A new majorization between functions, polynomials, and operator inequalities II

Abstract

Let PðIÞ be the set of all operator monotone functions defined on an interval I, and put P þ ðIÞ ¼ fh 2 PðIÞ : hðtÞ = 0; h 6 ¼ 0g and P À1 þ ðIÞ = {h : h is increasing on I; h À1 2 P þ ð0; 1Þ}. We will introduce a new set LP þ ðIÞ ¼ fh : hðtÞ > 0 on I; log h 2 PðIÞg and show LP þ ðIÞ Á P À1 þ ðIÞ & P À1 þ ðIÞ for every right open interval I. By making use of this result, we will establish an operator inequality that generalizes simultaneously two well known operator inequalities. We will also show that if pðtÞ is a real polynomial with a positive leading coefficient such that pð0Þ ¼ 0 and the other zeros of p are all in fz : Rz 5 0g and if qðtÞ is an arbitrary factor of pðtÞ, then pðAÞ 2 5 pðBÞ 2 for A; B = 0 implies A 2 5 B 2 and qðAÞ 2 5 qðBÞ 2 .

Operator monotone functions, positive definite kernels and majorization

“…g (t, s), it is positive definite too. The following is a part of the Product Lemma given in [16] and [17], but we give a simple proof for the sake of completeness. …”

“…Recall the symbol " " introduced in [16,17]: let h(t) be a non-decreasing continuous function on I and g(t) an increasing continuous function on I; then h is said to be majorized by g and denoted by h g on I if h(A) h(B) whenever g(A) g(B) for A, B whose spectra are both in I. It is clear that f (t) t on I means that f (t) is operator monotone on I.…”

Inequality between unitary orbits

“…We remark that the last statement does not imply that there is a common unitary U such that A a U * B a U for every a > 0, namely A u B. To extend this proposition, we use a few symbols induced in [16] + (I) the set of all increasing and continuous functions h(t) on I such that lim t→a+0 h(t) = 0, lim t→b−0 h(t) = ∞ and h −1 is operator monotone. Let LP + (I) denote the set of all functions h(t) such that h(t) > 0 on the interior of I and log h(t) is operator monotone there.…”

“…Then by ( 4) there is a unitary U such that A a e b A U * B a e bB U . Since for each i there is an operator monotone function φ i such that t a i e b i t = φ i (t a e bt ) (see Theorem 2.11 of [16]) we obtain:…”