MOND-PECARIC METHOD FOR A MEAN-LIKE TRANSFORMATION OF OPERATOR FUNCTIONS
Akemi Matsumoto, Masaru Tominaga · Scientiae mathematicae Japonicae · 2005
As a generalization of the quasi-arithmetic mean, we consider a mean-like transformation of operator functions. Let Φ be a unital positive linear map of B(H), the algebra of all bounded linear operators on a Hilbert space H , and f(t) (resp. g(t)) a continuous function on an interval [m,M ] (resp. f([m,M ])). Then it is defined by (g ◦ Φ ◦ f)(A) for a selfadjoint operator A with m ≤ A ≤ M . We give a lower bound of the difference between (g ◦ Φ ◦ f)(A) and Φ(A). Precisely we prove that if f(t) is concave on [m,M ] and g(t) is increasing and convex on f([m,M ]), then for each λ ∈ , (g ◦ Φ ◦ f)(A)− λΦ(A) ≥ mint∈[m,M ] {g(αf t+ βf )− λt} where αf := f(M)−f(m) M−m and βf := Mf(m)−mf(M) M−m . It is an extension of our previous estimation for Φ = ωx, the vector state for a unit vector x ∈ H .