Mitsuru Uchiyama

Subadditivity of eigenvalue sums

Let f(t) be a nonnegative concave function on 0 < t < ∞ with f(0) = 0, and let X, Y be n×n matrices. Then it is known that ||f(|X+Y|)|| 1 ≤ ||f(|X|)|| 1 + ||f(|Y|)|| 1 , where ||·|| 1 is the trace norm. We extend this result to all unitarily invariant norms and prove some inequalities of eigenvalue sums.

Operator monotone functions, positive definite kernels and majorization

Let be a real continuous function on an interval, and consider the operator function defined for Hermitian operators . We will show that if is increasing w.r.t. the operator order, then for the operator function is convex. Let and be functions defined on an interval . Suppose is non-decreasing and is increasing. Then we will define the continuous kernel function by , which is a generalization of the Lowner kernel function. We will see that it is positive definite if and only if whenever for Hermitian operators , and we give a method to construct a large number of infinitely divisible kernel functions.

Curvatures and similarity of operators with holomorphic eigenvectors

The curvature of the holomorphic vector bundle generated by eigenvectors of operators is estimated, and the necessary and sufficient conditions for contractions to be similar or quasi-similar with unilateral shifts are given.

Inverse functions of polynomials and orthogonal polynomials as operator monotone functions

We study the operator monotonicity of the inverse of every polynomial with a positive leading coefficient. Let {p n }∞n=o be a sequence of orthonormal polynomials and p n+ the restriction of p n to [a n , ∞), where a n is the maximum zero of p n . Then p -1 n+ and the composite p n-1 o p -1 n+ are operator monotone on [0, ∞). Furthermore, for every polynomial p with a positive leading coefficient there is a real number a so that the inverse function of p(t+a)-p(a) defined on [0, oo) is semi-operator monotone, that is, for matrices A, B > 0, (p(A+a)-p(a)) 2 ≤ ((p(B+a)-p(a)) 2 implies A 2 ≤ B 2 .

On some operator monotone functions

We try to find a continuous functionu defined on a real right half-line with the range (0, ∞) such thatu−1 is operator monotone. We then look for another functionv such thatv(u−1) is operator monotone, namely,u(A)≦u(B) impliesv(A)≦v(B) for self-adjoint operatorsA andB.

Further extension of the Heinz-Kato-Furuta inequality

Let T be a bounded operator on a Hilbert space H, and A, B positive definite operators. Kato has shown that if ‖Tx‖ ≤ ‖Ax‖ and ‖T ∗y‖ ≤ ‖By‖ for all x, y ∈ H, then |(Tx, y)| ≤ ||f(A)x|| ||g(B)y||, where f(t), g(t) are operator monotone functions defined on [0,∞) such that f(t)g(t) = t. Furuta has shown that |(T |T |α+β−1x, y)| ≤ ||Aαx|| ||Bβy||, where 0 ≤ α, β ≤ 1, 1 ≤ α + β. Let f(t), g(t) be any continuous operator monotone functions, and set h(t) = f(t)g(t)/t for t > 0. We will show that Th(|T |) is well defined and |(Th(|T |)x, y)| ≤ ||f(A)x|| ||g(B)y||. Moreover, we will extend this result for unbounded closed operators densely defined on H.

The principal inverse of the gamma function

Let Γ(x) be the gamma function in the real axis and α the maximal zero of Γ′(x). We call the inverse function of Γ(x)|(α,∞) the principal inverse and denote it by Γ−1(x). We show that Γ−1(x) has the holomorphic extension Γ−1(z) to C (−∞,Γ(α)], which maps the upper half-plane into itself, namely a Pick function, and that Γ(Γ−1(z)) = z on C (−∞,Γ(α)].

On Schwarz type inequalities

We show Schwarz type inequalities and consider their converses. A continuous function f : [0, oo) → [0, oo) is said to be semi-operator monotone on (a, b) if {f(t1/2)} 2 is operator monotone on (a 2 ,b 2 ). Let T be a bounded linear operator on a complex Hilbert space H and T = U|T| be the polar decomposition of T. Let 0 ≤ A, B E B(H) and ∥Tx∥ ≤ ∥Ax∥, ∥T*y∥ ≤ ∥By∥ for x,y E H. (1) If a non-zero function f is semi-operator monotone on (0,∞), then |(Tx,y)| ≤ ∥f(A)x∥ ∥g(B)y∥ for x,y E H, where g(t) = t/f(t). (2) If f,g are semi-operator monotone on (0, ∞), then |(Uf(|T|)g(|T|)x, y)| ≤ ∥f(A)x∥ ∥g(B)y∥ for x, y ∈ H. Also, we show converses of these inequalities, which imply that semi-operator monotonicity is necessary.

Strong monotonicity of operator functions

AbstractLetA, B be bounded selfadjoint operators on a Hilbert space. We will give a formula to get the maximum subspacen