Operator functions and the operator harmonic mean

Abstract

The objective of this paper is to investigate operator functions by making use of the operator harmonic mean ‘ ! \,!\, ’. For 0 > A ≦ B 0>A\leqq B , we construct a unique pair X X , Y Y such that 0 > X ≦ Y , A = X ! Y , B = X + Y 2 0>X\leqq Y, \; A=X\,!\,Y,\; B=\frac {X+Y}{2} . We next give a condition for operators A , B , C ≧ 0 A, B, C\geqq 0 in order that C ≦ A !   B C \leqq A\,!\ B and show that g ≠ 0 ge 0 is strongly operator convex on J J if and only if g ( t ) > 0 g(t)>0 and g ( A + B 2 ) ≦ g ( A ) ! g ( B ) g (\frac {A+B}{2}) \leqq g(A)\,!\,g(B) for A , B A, B with spectra in J J . This inequality particularly holds for an operator decreasing function on the right half line. We also show that f ( t ) f(t) defined on ( 0 , b ) (0, b) with 0 > b ≦ ∞ 0>b\leqq \infty is operator monotone if and only if f ( 0 + ) > ∞ , f ( A ! B ) ≦ 1 2 ( f ( A ) + f ( B ) ) f(0+)>\infty , \;f (A\,!\,B)\leqq \frac {1}{2}(f(A) + f(B)) . In particular, if f > 0 f>0 , then f f is operator monotone if and only if f ( A ! B ) ≦ f ( A ) ! f ( B ) f (A\, !\, B) \leqq f(A)\, !\, f(B) . We lastly prove that if a strongly operator convex function g ( t ) > 0 g(t)>0 on a finite interval ( a , b ) (a, b) is operator decreasing, then g g has an extension g ~ \tilde {g} to ( a , ∞ ) (a, \infty ) that is positive and operator decreasing.