Star partial order on B(H)

Partial orders onB(H)

“…In [4], using Definition 2.2, the authors proved the analogous result for the star partial order to the result concerning the minus partial order given in [12]: [4] …”

“…In the case when A ∈ B reg (H), using [12, Theorem 2], we can check that Šemrl's definition is equivalent to the definition (4) given by Hartwig. Inspired by Šemrl's definition of the minus partial order, Dolinar et al [4] introduced an equivalent definition for the star partial order on the algebra of all bounded linear operators on a Hilbert space, using orthogonal projections:…”

On mappings preserving the sharp and star orders

“…Following Šemrl's approach, Dolinar and Marovt extended in [16] the star partial order from M n (C) to B(H). From [16,Theorem 5], for T, S ∈ B(H), T ≤ * S if and only if, there exist two selfadjoint idempotent operators P, Q ∈ B(H), such that Im(P ) = Im(T ), Ker(Q) = Ker(T ), P T = P S, and T Q = SQ.…”

On disjoint range operators in a Hilbert space

“…In [39],Šemrl discussed possible extensions of the concept of the minus partial order from matrices to bounded linear operators acting on an infinite-dimensional Hilbert space. Dolinar and Marovt in [20], using orthogonal projectors, introduced the equivalent definition of the star partial order on B(H). And some properties of the generalized concept of order relations on B(H), defined with the help of idempotent operators, are investigated in [20].…”