Harmonious projections and Halmos' two projections theorem for Hilbert C⁎-module operators

Xu, Qingxiang; Yan, Guanjie

doi:10.1016/j.laa.2020.05.012

Cited by 5 publications

(2 citation statements)

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“…Let P, Q ∈ P(H) be such that (P, Q) is harmonious. In this case, the Halmos' two projections theorem [12,Theorem 3.3] indicates that up to unitary equivalence, P and Q have the block matrix forms…”

Section: Unitary Equivalences Associated With Two Projectionsmentioning

confidence: 99%

“…To check the validity of the Halmos' two projections theorem in the Hilbert C * -module case, the term of the harmonious pair of projections is introduced in [7, Section 4], and it is shown later in [12,Theorem 3.3] that for every pair (P, Q) of projections, the Halmos' two projections theorem is valid if and only if (P, Q) is harmonious. In view of the conditions stated in [7, Lemma 5.4] and [12,Theorem 3.3], we make a definition as follows.…”

Section: Introductionmentioning

confidence: 99%

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

Fu¹,

Xu²,

Gao³

2022

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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 ∧ Q exists as an element of L(H), where L(H) denotes the set of all adjointable operators on H. The C * -subalgebras of L(H) generated by elements in {P − P ∧ Q, Q − P ∧ Q, I} and {P, Q, P ∧ 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 ( π, X) of i(P, Q, H) such thatWhen (P, Q) is semi-harmonious, that is, R(P + Q) and 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 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.

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Section: Unitary Equivalences Associated With Two Projectionsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%