Properties of the star supremum for arbitrary Hilbert space operators

Simultaneous extension of two bounded operators between Hilbert spaces

“…The last section is comprised of several miniatures, putting our results in a somewhat different perspective: we will explain the relationship with the operator equation XA = B and Douglas' famous theorem, with Halmos' canonical decomposition for two subspaces, and with the theory of partial orders for Hilbert space operators initiated by Drazin. This last relationship was already suggested in [10].…”

“…In the following theorem we prove that two subspaces M and N have the simultaneous extension property for any two operators A and B if and only if M + N is closed. In fact, we will prove that if M + N is not closed, then for any K = {0} we can construct operators A, B ∈ B(H, K) such that C M,N is not bounded, which extends [10,Proposition 2.1]. Note also that "bounded" can be changed to "closable", and so also to "closed", and the theorem still holds.…”

“…For example, it was independently suggested by Gudder [15] as a natural order for bounded quantum observables, but it is also intimately related to a much older notion of *-orthogonality by Hestenes [17]. We will give a brief introduction here, and we refer the reader to [10,11] for more information and further references.…”

“…The star partial order * ≤ is defined as In addressing this problem, the first author introduced the notion of coherent pairs in [10]. For two closed subspaces M, N ⊆ H, and A, B ∈ B(H, K), the pairs (A, M) and (B, N ) are called coherent if C M,N (A, B) is bounded (in the notation of Section 3).…”

Operators with compatible ranges

“…The second reason for such generalization comes from the results of [13,11,14,12] which describe different properties of operators A and B which coincide on R(A * ) ∩ R(B * ) (a generalization of Werener's condition of weak complementarity, see [20]). Accordingly, we will present different properties of CoR operators, regarding range additivity, some additive results for the Moore-Penrose inverse, etc.…”