$(α,β)$-A-Normal Operators in Semi-Hilbertian Spaces

Numerical radius inequalities for products and sums of semi-Hilbertian space operators

New inequalities for the $A$-numerical radius of the products and sums of operators acting on a semi-Hilbert space, i.e. a space generated by a positive semidefinite operator $A$, are established. In particular, it is proved for operators $T$ and $S,$ having $A$-adjoint, that $$ ω_A(TS) \leq \frac{1}{2}ω_A(ST)+\frac{1}{4}\Big(\|T\|_A\|S\|_A+\|TS\|_A\Big),$$ where $ω_A(T)$ and $\|T\|_A$ denote the $A$-numerical radius and the $A$-operator seminorm of an operator $T$.

Davis-Wielandt shells of semi-Hilbertian space operators and its applications

In this paper we generalize the concept of Davis-Wielandt shell of operators on a Hilbert space when a semi-inner product induced by a positive operator $A$ is considered. Moreover, we investigate the parallelism of $A$-bounded operators with respect to the seminorm and the numerical radius induced by $A$. Mainly, we characterize $A$-normaloid operators in terms of their $A$-Davis-Wielandt radii. In addition, a connection between $A$-seminorm-parallelism to the identity operator and an equality condition for the $A$-Davis-Wielandt radius is proved. This generalizes the well-known results in \c…

半希尔伯特空间中算子的 A-q-数值域

On the $A$-$q$-Numerical Range of Operators in Semi-Hilbertian Spaces

本文在半希尔伯特空间中研究算子的 A-q-数值域，建立谱包含与圆盘并公式，推导数值半径界，并证明指数为 2 的 A-幂零算子其数值域为圆盘。

半希尔伯特空间算子的$A$-数值半径不等式的改进

Improvement of $A$-numerical radius inequalities of semi-Hilbertian space operators

该研究针对复希尔伯特空间上的正算子，得到半希尔伯特空间算子$A$-数值半径的新界，推广改进已有结果，还估计了对角分块为$A$的$2\times2$算子矩阵的$B$-算子半范数与$B$-数值半径的界，且界更优。

$A$-numerical radius inequalities for semi-Hilbertian space operators

Let $A$ be a positive bounded operator on a Hilbert space $\big(\mathcal{H}, \langle \cdot, \cdot\rangle \big)$. The semi-inner product ${\langle x, y\rangle}_A := \langle Ax, y\rangle$, $x, y\in\mathcal{H}$ induces a semi-norm ${\|\cdot\|}_A$ on $\mathcal{H}$. Let ${\|T\|}_A$ and $w_A(T)$ denote the $A$-operator semi-norm and the $A$-numerical radius of an operator $T$ in semi-Hilbertian space $\big(\mathcal{H}, {\|\cdot\|}_A\big)$, respectively. In this paper, we prove the following characterization of $w_A(T)$ \begin{align*} w_A(T) = \displaystyle{\sup_{α^2 + β^2 = 1}} {\left\|α\frac{T + T^…

关于半希尔伯特型算子的$A$-数值域的注记

A note on the $A$-numerical range of semi-Hilbertian operators

本文在半希尔伯特型结构下研究$A$-有界算子的$A$-数值域与$A$-谱的关系，引入$A$-正规算子新定义，证明其数值域闭包为谱的凸包，推广Anderson定理，并引入强$A$-数值闭类。

Spectral radius of semi-Hilbertian space operators and its applications

In this paper, we aim to introduce the notion of the spectral radius of bounded linear operators acting on a complex Hilbert space $\mathcal{H}$, which are bounded with respect to the seminorm induced by a positive operator $A$ on $\mathcal{H}$. Mainly, we show that $r_A(T)\leq ω_A(T)$ for every $A$-bounded operator $T$, where $r_A(T)$ and $ω_A(T)$ denote respectively the $A$-spectral radius and the $A$-numerical radius of $T$. This allows to establish that $r_A(T)=ω_A(T)=\|T\|_A$ for every $A$-normaloid operator $T$, where $\|T\|_A$ is denoted to be the $A$-operator seminorm of $T$. Moreover,…

The generalized Schwarz inequality for semi-Hilbertian space operators and Some $A$-numerical radius inequalities

In this work, the mixed Schwarz inequality for semi-Hilbertian space operators is proved. Namely, for every positive Hilbert space operator $A$. If $f$ and $g$ are nonnegative continuous functions on $\left[0,\infty\right)$ satisfying $f(t)g(t) =t$ $(t\ge0)$, then \begin{align*} \left| {\left\langle {T x,y} \right\rangle_A } \right| \le \left\| {f\left( {\left| T \right|_A x} \right)} \right\|_A \left\| {g\left( {\left| {T^{\sharp_A } } \right|_A y} \right)} \right\|_A \end{align*} for every Hilbert space operator $T$ such that the range of $T^* A$ is a subset in the range of $A$, such that $A…

半希尔伯特结构中算子的近似正交性与对称性研究

On approximate orthogonality and symmetry of operators in semi-Hilbertian structure

该研究在半希尔伯特结构中推广近似Birkhoff-James正交性与对称算子概念，定义两类$(ε,A)$-近似正交性并建立关联，刻画相关算子的正交性特征，推广已有希尔伯特空间框架下的结果。

Some results on higher order isosymmetries in Semi-Hilbertian Spaces

In this paper, we introduce the class of $(A,(m,n))$-isosymmetric operators and we study some of their properties, for a positive semi-definite operator $A$ and $ m,n\in\mathbb{ N}$, which extend, by changing the initial inner product with the semi-inner product induced by $A$, the well-known class of $(m,n)$-isosymmetric operators introduced by Mark Stankus (\cite{mark1}, \cite{mark}). In particular, we characterize a family of $A$-isosymmetric $(2\times2)$ upper triangular operator matrices. Moreover, we show that that if $T$ is $(A,(m,n))$-isosymmetric and if $Q$ is a nilpotent operator of …

Refinement of seminorm and numerical radius inequalities of semi-Hilbertian space operators

We give new inequalities for $A$-operator seminorm and $A$-numerical radius of semi-Hilbertian space operators and show that the inequalities obtained here generalize and improve on the existing ones. Considering a complex Hilbert space $\mathcal{H}$ and a non-zero positive bounded linear operator $A$ on $\mathcal{H},$ we show with among other seminorm inequalities, if $S,T,X\in \mathcal{B}_A(\mathcal{H})$, i.e., if $A$-adjoint of $S,T,X$ exist then $$2\|S^{\sharp_A}XT\|_A \leq \|SS^{\sharp_A}X+XTT^{\sharp_A}\|_A.$$ Further, we prove that if $T\in \mathcal{B}_A(\mathcal{H})$ then \begin{eqnarr…

半希尔伯特空间算子的$A$-数值半径正交性与平行性及其应用

$A$-Numerical radius orthogonality and parallelism of semi-Hilbertian space operators and their applications

本文引入并刻画了半希尔伯特空间中算子的$A$-数值半径正交性，证明了$A$-秩一算子的平行性刻画，并应用于算子矩阵的$\mathbb{A}$-数值半径不等式。

An extension of Birkhoff--James orthogonality relations in semi-Hilbertian space operators

Let $\mathbb{B}(\mathcal{H})$ denote the $C^{\ast}$-algebra of all bounded linear operators on a Hilbert space $\big(\mathcal{H}, \langle\cdot, \cdot\rangle\big)$. Given a positive operator $A\in\B(\h)$, and a number $λ\in [0,1]$, a seminorm ${\|\cdot\|}_{(A,λ)}$ is defined on the set $\B_{A^{1/2}}(\h)$ of all operators in $\B(\h)$ having an $A^{1/2}$-adjoint. The seminorm ${\|\cdot\|}_{(A,λ)}$ is a combination of the sesquilinear form ${\langle \cdot, \cdot\rangle}_A$ and its induced seminorm ${\|\cdot\|}_A$. A characterization of Birkhoff--James orthogonality for operators with respect to th…

Birkhoff--James orthogonality of operators in semi-Hilbertian spaces and its applications

In this paper, the concept of Birkhoff--James orthogonality of operators on a Hilbert space is generalized when a semi-inner product is considered. More precisely, for linear operators $T$ and $S$ on a complex Hilbert space $\mathcal{H}$, a new relation $T\perp^B_A S$ is defined if $T$ and $S$ are bounded with respect to the seminorm induced by a positive operator $A$ satisfying ${\|T + γS\|}_A\geq {\|T\|}_A$ for all $γ\in \mathbb{C}$. We extend a theorem due to R. Bhatia and P. Šemrl, by proving that $T\perp^B_A S$ if and only if there exists a sequence of $A$-unit vectors $\{x_n\}$ in $\math…

半希尔伯特空间中算子和的$A$-范数与$A$-数值半径的不等式及其应用

Inequalities for the $A$-Norm and $A$-Numerical Radius of Operator Sums in Semi-Hilbertian Spaces with Applications

本文对半希尔伯特空间中算子和的$A$-范数与$A$-数值半径建立新不等式，改进广义三角不等式并给出算子乘积与交换子的新边界，可应用于量子力学和偏微分方程，证明了不等式的最优性与优势。

半希尔伯特空间中算子的$A$-数值半径的若干上界及其应用

Certain Upper Bounds on the $A$-Numerical Radius of Operators in Semi-Hilbertian Spaces and Their Applications

本文在半希尔伯特空间框架下研究算子的$A$-数值半径，通过建立新不等式改进现有上界、细化三角不等式，并结合实例验证有效性，为数值半径不等式及算子理论应用提供更精确的理论支撑。

$A$-Davis-Wielandt 半希尔伯特空间算子的半径界限

$A$-Davis-Wielandt Radius Bounds of Semi-Hilbertian Space Operators

本文研究了复Hilbert空间中正算子诱导半范数下的 $A$-Davis-Wielandt 算子半径，提出了改进现有界限的新估计，并给出了 $2\times 2$ 非对角块矩阵的相应界限。

$A$-数值半径：新的不等式及等式的刻画

$A$-numerical radius : New inequalities and characterization of equalities

该研究提出半希尔伯特空间算子$A$-数值半径的新下界，借此推导算子交换子的$A$-数值半径上界（改进现有界），并给出现有$A$-数值半径不等式取等的刻画条件。

Some upper bounds for the $\mathbb{A}$-numerical radius of $2\times 2$ block matrices

Let $\mathbb{A}=\left( \begin{array}{cc} A & 0 \\ 0 & A \\ \end{array} \right)$ be the $2\times2$ diagonal operator matrix determined by a positive bounded operator $A$. For semi-Hilbertian operators $X$ and $Y$, we first show that \begin{align*} w^2_{\mathbb{A}}\left(\begin{bmatrix} 0 & X \\ Y & 0 \end{bmatrix}\right) &\leq \frac{1}{4}\max\Big\{{\big\|XX^{\sharp_A} + Y^{\sharp_A}Y\big\|}_{A}, {\big\|X^{\sharp_A}X + YY^{\sharp_A}\big\|}_{A}\Big\} + \frac{1}{2}\max\big\{w_{A}(XY), w_{A}(YX)\big\}, \end{align*} where $w_{\mathbb{A}}(\cdot)$, ${\|\cdot\|}_{A}$ and $w_{A}(\cdot)$ are the $\mathbb{…

半希尔伯特空间上 $A^{\frac{1}{2}}$-伴随算子的 $A$-谱半径公式

A formula of $A$-spectral radius for $A^{\frac{1}{2}}$-adjoint operators on semi-Hilbertian spaces

本文研究了半希尔伯特空间上的算子谱半径，证明了 $A^{\frac{1}{2}}$-伴随算子的 $A$-谱半径公式，并将其推广至多元算子的 $A$-Harte 谱半径不等式。

Operator inequalities and characterizations

In this survey, we shall present characterizations of some distinguished classes of Hilbertian bounded linear operators (namely, normal operators, selfadjoint operators, and unitary operators) in terms of operator inequalities related to the arithmetic-geometric mean inequality. For the class of all normal operators, we shall present new general characterizations.

Classification of contractively complemented Hilbertian operator spaces

We construct some separable infinite dimensional homogeneous Hilbertian operator spaces which generalize the row and column spaces R and C. We show that separable infinite-dimensional Hilbertian JC*-triples are completely isometric to an element of the set of (infinite) intersections of these spaces. This set includes the operator spaces R, C, their intersection, and the space spanned by creation operators on the full anti-symmetric Fock space. In fact, we show that these new spaces are completely isometric to the space of creation (resp. annihilation) operators on the anti-symmetric tensors o…

Some numerical radius inequalities for semi-Hilbert space operators

Let $A$ be a positive bounded linear operator acting on a complex Hilbert space $\big(\mathcal{H}, \langle \cdot\mid \cdot\rangle \big)$. Let $ω_A(T)$ and ${\|T\|}_A$ denote the $A$-numerical radius and the $A$-operator seminorm of an operator $T$ acting on the semi-Hilbertian space $\big(\mathcal{H}, {\langle \cdot\mid \cdot\rangle}_A\big)$ respectively, where ${\langle x\mid y\rangle}_A := \langle Ax\mid y\rangle$ for all $x, y\in\mathcal{H}$. In this paper, we show that \begin{equation*}\label{m1} \tfrac{1}{4}\|T^{\sharp_A} T+TT^{\sharp_A}\|_A\le ω_A^2\left(T\right) \le \tfrac{1}{2}\|T^{\sh…

半希尔伯特空间中算子矩阵的$A$-数值半径的更紧不等式及其应用

Tighter Inequalities for $A$-Numerical Radii of Operator Matrices and Their Applications

本文在半希尔伯特空间中利用$A$-Buzano不等式和$A$-伴随技巧建立了算子矩阵$A$-数值半径的更紧上界，推广并改进了经典数值半径不等式，并展示了其在量子力学、偏微分方程和控制理论中的应用。

Completely bounded maps into certain Hilbertian operator spaces

We prove a factorization of completely bounded maps from a $C^*$-algebra $A$ (or an exact operator space $E\subset A$) to $\ell_2$ equipped with the operator space structure of $(C,R)_θ$ ($0<θ<1$) obtained by complex interpolation between the column and row Hilbert spaces. More precisely, if $F$ denotes $\ell_2$ equipped with the operator space structure of $(C,R)_θ$, then $u: A \to F$ is completely bounded iff there are states $f,g$ on $A$ and $C>0$ such that \[ \forall a\in A\quad \|ua\|^2\le C f(a^*a)^{1-θ}g(aa^*)^θ.\] This extends the case $θ=1/2$ treated in a recent paper with Shlyakhtenk…

On A-numerical radius inequalities for $2 \times 2$ operator matrices

Let ($\mathcal{H}, \langle . , .\rangle )$ be a complex Hilbert space and $A$ be a positive bounded linear operator on it. Let $w_A(T)$ be the $A$-numerical radius and $\|T\|_A$ be the $A$-operator seminorm of an operator $T$ acting on the semi-Hilbertian space $(\mathcal{H}, \langle .,.\rangle_A),$ where $\langle x, y\rangle_A:=\langle Ax, y\rangle$ for all $x,y\in \mathcal{H}$. In this article, we establish several upper and lower bounds for $B$-numerical radius of $2\times 2$ operator matrices, where $B=\begin{bmatrix} A & 0 0 & A \end{bmatrix}$. Further, we prove some refinements of earlie…

Tropical reproducing kernels and optimization

Hilbertian kernel methods and their positive semidefinite kernels have been extensively used in various fields of applied mathematics and machine learning, owing to their several equivalent characterizations. We here unveil an analogy with concepts from tropical geometry, proving that tropical positive semidefinite kernels are also endowed with equivalent viewpoints, stemming from Fenchel-Moreau conjugations. This tropical analogue of Aronszajn's theorem shows that these kernels correspond to a feature map, define monotonous operators, and generate max-plus function spaces endowed with a repro…

半 Hilbert 空间中 $A$-正交性保持与 Blanco-Koldobsky-Turnšek 定理

On $A$-orthogonality preservation and Blanco-Koldobsky-Turnšek theorem in semi-Hilbert spaces

本文研究半 Hilbert 空间中 $A$-有界算子对点态 $A$-正交性的保持，给出完整刻画，并据此描述范数达到集及判定 $A$-等距。

Sobolev-to-Lipschitz Property on QCD-spaces and Applications

We prove the Sobolev-to-Lipschitz property for metric measure spaces satisfying the quasi curvature dimension condition recently introduced in E. Milman, The Quasi Curvature-Dimension Condition with applications to sub-Riemannian manifolds, Comm. Pure Appl. Math. (to appear, arXiv:1908.01513v5). We provide several applications to properties of the corresponding heat semigroup. In particular, under the additional assumption of infinitesimal Hilbertianity, we show the Varadhan short-time asymptotics for the heat semigroup with respect to the distance, and prove the irreducibility of the heat sem…

Some bounds for the $\mathbb{A}$-numerical radius of certain $2 \times 2$ operator matrices

For a given bounded positive (semidefinite) linear operator $A$ on a complex Hilbert space $\big(\mathcal{H}, \langle \cdot\mid \cdot\rangle \big)$, we consider the semi-Hilbertian space $\big(\mathcal{H}, \langle \cdot\mid \cdot\rangle_A \big)$ where ${\langle x\mid y\rangle}_A := \langle Ax\mid y\rangle$ for every $x, y\in\mathcal{H}$. The $A$-numerical radius of an $A$-bounded operator $T$ on $\mathcal{H}$ is given by \begin{align*} ω_A(T) = \sup\Big\{\big|{\langle Tx\mid x\rangle}_A\big|\,; \,\,x\in \mathcal{H}, \,{\langle x\mid x\rangle}_A= 1\Big\}. \end{align*} Our aim in this paper is t…

A convex extension of lower semicontinuous functions defined on normal Hausdorff space

We prove that, any problem of minimization of proper lower semicontinuous function defined on a normal Hausdorff space, is canonically equivalent to a problem of minimization of a proper weak * lower semicontinuous convex function defined on a weak * convex compact subset of some dual Banach space. We estalish the existence of an bijective operator between the two classes of functions which preserves the problems of minimization.

关于算子A-数值半径的不等式

On inequalities for A-numerical radius of operators

本文研究复Hilbert空间上正算子A对应的A-数值半径不等式，给出改进并推广现有结果的上、下界，并进一步得到2×2算子矩阵B-数值半径不等式及算子乘积A-数值半径的改进上界。

Hermitian operators and isometries on symmetric operator spaces

Let $\mathcal{M}$ be an atomless semifinite von Neumann algebra (or an atomic von Neumann algebra with all atoms having the same trace) acting on a (not necessarily separable) Hilbert space $H$ equipped with a semifinite faithful normal trace $τ$. Let $E(\mathcal{M},τ) $ be a symmetric operator space affiliated with $ \mathcal{M} $, whose norm is order continuous and is not proportional to the Hilbertian norm $\left\|\cdot\right\|_2$ on $L_2(\mathcal{M},τ)$. We obtain general description of all bounded hermitian operators on $E(\mathcal{M},τ)$. This is the first time that the description of he…

巴拿赫空间中线性随机发展方程的Bismut-Elworthy-Li公式

A Bismut-Elworthy-Li formula for linear stochastic evolution equations in Banach spaces

该研究针对巴拿赫空间中线性随机发展方程，证明了Bismut-Elworthy-Li公式，在特定平方函数条件下实现了半群导数的随机积分表示，为固定时间高斯转移测度的Cameron–Martin公式提供了具体控制实现。

The maximal operator of a normal Ornstein--Uhlenbeck semigroup is of weak type $(1,1)$

Consider a normal Ornstein--Uhlenbeck semigroup in $\Bbb{R}^n$, whose covariance is given by a positive definite matrix. The drift matrix is assumed to have eigenvalues only in the left half-plane. We prove that the associated maximal operator is of weak type $(1,1)$ with respect to the invariant measure. This extends earlier work by G. Mauceri and L. Noselli. The proof goes via the special case where the matrix defining the covariance is $I$ and the drift matrix is diagonal.