Lipschitz type inequalities for noncommutative perspectives of operator monotone functions in Hilbert spaces

摘要

Assume that f : 1/20; 1THORN ! R is a continuous function. We can define the perspective Pf oB; ATHORN by setting Pf oB; ATHORN :1/4 A1=2f A similar to 1=2BA similar to 1=2 similar to similar to A1=2; where A, B[ 0: We show in this paper among others that Pf B; Po THORN similar to P f A; Po THORN similar to similar to similar to similar to similar to kPk2 kB similar to Ak p2 Pf m2; po THORN similar to P f om1; pTHORN m2 similar to m1 similar to similar to if m1 6 1/4 m2; f 0 m p similar to similar to if m1 1/4 m2 1/4 m8>>><>>>: for all A similar to m1[ 0; B similar to m2[ 0 and P similar to p[ 0. If f is operator monotone on 1/20; 1THORN, then for all C similar to n1[ 0; D similar to n2[ 0; Q[ q[ 0 we also have