Jana Hamza, Hassan Issa
In this paper, we present various inequalities for generalized numerical radius of $2 \times 2$ block matrices for Schatten p-norm. Moreover, we give a refinement of the triangle inequality for the Schatten p-generalized numerical radius.
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We generalize von Neumann's well-known trace inequality, as well as related eigenvalue inequalities for hermitian matrices, to Schatten-class operators between complex Hilbert spaces of infinite dimension. To this end, we exploit some recent results on the $C$-numerical range of Schatten-class operators. For the readers' convenience, we sketched the proof of these results in the Appendix.
Sharp Concave-Function Transfer for Lee-Type Schatten Norm Inequalities
该研究证明李型Schatten范数不等式的sharp线性结果，可无损失传递至任意非负凹函数类，明确了非线性问题与对应线性问题的最优常数一致。
On inequalities involving the spherical operator transforms
本文利用广义球面Aluthge变换和球面Heinz变换改进算子范数不等式，引入球面Schatten p-范数、Schatten p-数值半径等新概念，并建立相关不等式及等号条件。
Clarkson--McCarthy type inequalities, part I: $\ell_p$--$\ell_p$ and $\ell_q$--$\ell_p$ Schatten $p$-estimates
本文基于矩阵U的特性，推导出Clarkson-McCarthy型不等式，涵盖ℓ_p--ℓ_p和ℓ_q--ℓ_p Schatten p-估计，并扩展了Ball-Carlen-Lieb不等式及Audenaert猜想的弱界。
On weighted Schatten p-numerical radii via generalized Aluthge transformation
该研究针对希尔伯特空间算子，借助Schatten p-范数与广义Aluthge变换推导加权p-数值半径不等式，改进了已有结果，并通过实例分析了不同矩阵加权数值半径的上界。
We develop a new method for obtaining bounds on the negative eigenvalues of self-adjoint operators B in terms of a Schatten norm of the difference of the semigroups generated by A and B, where A is an operator with non-negative spectrum. Our method is based on the application of the Jensen identity of complex function theory to a suitably constructed holomorphic function, whose roots are in one-to-one correspondence with the negative eigenvalues of B. Applying our abstract results, together with bounds on Schatten norms of semigroup differences obtained by Demuth and Van Casteren, to Schroedin…
Let $\mathcal{C}_p$ be the Schatten $p$-class for $p>0$. Generalizations of the parallelogram law for the Schatten 2-norms have been given in the following form: If $\mathbf{A}=\{A_1,A_2,...,A_n\}$ and $\mathbf{B}=\{B_1,B_2,...,B_n\}$ are two sets of operators in $\mathcal{C}_2$, then $$\sum_{i,j=1}^n\|A_i-A_j\|_2^2 + \sum_{i,j=1}^n\|B_i-B_j\|_2^2 = 2\sum_{i,j=1}^n\|A_i-B_j\|_2^2 - 2\Norm{\sum_{i=1}^n(A_i-B_i)}_2^2.$$ In this paper, we give generalizations of this as pairs of inequalities for Schatten $p$-norms, which hold for certain values of $p$ and reduce to the equality above for $p=2$. M…
In this note we prove that the ball is a maximiser of some Schatten $p$-norms of the Riesz potential operators among all domains of a given measure in $\mathbb R^{d}$. In particular, the result is valid for the polyharmonic Newton potential operator, which is related to a nonlocal boundary value problem for the poly-Laplacian extending the one considered by M. Kac in the case of the Laplacian, so we obtain and isoperimetric inequalities for its eigenvalues as well, namely, analogues of Rayleigh-Faber-Krahn and Hong-Krahn-Szegö inequalities.
Upper and lower estimates are obtained for the Schatten-von Neumann norms of the Hardy-Steklov operator in Lebesgue function spaces on the semi-axis.
A refinement of a trace inequality of McCarthy establishing the uniform convexity of the Schatten $p$-classes for p>2 is proved
In the present paper, we provide several inequalities for the generalized numerical radius of operator matrices as introduced by A. Abu-omar and F. Kittaneh in [3]. We generalize the convexity and the log-convexity results obtained by M. Sababheh in [12] for the case of the numerical radius to the case of the generalized numerical radius. We illustrate our work by providing a positive answer for the question addressed in [12] for the convexity of a certain matrix operator function. Moreover, and motivated by the results of A. Aldalabih and F. Kittaneh in [2] for the case of Hilbert-Schmidt num…
Let $A_1, ... A_n$ be operators acting on a separable complex Hilbert space such that $\sum_{i=1}^n A_i=0$. It is shown that if $A_1, ... A_n$ belong to a Schatten $p$-class, for some $p>0$, then 2^{p/2}n^{p-1} \sum_{i=1}^n \|A_i\|^p_p \leq \sum_{i,j=1}^n\|A_i\pm A_j\|^p_p for $0<p\leq 2$, and the reverse inequality holds for $2\leq p<\infty$. Moreover, \sum_{i,j=1}^n\|A_i\pm A_j\|^2_p \leq 2n^{2/p} \sum_{i=1}^n \|A_i\|^2_p for $0<p\leq 2$, and the reverse inequality holds for $2\leq p<\infty$. These inequalities are related to a characterization of inner product spaces due to E.R. Lorch.
In this paper, we fully characterize the duality mapping over the space of matrices that are equipped with Schatten norms. Our approach is based on the analysis of the saturation of the Hölder inequality for Schatten norms. We prove in our main result that, for $p\in (1,\infty)$, the duality mapping over the space of real-valued matrices with Schatten-$p$ norm is a continuous and single-valued function and provide an explicit form for its computation. For the special case $p = 1$, the mapping is set-valued; by adding a rank constraint, we show that it can be reduced to a Borel-measurable singl…
They proved several properties and introduced some inequalities. We continue with the study of this generalized numerical radius and we develop diverse inequalities involving w_N. We also study particular cases with a fixed N(.), for instance the p-Schatten norms. In ["A generalization of the numerical radius". Linear Algebra Appl. 569 (2019)], Abu Omar and Kittaneh defined a new generalization of the numerical radius. That is, given a norm $N(\cdot)$ on $\bh$, the space of bounded linear operators over a Hilbert space H, and A in B(H) w_N(A)=sup_{θ\in \R}N(Re(e^{iθ}A)). They proved several pr…
In this paper we prove that the disc is a maximiser of the Schatten $p$-norm of the logarithmic potential operator among all domains of a given measure in $\mathbb R^{2}$, for all even integers $2\leq p<\infty$. We also show that the equilateral triangle has the largest Schatten $p$-norm among all triangles of a given area. For the logarithmic potential operator on bounded open or triangular domains, we also obtain analogies of the Rayleigh-Faber-Krahn or P{ó}lya inequalities, respectively. The logarithmic potential operator can be related to a nonlocal boundary value problem for the Laplacian…
Schatten $p$-norm and numerical radius inequalities with applications
本文精细化 Kato 不等式，研究算子数值半径、分块矩阵正定性及矩阵和的 Schatten $p$-范数界，并将其应用于图能量下界。
Essential norm and Schatten class of Hankel operators between weighted Fock spaces
本文研究加权Fock空间到加权Lebesgue空间的Hankel算子，完整刻画了其在所有 $1 \leq p, q < \infty$ 下的本质范数，并给出了由连续递增凸函数 $h$ 定义的Schatten-$h$ 类隶属准则。
We generalize the theorems of Stein--Tomas and Strichartz about surface restrictions of Fourier transforms to systems of orthonormal functions with an optimal dependence on the number of functions. We deduce the corresponding Strichartz bounds for solutions to Schrödinger equations up to the endpoint, thereby solving an open problem of Frank, Lewin, Lieb and Seiringer. We also prove uniform Sobolev estimates in Schatten spaces, extending the results of Kenig, Ruiz, and Sogge. We finally provide applications of these results to a Limiting Absorption Principle in Schatten spaces, to the well-pos…
Counterexamples to a conjectured Schatten norm inequality
针对Tang和Zhang从公共左等角秩1族得到的c_p(m)闭式公式猜想，证明对所有m≥2及1<p<2均不成立，给出了相关Schatten范数不等式的反例。
In this paper we prove that the ball is a maximizer of the Schatten $p$-norm of some convolution type integral operators with non-increasing kernels among all domains of a given measure in $\mathbb R^{d}$. We also show that the equilateral triangle has the largest Schatten $p$-norm among all triangles of a given area. Some physical motivations for our results are also presented.
Extended Cauchy-Schwarz inequalities for $σ$-elementary transformers in Schatten-von Neumann norm ideals
本文针对 Schatten-von Neumann 范数理想中的 σ-初等变换子，建立了扩展 Cauchy-Schwarz 不等式，并给出等价形式及对非双重平方可和算子族的应用，显著推广了作者已有结果。
We consider norms on a complex separable Hilbert space such that $\langle aξ,ξ\rangle\leq\|ξ\|^2\leq\langle bξ,ξ\rangle$ for positive invertible operators $a$ and $b$ that differ by an operator in the Schatten class. We prove that these norms have unitarizable isometry groups, our proof uses a generalization of a fixed point theorem for isometric actions on positive invertible operators. As a result, if the isometry group does not leave any finite dimensional subspace invariant, then the norm must be Hilbertian. That is, if a Hilbertian norm is changed to a close non-Hilbertian norm, then the …
Jensen inequalities for positive linear maps of Choi and Hansen-Pedersen type are established for a large class of operator/matrix means. These results are also extensions of the Minkowski determinantal inequality. To this end we develop the study of anti-norms, a notion parallel to the symmetric norms in matrix analysis, including functionals like Schatten q-norms for a parameter q<1 and the Minkowski functional. An interpolation theorem for the Schur multiplication is given in this setting. Two sections have been added to the previous version, devoted to means of severable variables and anti…
Essential norm and Schatten class difference of weighted composition operators over the ball
研究单位球上加权 Bergman 空间中两个加权复合算子之差，利用 Carleson 测度和 $R$-Berezin 变换估计其本性范数，并刻画其属于 Schatten 类的条件。
On the Relation of Schatten Norms and the Thompson Metric
本文研究了Schatten范数与Thompson度量之间的关系，提出基于Thompson度量的Schatten范数上界及Frobenius范数更紧的上界。
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…
The paper contains three results, the common feature of which is that they deal with the Schatten $p$ class. The first is a presentation of a new complemented subspace of $C_p$ in the reflexive range (and $p\not= 2$). This construction answers a question of Arazy and Lindestrauss from 1975. The second result relates to tight embeddings of finite dimensional subspaces of $C_p$ in $C_p^n$ with small $n$ and shows that $\ell_p^k$ nicely embeds into $C_p^n$ only if $n$ is at least proportional to $k$ (and then of course the dimension of $C_p^n$ is at least of order $k^2$). The third result concern…
Operator symmetric moduli and sharp triangle inequalities
本文研究了算术对称模和二次对称模与常规算子模之间的关系，推导了单位不变范数下的精确等价常数及三角不等式，给出了最优Schatten $p$-范数界和相变现象，并通过实例验证了常数的最优性。
We study perturbations of functions $f(A,B)$ of noncommuting self-adjoint operators $A$ and $B$ that can be defined in terms of double operator integrals. We prove that if $f$ belongs to the Besov class $B_{\be,1}^1(\R^2)$, then we have the following Lipschitz type estimate in the Schatten--von Neumann norm $\bS_p$, $1\le p\le2$ norm: $\|f(A_1,B_1)-f(A_2,B_2)\|_{\bS_p}\le\const(\|A_1-A_2\|_{\bS_p}+\|B_1-B_2\|_{\bS_p})$. However, the condition $f\in B_{\be,1}^1(\R^2)$ does not imply the Lipschitz type estimate in $\bS_p$ with $p>2$. The main tool is Schatten--von Neumann norm estimates for trip…
For a positive integer $n$, let $M_n$ be the set of $n\times n$ complex matrices. Suppose $\|\cdot\|$ is the Ky Fan $k$-norm with $1 \le k \le mn$ or the Schatten $p$-norm with $1 \le p \le \infty$ ($p\ne 2$) on $M_{mn}$, where $m,n\ge 2$ are positive integers. It is shown that a linear map $ϕ: M_{mn} \rightarrow M_{mn}$ satisfying $$\|A\otimes B\| = \|ϕ(A\otimes B)\| \quad \hbox{for all} A \in M_m \hbox{and} B \in M_n$$ if and only if there are unitary $U, V \in M_{mn}$ such that $ϕ$ has the form $A\otimes B \mapsto U(φ_1(A) \otimes φ_2(B))V$, where $φ_s(X)$ is either the identity map $X \map…
On Quantum Sobolev Inequalities
本文研究相空间中量子Sobolev不等式，以交换子Schatten范数定义量子范数，提供Wigner-Yanase斜信息的不确定性原理，并导出Weyl量子化新界，引入量子Besov空间并给出最优常数显式估计。
We present some properties of orthogonality and relate them with support disjoint and norm inequalities in p Schatten ideals. In addition, we investigate the problem of characterization of norm parallelism for bounded linear operators. We consider the characterization of norm parallelism problem in p Schatten ideals and locally uniformly convex spaces. Later on, we study the case when an operator is norm parallel to the identity operator. Finally, we give some equivalence assertions about the norm parallelism of compact operators. Some applications and generalizations are discussed for certain…
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 generalize the $C$-numerical range $W_C(T)$ from trace-class to Schatten-class operators, i.e. to $C\in\mathcal B^p(\mathcal H)$ and $T\in\mathcal B^q(\mathcal H)$ with $1/p + 1/q = 1$, and show that its closure is always star-shaped with respect to the origin. For $q \in (1,\infty]$, this is equivalent to saying that the closure of the image of the unitary orbit of $T\in\mathcal B^q(\mathcal H)$ under any continous linear functional $L\in(\mathcal B^q(\mathcal H))'$ is star-shaped with respect to the origin. For $q=1$, one has star-shapedness with respect to $\operatorname{tr}(T)W_e(L)$, w…
Schatten Properties of Calderón--Zygmund Singular Integral Commutator on stratified Lie groups
本文完整刻画了分层李群上Calderón--Zygmund奇异积分交换子的Schatten性质，证明了在临界指标处范数等价性，并提出了可推广的新方法。
Schatten--Lorentz characterization of Riesz transform commutator associated with Bessel operators
本文用鞅和二进分析刻画了与Bessel算子相关的Riesz变换换位子的Schatten--Lorentz范数，建立了其与符号振荡空间及Besov范数的等价关系，并确定了临界指标为Bessel测度的下维数。
We study the Schatten class membership of generalized Volterra companion integral operators on the standard Fock spaces $\mathcal{F}_α^2$. The Schatten $\mathcal{S}_p(\mathcal{F}_α^2)$ membership of the operators are characterized in terms of $L^{p/2}$ integrability of certain generalized Berezin type integral transforms on the complex plane. We also give a more simplified and easy to apply description in terms of $L^p$ integrability of the symbols inducing the operators against a supper exponentially decreasing weights. An asymptotic estimates for the $\mathcal{S}_p(\mathcal{F}_α^2)$ norms of…
We reprove an inequality for Rademacher series with coefficients in the Schatten class $S_1$. Our method yields the same estimate for coefficients after suitable gaps in $S_1$-valued trigonometric series; this was known for scalar-valued functions. A very similar method gives a new proof of the extension to $S_1$-valued $H^1$ functions of Paley's theorem about lacunary coefficients.
Some Properties of Products of Linear Fractional Composition Operators on Weighted Dirichlet Spaces
本文研究了加权Dirichlet空间上线性分数复合算子乘积的Schatten类成员资格和本质范数，给出了基于符号边界行为的完整刻画及精确或估计的范数公式。
We obtain a number of explicit estimates for quasi-norms of pseudo-differential operators in the Schatten-von Neumann classes $S_q$ with $0<q\le 1$. The estimates are applied to derive semi-classical bounds for operators with smooth or non-smooth symbols.
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',\ldot…
Sharp Quasi-Reverse Minkowski Inequality for Schatten Norms
该论文针对Schatten p-范数，证明2≤p<∞时存在严格拟逆Minkowski不等式，对1<p<2则说明最优常数公式不成立并给出反例与解析构造。
In this paper, motivated by perturbation theory of operators, we present some upper bounds for $|||f(A)Xg(B)+ X|||$ in terms of $|||\,|AXB|+|X|\,|||$ and $|||f(A)Xg(B)- X|||$ in terms of $|||\,|AX|+|XB|\,|||$, where $A, B$ are $G_{1}$ operators, $|||\cdot|||$ is a unitarily invariant norm and $f, g$ are certain analytic functions. Further, we find some new upper bounds for the the Schatten $2$-norm of $f(A)X\pm Xg(B)$. Several special cases are discussed as well.
For $1< p <2$ we obtain sharp inequalities for the supremum of products of homogeneous polynomials on $L_p(μ)$, whenever the number of factors is no greater than the dimension of these Banach spaces (a condition readily satisfied in the infinite dimensional settings). The results also holds for the Schatten classes $\mathcal S_p$. For $p>2$ we present some estimates on the involved constants.
Weak Log-Majorization for Negative Lim-Pálfia Power Means
本文研究负阶Lim-Pálfia幂均值的弱对数优超性，证明Ando的酉不变范数不等式对0<q<1的Schatten q-拟范数成立，解决了Hiai与Lim提出的问题，得到有限族正定矩阵的相关Schatten拟范数不等式。
It was shown in a previous work of the first named author with De Chiffre, Glebsky and Thom that there exists a finitely presented group which cannot be approximated by almost-homomorphisms to the unitary groups $U(n)$ equipped with the Frobenius norms (a.k.a as $L^2$ norm, or the Schatten-2-norm). In his ICM18 lecture, Andreas Thom asks if this result can be extended to general Schatten-p-norms. We show that this is indeed the case for $1<p< \infty$.
$A_\infty$-invariance of oscillatory norms, and Schatten characterisations of commutators
本文扩展已有交换子Schatten类刻画的抽象框架，引入$A_\infty$等价测度对放宽正则性假设，用实变调和分析方法复现并简化了Bessel-Riesz变换交换子的相关刻画结果。
Positive Toeplitz operators on pluriharmonic Fock spaces: Schatten class criteria, sharp norm comparisons, and generalized weights
该研究针对多重调和Fock空间上的正Toeplitz算子，建立了其Schatten类准则，解决了相关猜想，得到了与对应全纯Toeplitz算子的精确范数比较，还获得了对称范数理想的精确比较及迹恒等式。
Sharp Hilbert transform norms on Schatten classes: classification and optimal spectral corrections
本文证明Hilbert变换在Schatten类上的范数为$C_p$，解决多个猜想，并通过谱缺陷恒等式分类迹不等式，提出最优谱修正。
Let $\mathcal{H}$ be a Hilbert space, and let $K(\mathcal{H})$ be the $C^*$-algebra of compact operators on $\mathcal{H}$. In this paper, we present some characterizations of the norm-parallelism for elements of a Hilbert $K(\mathcal{H})$-module by employing the Birkhoff--James orthogonality. Among other things, we present a characterization of transitive relation of the norm-parallelism for elements in a certain Hilbert $K(\mathcal{H})$-module. We also give some characterizations of the Schatten $p$-norms and the operator norm-parallelism for matrices.
Additivity and chain rules for quantum entropies via multi-index Schatten norms
通过推广多指标Schatten范数，建立了量子信道优化夹层Rényi熵的通用可加性，并推导了Rényi条件熵的链式法则，用于分析时间自适应量子密码协议。
Two-Indexed Schatten Quasi-Norms with Applications to Quantum Information Theory
本文定义了双重索引(q,p)-Schatten准范数，用于描述张量积希尔伯特空间上的算子，扩展了Pisier的算子值Schatten空间理论，并引入了完全有界准范数和共准范数，探讨了其在量子信息度量中的应用。
We introduce some symmetric homogeneous means, and then show unitarily invariant norm inequalities for them, applying the method established by Hiai and Kosaki. Our new inequalities give the tighter bounds of the logarithmic mean than the inequalities given by Hiai and Kosaki. Some properties and norm continuities in parameter for our means are also discussed.
Let ${\mathfrak A}$ be a $C^*$-algebra, $T$ be a locally compact Hausdorff space equipped with a probability measure $P$ and let $(A_t)_{t\in T}$ be a continuous field of operators in ${\mathfrak A}$ such that the function $t \mapsto A_t$ is norm continuous on $T$ and the function $t \mapsto \|A_t\|$ is integrable. Then the following equality including Bouchner integrals holds \begin{eqnarray}\label{oi} \int_T|A_t - \int_TA_s{\rm d}P|^2 {\rm d}P=\int_T|A_t|^2{\rm d}P - |\int_TA_t{\rm d}P|^2 . \end{eqnarray} This equality is related both to the notion of variance in statistics and to a characte…
We consider the difference $f(H_1)-f(H_0)$ for self-adjoint operators $H_0$ and $H_1$ acting in a Hilbert space. We establish a new class of estimates for the operator norm and the Schatten class norms of this difference. Our estimates utilise ideas of scattering theory and involve conditions on $H_0$ and $H_1$ in terms of the Kato smoothness. They allow for a much wider class of functions $f$ (including some unbounded ones) than previously available results do. As an important technical tool, we propose a new notion of Schatten class valued smoothness and develop a new framework for double op…
A Counterexample to the Tang Zhang Schatten Norm Conjecture and Sharp Positive Results
该研究在 $p=3/2$ 时构造反例否定 Tang-Zhang Schatten 范数猜想，同时证明 $2\leq p<\infty$ 等情形下该猜想的严格界成立，并给出 $m=2$、$p=4$ 复矩阵情形的严格常数。
One of the long standing questions in the theory of Schatten-von Neumann ideals of compact operators is whether their norms have the same differentiability properties as the norms of their commutative counterparts. We answer this question in the affirmative. A key technical observation underlying our proof is a discovery of connection between this question and recent affirmative resolution of L.S. Koplienko's conjecture concerning existence of higher order spectral shift functions.
Some subadditivity results involving symmetric (unitarily invariant) norms are obtained. For instance, if $g(t)=\sum_{k=0}^m a_kt^k$ is a polynomial of degree $m$ with non-negative coefficients, then, for all positive operators $A,\,B$ and all symmetric norms, $\|g(A+B)\|^{1/m} \le \|g(A)\|^{1/m} + \|g(B)\|^{1/m}$. To give parallel superadditivity results, we investigate anti-norms, a class of functionals containing the Schatten $q$-norms for $q\in(0,1]$ and $q<0$. The results are extensions of the Minkowski determinantal inequality. A few estimates for block-matrices are derived. For instance…
We derive strong estimates for Schatten norms of operator derivatives along paths of contractions and apply them to prove existence of higher order spectral shift functions for pairs of contractions.
In this paper we study properties of the triangular projection ${\mathcal P}_n$ on the space of $n\times n$ matrices. The projection ${\mathcal P}_n$ annihilates the entries of an $n\times n$ matrix below the main diagonal and leaves the remaining entries unchanged. We estimate the $p$-norms of ${\mathcal P}_n$ as an operator on the Schatten--von Neumann class $\boldsymbol{S}_p$ for $0<p<1$. The main result of the paper shows that for $p\in(0,1)$, the $p$-norms of ${\mathcal P}_n$ on $\boldsymbol{S}_p$ behave as $n\to\infty$ as $n^{1/p-1}$. This solves a problem posed by B.S. Kashin. Among oth…
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