Halmos' two projections theorem for Hilbert C⁎-module operators and the Friedrichs angle of two closed submodules

Abstract

Halmos' two projections theorem for Hilbert space operators is one of the fundamental results in operator theory. In this paper, we introduce the term of two harmonious projections in the context of adjointable operators on Hilbert C * -modules, extend Halmos' two projections theorem to the case of two harmonious projections. We also give some new characterizations of the closed submodules and their associated projections. As an application, a norm equation associated to a characterization of the Friedrichs angle is proved to be true in the framework of Hilbert C * -modules.where the notation P E stands for the projection from H onto its closed subspace E. Given any projection P on H, let R(P ) and N (P ) denote the range and the null space of P , respectively. It is proved in [6] that for every two projections P and Q on H,2) which gives a characterization of the Friedrichs angle as c(M, N) = c(M ⊥ , N ⊥ ). (1.3) 2010 Mathematics Subject Classification. 46L08, 47A05.

The Friedrichs angle and alternating projections in Hilbert $C^{*}$-modules

“…To our knowledge, it is an open question whether the intersection of complemented submodules is again complemented. In [LMX,Section 3] it was shown that even in case all the projections exist, (1.1) need not not hold (see Remark 1.2 below).…”

“…The pair (M, N ) being concordant is strictly stronger than the requirement that M ∩N be complemented. In [LMX,Section 3] it is shown that for…”

“…as explained in the discussion after [LMX,Definition 4.1]. Thus (M, N ) is harmonious if and only if each of the pairs (M, N ), (M, N ⊥ ), (M ⊥ , N ) and…”

“…From this point of view, the counterexample of [LMX,Section 3] discussed in Remark 1.2 above arises from the universal example. This shows that specific properties such as being concordant or harmonious hold in some representations of C * (p, q), but not in all of them.…”

Weighted Moore–Penrose Inverses Associated with Weighted Projections on Indefinite Inner Product Spaces

“…It was shown in [4,Theorem 13] that P Q − Q P is Moore-Penrose invertible if and only if both P Q P and P − Q are Moore-Penrose invertible. Later, it was proved in [5,Theorem 11 and Corollary 12] that P Q − Q P is Moore-Penrose invertible if and only if (P Q) k ±(Q P) k are Moore-Penrose invertible for every k ∈ N. An interpretation of the latter characterization was given in [5,Section 3] using Halmos two projections theorem [1,6].…”