More on $ω$-orthogonality and $ω$-parallelism

The operator--valued parallelism and norm-parallelism in matrices

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.

关于算子空间的一些几何性质

On some geometric properties of operator spaces

本文刻画了有界线性算子空间中平行性与 Birkhoff-James 正交性，并引入半圆点与半圆空间概念，证明在有限维且值域严格凸条件下算子空间是半圆但非严格凸的。

Norm-parallelism and the Davis--Wielandt radius of Hilbert space operators

We present a necessary and sufficient condition for the norm-parallelism of bounded linear operators on a Hilbert space. We also give a characterization of the Birkhoff--James orthogonality for Hilbert space operators. Moreover, we discuss the connection between norm-parallelism to the identity operator and an equality condition for the Davis--Wielandt radius. Some other related results are also discussed.

Norm-parallelism in the geometry of Hilbert $C^*$-modules

Utilizing the Birkhoff--James orthogonality, we present some characterizations of the norm-parallelism for elements of $\mathbb{B}(\mathscr{H})$ defined on a finite dimensional Hilbert space, elements of a Hilbert $C^*$-module over the $C^*$-algebra of compact operators and elements of an arbitrary $C^*$-algebra. We also consider the characterization of norm parallelism problem for operators on a finite dimensional Hilbert space when the operator norm is replaced by the Schatten $p$-norm. Some applications and generalizations are discussed for certain elements of a Hilbert $C^*$-module.

与积分正交性相关的近似正交性的一些研究

Some study of the approximate orthogonality connected with integral orthogonalities

本文提出一种基于积分正交性的新型近似正交性，建立其基本性质，考察其保持正交性映射的能力，并探讨其与其他近似正交性的关系。

An extension of orthogonality relations based on norm derivatives

We introduce the relation $ρ_λ$-orthogonality in the setting of normed spaces as an extension of some orthogonality relations based on norm derivatives, and present some of its essential properties. Among other things, we give a characterization of inner product spaces via the functional $ρ_λ$. Moreover, we consider a class of linear mappings preserving this new kind of orthogonality. In particular, we show that a linear mapping preserving $ρ_λ$-orthogonality has to be a similarity, that is, a scalar multiple of an isometry.

Numerical radius orthogonality in $C^*$-algebras

In this paper we characterize the Birkhoff--James orthogonality with respect to the numerical radius norm $v(\cdot)$ in $C^*$-algebras. More precisely, for two elements $a, b$ in a $C^*$-algebra $\mathfrak{A}$, we show that $a\perp_{B}^{v} b$ if and only if for each $θ\in [0, 2π)$, there exists a state $φ_{_θ}$ on $\mathfrak{A}$ such that $|φ_{_θ}(a)| = v(a)$ and $\mbox{Re}\big(e^{iθ}\overline{φ_{_θ}(a)}φ_{_θ}(b)\big)\geq 0$. Moreover, we compute the numerical radius derivatives in $\mathfrak{A}$. In addition, we characterize when the numerical radius norm of the sum of two (or three) elements…

Stampacchia's property, self-duality and orthogonality relations

We show that if the conclusion of the well known Stampacchia Theorem, on variational inequalities, holds on a Banach space X, then X is isomorphic to a Hilbert space. Motivated by this we obtain a relevant result concerning self-dual Banach spaces and investigate some connections between existing notions of orthogonality and self-duality. Moreover, we revisit the notion of the cosine of a linear operator and show that it can be used to characterize Hilbert space structure. Finally, we present some consequences of our results to quadratic forms and to evolution triples.

Numerical radius parallelism of Hilbert space operators

In this paper, we introduce a new type of parallelism for bounded linear operators on a Hilbert space $\big(\mathscr{H}, \langle \cdot ,\cdot \rangle\big)$ based on numerical radius. More precisely, we consider operators $T$ and $S$ which satisfy $ω(T + λS) = ω(T)+ω(S)$ for some complex unit $λ$. We show that $T \parallel_ω S$ if and only if there exists a sequence of unit vectors $\{x_n\}$ in $\mathscr{H}$ such that \begin{align*} \lim_{n\rightarrow\infty} \big|\langle Tx_n, x_n\rangle\langle Sx_n, x_n\rangle\big| = ω(T)ω(S). \end{align*} We then apply it to give some applications.

Orthogonality Hilbert A-modules and operators preserving multi-A-linearity

In this paper we present results concerning orthogonality in Hilbert $C^*$-modules. Moreover, for a $C^*$-algebra $\mathscr{A}$, we prove theorems concerning the multi-$\mathscr{A}$-linearity and its preservation by $\mathscr{A}$-linear operators. New version of solution of the orthogonality equation on Hilbert $C^*$-modules and mappings preserving orthogonality are also investigated.

一种用于Birkhoff-James正交性的数值域方法及其应用

A numerical range approach to Birkhoff-James orthogonality with applications

本文利用抽象数值域工具给出多类Banach空间中Birkhoff-James正交性的刻画，推广了Bhatia-Šemrl定理，并将结果应用于spear向量与算子研究，部分解决了相关公开问题。

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…

Characterizations of operator Birkhoff-James orthogonality

In this paper, we obtain some characterizations of the (strong) Birkhoff--James orthogonality for elements of Hilbert $C^*$-modules and certain elements of $\mathbb{B}(\mathscr{H})$. Moreover, we obtain a kind of Pythagorean relation for bounded linear operators. In addition, for $T\in \mathbb{B}(\mathscr{H})$ we prove that if the norm attaining set $\mathbb{M}_T$ is a unit sphere of some finite dimensional subspace $\mathscr{H}_0$ of $\mathscr{H}$ and $\|T\|_{{\mathscr{H}_0}^\perp} < \|T\|$, then for every $S\in\mathbb{B}(\mathscr{H})$, $T$ is the strong Birkhoff--James orthogonal to $S$ if a…

Characterization of numerical radius parallelism in $C^*$-algebras

Let $v(x)$ be the numerical radius of an element $x$ in a $C^*$-algebra $\mathfrak{A}$. First, we prove several numerical radius inequalities in $\mathfrak{A}$. Particularly, we present a refinement of the triangle inequality for the numerical radius in $C^*$-algebras. In addition, we show that if $x\in\mathfrak{A}$, then $v(x) = \frac{1}{2}\|x\|$ if and only if $\|x\| = \|\mbox{Re}(e^{iθ}x)\| + \|\mbox{Im}(e^{iθ}x)\|$ for all $θ\in \mathbb{R}$. Among other things, we introduce a new type of parallelism in $C^*$-algebras based on numerical radius. More precisely, we consider elements $x$ and $…