On the $$ \mathop \leqslant \limits^\# $$ -order on the set of linear bounded operators in Banach space

Generalization of sharp and core partial order using annihilators

“…We are now in a position to restate Theorem 3.5. The following result is proved in Theorem 1 in [7]. The only difference is that we have 3 × 3 matrix representations in contrast to 2 × 2 matrix representations obtained in [7].…”

“…The following result is proved in Theorem 1 in [7]. The only difference is that we have 3 × 3 matrix representations in contrast to 2 × 2 matrix representations obtained in [7]. We now proceed to one-sided sharp orders.…”

“…Šemrl proved that his relation coincides with Hartwig's minus partial order when dim X < ∞ and that ≤ − is a partial order relation on B(H). Following Šemrl's approach, Efimov [7] extended the notion of sharp order to B(X) the algebra of bounded linear operators on Banach space X. Let…”

“…The sharp order ≤ is a partial order relation on I X , [7] and this relation is an extension of the sharp partial order from I 1,n to I X .…”