Searcharxiv⌕ Search

arXiv subjects

Chunhong Fu

Publications and source records attributed to Chunhong Fu.

6 recordsLinked to original sources
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↗
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↗
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 $\alpha$ between the minimum value and the maximum value, there exists a projection $P$ whose Frobenius distance away from $Q$ takes the value $\alpha$. 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↗
arXiv2023

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.

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 $(\pi, X)$ of $o(P,Q,H)$ can induce a faithful representation $(\widetilde{\pi}, X)$ of $i(P,Q,H)$ such that \begin{align*}&\widetilde{\pi}(P-P\wedge Q)=\pi(P)-\pi(P)\wedge \pi(Q),\\ &\widetilde{\pi}(Q-P\wedge Q)=\pi(Q)-\pi(P)\wedge \pi(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↗