Grant-in-Aid for Scientific Research (C)
|Allocation Type||Single-year Grants |
|Research Institution||FUKUOKA UNIVERSITY OF EDUCATION |
UCHIYAMA Mitsuru FUKUOKA UNIVERSITY OF EDUCATION,FACULTY OF EDU,PROFESSOR, 教育学部, 教授 (60112273)
HARA Takuya FUKUOKA UNIVERSITY OF EDUCATION,FACULTY OF EDU,ASSOCIATE PROFESSOR, 教育学部, 助教授 (50263984)
FUKUTAKE Takayoshi FUKUOKA UNIVERSITY OF EDUCATION,FACULTY OF EDU,PROFESSOR, 教育学部, 教授 (60036887)
|Project Period (FY)
1997 – 1998
Completed (Fiscal Year 1998)
|Budget Amount *help
¥2,400,000 (Direct Cost: ¥2,400,000)
Fiscal Year 1998: ¥1,000,000 (Direct Cost: ¥1,000,000)
Fiscal Year 1997: ¥1,400,000 (Direct Cost: ¥1,400,000)
|Keywords||Operator monotone function / Lowner-Heinz inequality / positive definite operator / norm / operator / matrix / determinant / Korovkin theory / Pick 関数 / 作用素単調関数 / Korovkinの定理 / C^*-環 / 作用素不等式 / 数域半径 / シュワルツノルム|
The research results which we have gotten with a support of GRANT TN AID for SEIENTIFIC RESEARCH (C) are 5 published papers, 3 accepted papers and I submitted paper. The content of them are as follows :
1. For non-negative operators (or matrices) A, B, and for an operator nonotone function f, we had
llf (A) f (B) ll * f (ROO<>llABll)^2
Especially, for the norms of products of logarithmic funtions we had
lllog (1 A) log (1+ B) ll * log (1+ ROO<>llABll)^2
Moreover these are extended to the case of real number powers, so it may be called Minkowski-type inequality. Furthur we investigated and showed that similar Minkowski-type inequality holds for determinants.
Mathematical Inequality and Appl.Vol.1(2)(1998)279-284.
2. Furuta extended the Heinz-Kato Inequality. We extended it as follows :
For operator monotone functions f(t), g(t) >0, T(fg/t)(ITI) is well defined for every T and satisfies T(fg/t)(ITI)x, y ) * (f(ITI)x, x)(g(ITI )y.y) for all vectors x, y
Proc. Amer.Math. Soc.
3. We studied Korovkin theory in C^*-algebras. We found a new inequality whch is very useful to study Korovkin theory. By making use of it, we made clear the the proofs of known theorems and got new Korovkin sets.
4. The function f is called an operator monotone function if for operators A, B
0* f(A) * f(B) whenever 0* A * B
We showed that if f is an operator monotone function and if f is not rational, then f is strongly monotone.