Mitsuru Uchiyama
Shimane University
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Featured researches published by Mitsuru Uchiyama.
Proceedings of the American Mathematical Society | 2005
Mitsuru Uchiyama
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.
Proceedings of the American Mathematical Society | 2010
Mitsuru Uchiyama
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.
Transactions of the American Mathematical Society | 1990
Mitsuru Uchiyama
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.
Transactions of the American Mathematical Society | 2003
Mitsuru Uchiyama
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 .
Integral Equations and Operator Theory | 2002
Mitsuru Uchiyama; Morisuke Hasumi
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.
Proceedings of the American Mathematical Society | 1999
Mitsuru Uchiyama
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.
Proceedings of the American Mathematical Society | 2012
Mitsuru Uchiyama
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)].
Proceedings of the American Mathematical Society | 2003
Kotaro Tanahashi; Atsushi Uchiyama; Mitsuru Uchiyama
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.
Integral Equations and Operator Theory | 2000
Mitsuru Uchiyama
AbstractLetA, B be bounded selfadjoint operators on a Hilbert space. We will give a formula to get the maximum subspacen
American Mathematical Monthly | 2003
Mitsuru Uchiyama