formulated in terms of the norm, we construct a sequence of contours of the power type on the contrary to the results by Lidskii, where a sequence of contours of the exponential type was used.
设$T$是可分Hilbert空间$H$上的紧算子.证明了对于$2\le p<\infty$，$T$属于Schatten类$S_p$当且仅当$\{\|Tf_n\|\}\in \ell^p$ for \emph{every} frame $\{f_n\}$ in $H$ ;对于$0<p\le2$，$T$属于$S_p$当且仅当$\{\|Tf_n\|\}\in\ell^p$ for \emph{some} frame $\{f_n\}$ in $H$ .根据序列$\{<Tf_n,f_n>\}$和双索引序列$\{<Tf_n,f_m>\}$也获得类似的条件。
A Helson matrix is an infinite matrix $A = (a_{m,n})_{m,n\geq1}$ such that the entry $a_{m,n}$ depends only on the product $mn$. We demonstrate that the orthogonal projection from the Hilbert--Schmidt class $\mathcal{S}_2$ onto the subspace of Hilbert--Schmidt Helson matrices does not extend to a bounded operator on the Schatten class $\mathcal{S}_q$ for $1 \leq q \neq 2 < \infty$. In fact, we prove a more general result showing that a large class of natural projections onto Helson matrices are unbounded in the $\mathcal{S}_q$-norm for $1 \leq q \neq 2 < \infty$. Two additional results are als…
In this paper we study non-selfadjoint operators using the methods of the spectral theory. The main challenge is to represent a complete description of an operator belonging to the Schatten-von Neumann class having used the order of the Hermitian real component. The latter fundamental result is advantageous since many theoretical statements based upon it and one of them is a concept on the root series expansion to say nothing on a wide spectrum of applications in the theory of evolution equations. In this regard the evolution equations of fractional order with the operator function in the term…
We define Schatten classes of adjointable operators on Hilbert modules over abelian $C^*$-algebras. Many key features carry over from the Hilbert space case. In particular, the Schatten classes form two-sided ideals of compact operators and are equipped with a Banach norm and a $C^*$-valued trace with the expected properties. For trivial Hilbert bundles, we show that our Schatten-class operators can be identified bijectively with Schatten-norm-continuous maps from the base space into the Schatten classes on the Hilbert space fiber, with the fiberwise trace. As applications, we introduce the $C…
Let $G$ be a compact Lie group of dimension $n.$ In this work we characterise the membership of classical pseudo-differential operators on $G$ in the trace class ideal $S_{1}(L^2(G)),$ as well as in the setting of the Schatten ideals $S_{r}(L^2(G)),$ for all $r>0.$ In particular, we deduce Schatten characterisations of elliptic pseudo-differential operators of $(ρ,δ)$-type for the large range $0\leq δ<ρ\leq 1.$ Additional necessary and sufficient conditions are given in terms of the matrix-valued symbols of the operators, which are global functions on the phase space $G\times \widehat{G},$ wit…
We study composition operators on global classes of ultradifferentiable functions of Beurling type invariant under Fourier transform. In particular, for the classical Gelfand-Shilov classes $Σ_d,\ d > 1,$ we prove that a necessary condition for the composition operator $f\mapsto f\circ ψ$ to be well defined is the boundedness of $ψ'.$ We find the optimal index $d'$ for which $C_ψ(Σ_d({\mathbb R}))\subset Σ_{d'}({\mathbb R})$ holds for any non-constant polynomial $ψ.$
We characterize the Hilbert--Schmidt class membership of commutator with the Hilbert transform in the two weight setting. The characterization depends upon the symbol of the commutator being in a new weighted Besov space. This follows from a Schatten class $S_p$ result for dyadic paraproducts, where $1< p < \infty $. We discuss the difficulties in extending the dyadic result to the full range of Schatten classes for the Hilbert transform.
On the thin-shell conjecture for the Schatten classes
研究了Schatten类的薄壳猜想。特别地，我们建立了算子范数的猜想;我们还改进了Barthe和Cordero-Erausquin [F. Barthe和D. Cordero-Erausquin，“Invariances in variance estimates”，Proceedings of the London Mathematical Society 106（2013）：33-64]，对于更多的情况。此外，我们证明了对于任何Schatten类，猜想为真的一个必要条件是相当强的负相关性质：这当然意味着，对于我们已经有猜想的情况（例如算子范数），但实际上也对于我们可以得到比[F]中更好的估计的所有情况。Barthe和D. Cordero-Erausquin，“方差估计的不变性”]，这种负相关性质如下。对于证明，我们的出发点是在[H.柯尼希湾Meyer和A. Pajor，“Schatten类的各向同性常数是有界的”，Mathematische Annalen 312（1998）：773-783]和[O. Guedon和G. Paouris，“Concentration of mass on the Schatten classes”，Annales de l'Institut Henri Poincaré，Probabilité et Staubiques 43（2007）：87…
Let $H=H_+\oplus H_-$ be a fixed orthogonal decomposition of the complex Hilbert space $H$ in two infinite dimensional subspaces. We study the geometry of the set $P^p$ of selfadjoint projections in the Banach algebra $$ {\cal A}^p=\{A\in B(H): [A,E_+]\in B_p(H)\}, $$ where $E_+$ is the projection onto $H_+$ and $B_p(H)$ is the Schatten ideal of $p$-summable operators ($1\le p <\infty$). The norm in ${\cal A}^p$ is defined in terms of the norms of the matrix entries of the operators given by the above decomposition. The space $P^p$ is shown to be a differentiable $C^\infty$ submanifold of ${\c…
Schatten classes of integration operators on Dirichlet spaces
我们解决的问题，描述成员的Schatten-Von Neumann理想$\mathcal{S}_ p$的积分算子$(T_ g f)(z)=\int_{0}^{z}f(ζ)\,g'(ζ)\,dζ$作用于Dirichlet型空间。我们还研究了这个问题的乘法，汉克尔和Toeplitz运营商。特别地，我们推广了Luecking关于Toeplitz算子的结果.
The optimal 2-uniform convexity of Schatten classes $S_p, 1<p\le 2$ was first proved by Ball, Carlen and Lieb \cite{BCL94}. In this note we revisit this result using multiple operator integrals and generalized monotone metrics in quantum information theory.
We study commutators with the Riesz transforms on the Heisenberg group. The Schatten norm of these commutators is characterized in terms of Besov norms of the symbol. This generalizes the classical Euclidean results of Peller, Janson--Wolff and Rochberg--Semmes. The method of proof extends the earlier methods, allowing us to address not just the Riesz transforms, but also the Cauchy--Szego projection and second order Riesz transforms on the Heisenberg group, among other settings.
We characterize the membership of Hankel operators with general symbols in the Schatten Classes $S^p,\, p\in(0,1),$ of the large Bergman spaces $A^2_ω$. The case $p\geq 1$ was proved by Lin and Rochberg.
In this work we establish sharp kernel conditions ensuring that the corresponding integral operators belong to Schatten-von Neumann classes. The conditions are given in terms of the spectral properties of operators acting on the kernel. As applications we establish several criteria in terms of different types of differential operators and their spectral asymptotics in different settings: compact manifolds, operators on lattices, domains in ${\mathbb R}^n$ of finite measure, and conditions for operators on ${\mathbb R}^n$ given in terms of anharmonic oscillators. We also give examples in the se…
Let $\{λ_n\}_n \in \ell^\infty(\mathbb{N})$. In 1960, R. Schatten \cite{SCHATTEN} studied operators of the form $\sum_{n=1}^{\infty}λ_n (x_n\otimes \bar{y_n})$, where $\{x_n\}_n$, $\{y_n\}_n$ are orthonormal sequences in a Hilbert space. In 2007, P. Balazs \cite{BALAZS3} generalized this by replacing $\{x_n\}_n$ and $\{y_n\}_n$ by Bessel sequences. In this paper, we generalize this by studying the operators of the form $\sum_{n=1}^{\infty}λ_n (A^*_nx_n\otimes \bar{B^*_ny_n})$, where $\{A_n\}_n$ and $\{B_n\}_n$ are operator-valued Bessel sequences and $\{x_n\}_n$, $\{y_n\}_n$ are sequences in t…
