arXiv subjects

Qingxiang Xu

Publications and source records attributed to Qingxiang Xu.

At least 19 recordsLinked to original sources

arXiv2026

Norm attainment of a class of block operator matrices

Given complex numbers $a, b, c$ and a non-negative continuous function $φ$ defined on $[0, +\infty)$, consider the $2 \times 2$ matrix $$ M_t = \begin{pmatrix} a & t \\ ct & bφ(t) \end{pmatrix}, \quad t \in [0, +\infty). $$ We establish conditions for the strict monotonicity of the norm function $t \mapsto \|M_t\|$. As an application, we characterize the norm attainment of the corresponding block operator matrix $$ T = \begin{pmatrix} aI_H & A \\ cA^* & bφ(|A|) \end{pmatrix},$$ where $I_H$ is the identity operator on a Hilbert space $H$ and $A$ is a bounded linear operator from another Hilbert space to $H$.

math.FA↗

arXiv2026

Some applications of the matched projections of idempotents

For every idempotent $Q$ on a Hilbert space $H$, the matched projection $m(Q)$ is a well-established concept. This paper explores several applications of the matched projections. The first application addresses the distances from projections on $H$ to a given idempotent $Q$. Using $m(Q)$, a complete characterization of these distances is established, covering the minimum, maximum, and intermediate values. The second application focuses on the $C^*$-algebra $C^*\{Q\}$ generated by a single non-projection idempotent $Q$. A new $4\times 4$ block matrix representation of $Q$, induced by $m(Q)$, yields novel formulas for $Q$, leading to a full characterization of all elements in $C^*\{Q\}$ via explicit $4\times 4$ block matrices. Furthermore, for each $r>1$, a family of universal $r$-idempotents is introduced. These idempotents possess a universal property distinct from known properties of projection pairs. Some necessary and sufficient conditions are provided for such universal $r$-idempotents. The third application presents new characterizations of the numerical ranges. An operator version of the elliptical range theorem is established. Using a general non-projection idempotent $Q$ and its matched projection $m(Q)$, a non-quadratic operator is constructed, and its numerical range is described in detail. Additionally, another operator is introduced whose numerical range closure is not an elliptical disk, and the numerical range itself is neither closed nor open.

math.FA↗

arXiv2025

A simplified formula for the matched projection of an idempotent

Let $\mathcal{L}(H)$ be the set of all adjointable operators on a Hilbert $C^*$-module $H$. For each $T\in\mathcal{L}(H)$, $T^*$ denotes its adjoint operator, and $|T^*|$ is the positive square root of $TT^*$. We establish a simplified formula for the matched projection $m(Q)$ of an idempotent $Q\in\mathcal{L}(H)$ as $$m(Q)=\frac{I+|Q^*|-|I-Q^*|}{2},$$ where $I$ is the identity operator on $H$. This explicit expression allows for the direct derivation of some basic properties of $m(Q)$.

math.FA↗

arXiv2025

The numerical ranges of the generalized quadratic operators

We investigate the generalized quadratic operator defined by $$T =\left( \begin{array}{cc} a I_H & A \\ c A^* & bI_K \end{array} \right) ,$$ where $H$ and $K$ are Hilbert spaces, $A:K\to H$ is a bounded linear operator, $I_H$ and $I_K$ denote the identity operators on $H$ and $K$, respectively, and $a,b,c$ are complex numbers. It is shown that $T$ attains its norm if and only if $A$ attains its norm. Furthermore, a complete characterization of the numerical range of $T$ is provided by a new approach.

math.FA↗

arXiv2025

The block matrix representations for the quasi-projection pairs on Hilbert $C^*$-modules

A quasi-projection pair consists of two operators $P$ and $Q$ acting on a Hilbert $C^*$-module $H$, where $P$ is a projection and $Q$ is an idempotent satisfying $Q^*=(2P-I)Q(2P-I)$, in which $Q^*$ denotes the adjoint operator of $Q$, and $I$ is the identity operator on $H$. Such a pair is said to be harmonious if both $P(I-Q)$ and $(I-P)Q$ admit polar decompositions. The primary goal of this paper is to present the block matrix representations for a harmonious quasi-projection pair $(P,Q)$ on a Hilbert $C^*$-module, and additionally to derive new block matrix representations for the matched projection, the range projection, and the null space projection of $Q$. Several applications of these newly obtained block matrix representations are also explored.

math.FA↗

arXiv2025

Characterizations of the semi-harmonious and harmonious quasi-projection pairs on Hilbert $C^*$-modules

For each adjointable idempotent $Q$ on a Hilbert $C^*$-module $H$, a specific projection $m(Q)$ called the matched projection of $Q$ was introduced recently due to the characterization of the minimum value among all the distances from projections to $Q$. Inspired by the relationship between $m(Q)$ and $Q$, another term called the quasi-projection pair $(P,Q)$ was also introduced recently, where $P$ is a projection on $H$ satisfying $Q^*=(2P-I)Q(2P-I)$, in which $Q^*$ is the adjoint operator of the idempotent $Q$ and $I$ is the identity operator on $H$. This paper aims to make systematical characterizations of the semi-harmonious and harmonious quasi-projection pairs on Hilbert $C^*$-modules, and meanwhile to provide examples illustrating the non-triviality of the associated characterizations.

math.OA↗

arXiv2025

The matched projections of idempotents on Hilbert $C^*$-modules

The aim of this paper is to give new characterizations of some fundamental issues about idempotents. In the general setting of adjointable operators on Hilbert $C^*$-modules, a new term of quasi-projection pair is introduced. For each idempotent $Q$, a projection $m(Q)$, called the matched projection of $Q$, is constructed. It is shown that $Q$ and $m(Q)$ as idempotents are homotopic, and $\big(m(Q),Q\big)$ is a quasi-projection pair. Some formulas for $m(Q)$ are derived. Based on these formulas, representations and norm estimations associated with $m(Q)$ are dealt with.

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arXiv2025

Convex inequalities in Hilbert $C^*$-modules

The H$\ddot{\rm o}$lder-McCarty inequalities are originally derived in the Hilbert space case and have been generalized via a convex inequality. The main purpose of this paper is to extend this convex inequality to the Hilbert $C^*$-module case, and meanwhile to make some investigations on the H$\ddot{\rm o}$lder-McCarty inequalities in the Hilbert $C^*$-module case.

math.FA↗

arXiv2025

A new formula for the weighted Moore-Penrose inverse and its applications

In the general setting of the adjointable operators on Hilbert $C^*$-modules, this paper deals mainly with the weighted Moore-Penrose (briefly weighted M-P) inverse $A^†_{MN}$ in the case that the weights $M$ and $N$ are self-adjoint invertible operators, which need not to be positive. A new formula linking $A^†_{MN}$ to $A$, $A^†$, $M$ and $N$ is derived, in which $A^†$ denotes the M-P inverse of $A$. Based on this formula, some new results on the weighted M-P inverse are obtained. Firstly, it is shown that $A^†_{MN}=A^†_{ST}$ for some positive definite operators $S$ and $T$. This shows that $A^†_{MN}$ is essentially an ordinary weighted M-P inverse. Secondly, some limit formulas for the ordinary weighted M-P inverse originally known for matrices are generalized and improved. Thirdly, it is shown that when $A,M$ and $N$ act on the same Hilbert $C^*$-module, $A^†_{MN}$ belongs to the $C^*$-algebra generated by $A$, $M$ and $N$. Finally, some characterizations of the continuity of the weighted M-P inverse are provided.

math.FA↗

arXiv2024

Some aspects of semi-harmonious quasi-projection pairs

A term called the quasi-projection pair $(P,Q)$ was introduced recently by the authors, where $P$ is a projection and $Q$ is an idempotent on a Hilbert $C^*$-module $H$ satisfying $Q^*=(2P-I)Q(2P-I)$, in which $Q^*$ is the adjoint operator of the idempotent $Q$ and $I$ is the identity operator on $H$. Some fundamental issues on quasi-projection pairs, such as the block matrix representations for quasi-projection pairs and the $C^*$-morphisms associated with quasi-projection pairs, are worthwhile to be investigated. This paper aims to make some preparations. One object called the semi-harmonious quasi-projection pair is introduced in the general setting of the adjointable operators on Hilbert $C^*$-modules. Some related operator theories on the common similarity of operators and a norm equation associated with the Friedrichs angle are dealt with.

math.FA↗

arXiv2024

Some inequalities for adjointable operators on Hilbert $C^*$-modules

The main purpose of this paper is, in the general setting of the adjointable operators on Hilbert $C^*$-modules, to develop two new tools that can be applied to deal with the positive solutions of certain operator equations, the operator norm as well as the numerical radius, respectively. Among other things, the positivity of a $2\times 2$ block operator matrix is clarified without any preconditions on its entries, and a generalized version of the mixed Schwarz inequality with a parameter is derived. Numerical examples are provided to illustrate the non-triviality of this newly obtained inequality.

math.FA↗

arXiv2024

The generalized polar decomposition, the weak complementarity and the parallel sum for adjointable operators on Hilbert $C^*$-modules

This paper deals mainly with some aspects of the adjointable operators on Hilbert $C^*$-modules. A new tool called the generalized polar decomposition for each adjointable operator is introduced and clarified. As an application, the general theory of the weakly complementable operators is set up in the framework of Hilbert $C^*$-modules. It is proved that there exists an operator equation which has a unique solution, whereas this unique solution fails to be the reduced solution. Some investigations are also carried out in the Hilbert space case. It is proved that there exist a closed subspace $M$ of certain Hilbert space $K$ and an operator $T\in \mathbb{B}(K)$ such that $T$ is $(M,M)$-weakly complementable, whereas $T$ fails to be $(M,M)$-complementable. The solvability of the equation $$A:B=X^*AX+(I-X)^*B(I-X) \quad (X\in\mathbb{B}(H))$$ is also dealt with in the Hilbert space case, where $A,B\in \mathbb{B}(H)$ are two general positive operators, and $A:B$ denotes their parallel sum. Among other things, it is shown that there exist certain positive operators $A$ and $B$ on the Hilbert space $\ell^2(\mathbb{N})\oplus \ell^2(\mathbb{N})$ such that the above equation has no solution.

math.FA↗

arXiv2023

The Frobenious distances from projections to an idempotent matrix

For each pair of matrices $A$ and $B$ with the same order, let $\|A-B\|_F$ denote their Frobenius distance. This paper deals mainly with the Frobenius distances from projections to an idempotent matrix. For every idempotent $Q\in \mathbb{C}^{n\times n}$, a projection $m(Q)$ called the matched projection can be induced. It is proved that $m(Q)$ is the unique projection whose Frobenius distance away from $Q$ takes the minimum value among all the Frobenius distances from projections to $Q$, while $I_n-m(Q)$ is the unique projection whose Frobenius distance away from $Q$ takes the maximum value. Furthermore, it is proved that for every number $α$ between the minimum value and the maximum value, there exists a projection $P$ whose Frobenius distance away from $Q$ takes the value $α$. Based on the above characterization of the minimum distance, some Frobenius norm upper bounds and lower bounds of $\|P-Q\|_F$ are derived under the condition of $PQ=Q$ on a projection $P$ and an idempotent $Q$.

math.FA↗

arXiv2022

The polar decomposition of the product of three operators

In the setting of adjointable operators on Hilbert $C^*$-modules, this paper deals with the polar decomposition of the product of three operators. The relationship between the polar decompositions associated with three operators is clarified. Based on this relationship, a formula for the polar decomposition of a multiplicative perturbation of an operator is provided. In addition, some characterizations of the polar decomposition associated with three operators are provided.

math.FA↗

arXiv2022

The $(n+1)$-centered operator on a Hilbert $C^*$-module

Let $T$ be an adjointable operator on a Hilbert $C^*$-module such that $T$ has the polar decomposition $T=UT|$. For each natural number $n$, $T$ is called an $(n+1)$-centered operator if $T^k=U^k|T^k|$ is the polar decomposition for $1\le k\le n+1$. This paper initiates the study of the $(n+1)$-centered operator via the generalized Aluthge transform and the generalized iterative Aluthge transform. Some new characterizations of the $(n+1)$-centered operator are provided.

math.OA↗

arXiv2022

$C^$-isomorphisms associated with two projections on a Hilbert $C^$-module

Motivated by two norm equations used to characterize the Friedrichs angle, this paper studies $C^*$-isomorphisms associated with two projections by introducing the matched triple and the semi-harmonious pair of projections. A triple $(P,Q,H)$ is said to be matched if $H$ is a Hilbert $C^*$-module, $P$ and $Q$ are projections on $H$ such that their infimum $P\wedge Q$ exists as an element of $\mathcal{L}(H)$, where $\mathcal{L}(H)$ denotes the set of all adjointable operators on $H$. The $C^*$-subalgebras of $\mathcal{L}(H)$ generated by elements in $\{P-P\wedge Q, Q-P\wedge Q, I\}$ and $\{P,Q,P\wedge Q,I\}$ are denoted by $i(P,Q,H)$ and $o(P,Q,H)$, respectively. It is proved that each faithful representation $(π, X)$ of $o(P,Q,H)$ can induce a faithful representation $(\widetildeπ, X)$ of $i(P,Q,H)$ such that \begin{align*}&\widetildeπ(P-P\wedge Q)=π(P)-π(P)\wedge π(Q),\\ &\widetildeπ(Q-P\wedge Q)=π(Q)-π(P)\wedge π(Q). \end{align*} When $(P,Q)$ is semi-harmonious, that is, $\overline{\mathcal{R}(P+Q)}$ and $\overline{\mathcal{R}(2I-P-Q)}$ are both orthogonally complemented in $H$, it is shown that $i(P,Q,H)$ and $i(I-Q,I-P,H)$ are unitarily equivalent via a unitary operator in $\mathcal{L}(H)$. A counterexample is constructed, which shows that the same may be not true when $(P,Q)$ fails to be semi-harmonious. Likewise, a counterexample is constructed such that $(P,Q)$ is semi-harmonious, whereas $(P,I-Q)$ is not semi-harmonious. Some additional examples indicating new phenomena of adjointable operators acting on Hilbert $C^*$-modules are also provided.

math.OA↗

arXiv2020

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 $\mathbb{A}$-numerical radius, $A$-operator seminorm and $A$-numerical radius, respectively. We then apply the above inequality to find some upper bounds for the $\mathbb{A}$-numerical radius of certain $2\times 2$ operator matrices. In particular, we obtain some refinements of earlier $A$-numerical radius inequalities for semi-Hilbertian operators. An upper bound for the $\mathbb{A}$-numerical radius of $2\times 2$ block matrices of semi-Hilbertian space operators is also given.

math.FA↗

arXiv2019

Douglas factorization theorem revisited

Inspired by the Douglas lemma, we investigate the solvability of the operator equation $AX=C$ in the framework of Hilbert C*-modules. Utilizing partial isometries, we present its general solution when $A$ is a semi-regular operator. For such an operator $A$, we show that the equation $AX=C$ has a positive solution if and only if the range inclusion ${\mathcal R}(C) \subseteq {\mathcal R}(A)$ holds and $CC^*\le t\, CA^*$ for some $t>0$. In addition, we deal with the solvability of the operator equation $(P+Q)^{1/2}X=P$, where $P$ and $Q$ are projections. We provide a counterexample to show that there exists a $C^*$-algebra $\mathfrak{A}$, a Hilbert $\mathfrak{A}$-module $\mathscr{H}$ and projections $P$ and $Q$ on $\mathscr{H}$ such that the operator equation $(P+Q)^{1/2}X=P$ has no solution. Moreover, we give a perturbation result related to the latter equation.

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