Bounds for the diamond partial order in ℬ(ℋ)
Lili Yang, Guoxing Ji · Linear and Multilinear Algebra · 2019
Let H be a complex Hilbert space with dim H≥2 and B(H) the algebra of all bounded linear operators on H. Let ≤⋄ be the diamond order on B(H), that is, for A,B∈B(H), we say that A≤⋄B if R(A)¯⊆R(B)¯,R(A∗)¯⊆R(B∗)¯ and AA∗A=AB∗A. We consider the minimal upper bounds and maximal lower bounds of a subset in B(H) with respect to the diamond partial order. It is proved that there exist minimal upper bounds and maximal lower bounds for a nonempty subset with an upper bound. We also give necessary and sufficient conditions for monotone bounded nets to have the supremum as well as infimum.