A quasi-projection pair consists of two operators P and Q acting on a Hilbert <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$C^$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>C</mi> <mo>∗</mo> </msup> </math></EquationSource> </InlineEquation>-module H, where P is a projection and Q is an idempotent satisfying <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$Q^=(2P-I)Q(2P-I)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi>Q</mi> <mo>∗</mo> </msup> <mo>=</mo> <mrow> <mo stretchy="false">(</mo> <mn>2</mn> <mi>P</mi> <mo>-</mo> <mi>I</mi> <mo stretchy="false">)</mo> </mrow> <mi>Q</mi> <mrow> <mo stretchy="false">(</mo> <mn>2</mn> <mi>P</mi> <mo>-</mo> <mi>I</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>, in which <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$Q^$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>Q</mi> <mo>∗</mo> </msup> </math></EquationSource> </InlineEquation> denotes the adjoint operator of Q, and I is the identity operator on H. Such a pair is said to be harmonious if both <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$P(I-Q)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>P</mi> <mo stretchy="false">(</mo> <mi>I</mi> <mo>-</mo> <mi>Q</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$(I-P)Q$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <mi>I</mi> <mo>-</mo> <mi>P</mi> <mo stretchy="false">)</mo> <mi>Q</mi> </mrow> </math></EquationSource> </InlineEquation> admit polar decompositions. The primary goal of this paper is to present block matrix representations for a harmonious quasi-projection pair (P, Q) on a Hilbert <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$C^$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>C</mi> <mo>∗</mo> </msup> </math></EquationSource> </InlineEquation>-module, and additionally to derive new block matrix representations for the matched projection, range projection, and null space projection of Q. Several applications of these newly obtained block matrix representations are also explored.

Block matrix representations for quasi-projection pairs on Hilbert $C^*$-modules

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2026年6月2日

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Xiaoyi Tian,
Qingxiang Xu,
Chunhong Fu

摘要

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

Block matrix representations for quasi-projection pairs on Hilbert \(C^*\)-modules

摘要