SCHWARZ NORMS IN OPERATOR ALGEBRAS AND CONTRACTIONS, AND ITS APPLICATIONS TO DIFFERENTIAL OPERATORS

算子代数和收缩中的施瓦茨范数及其在微分算子中的应用

• 批准号: 09640200
• 负责人: UCHIYAMA Mitsuru
• 金额: $ 1.54万
• 依托单位: FUKUOKA UNIVERSITY OF EDUCATION
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (C)
• 财政年份: 1997
• 资助国家: 日本
• 起止时间: 1997 至 1998
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-09640200/
• 关键词: 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 hadllf (A) f (B) ll * f (ROO<>llABll)^2Especially, for the norms of products of logarithmic funtions we hadlllog (1 A) log (1+ B) ll * log (1+ ROO<>llABll)^2Moreover 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, yProc. 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.Mathinatishe Zeitshrift4. The function f is called an operator monotone function if for operators A, B0* f(A) * f(B) whenever 0* A * BWe showed that if f is an operator monotone function and if f is not rational, then f is strongly monotone.

我们得到了科学研究资助（C）的研究成果是5篇发表的论文，3篇接受的论文和我提交的论文。主要内容如下：1.对于非负算子（或矩阵）A，B，以及算子非单调函数f，我们有ll f（A）f（B）ll * f（ROO<>llABll）^2特别地，对于对数函数乘积的范数，我们有ll log（1 A）log（1+ B）ll * log（1+ ROO<>llABll）^2而且这些不等式被推广到真实的数幂的情形，因此它可称为Minkowski型不等式。进一步研究并证明了类似的Minkowski型不等式对行列式也成立.数学不等式与应用第1卷（2）（1998）279-284.2.古田推广了Heinz-Kato不等式。对算子单调函数f（t），g（t）>0，T（fg/t）（ITI）对每个T都有良好的定义，且对所有向量x，yProc都满足T（fg/t）（ITI）x，y）*（f（ITI）x，x）（g（ITI）y，y）. Amer. Math.Soc.3.我们研究了C^*-代数中的Korovkin理论。我们发现了一个新的不等式，它对研究Korovkin理论非常有用.利用它，我们澄清了已知定理的证明，并得到了新的Korovkin集。称函数f为算子单调函数，如果对算子A，B 0 * f（A）* f（B），只要0* A * B，我们证明了如果f是算子单调函数，且f不是有理函数，则f是强单调的.

项目成果

M.Uchiyama: "Norms and determinants of products of logarithmic functions" Math. Inequal.and Appl.1. 279-284 (1998)

M.Uchiyama：“对数函数乘积的范数和行列式”数学。

M.Uchiyama: "Korovkin-type theorems for Schwarz maps and operator monotone functions" Math.Z.(印刷中).

M. Uchiyama：“Schwarz 映射和算子单调函数的 Korovkin 型定理”Math.Z（正在出版）。

M.Uchiyama: "Further Extension of Heinz-Kato-Furuta Ineguality" Proc.American Math.Soc.校正済み.

M. Uchiyama：“Heinz-Kato-Furuta 不等式的进一步扩展”Proc。

M.Uchiyama: "Heinz-Kato-Furuta lnequalitty" Proc,Amer.Math.SOC.(受理).

M.Uchiyama：“Heinz-Kato-Furuta 不等式”Proc，Amer.Math.SOC.（已接受）。

T.Hara,M.Uchiyama: "A retinement of Vari'ous Mean Inegualities" Journal of Inequalities and Applications. 2巻. 387-395 (1998)

T.Hara，M.Uchiyama：“各种平均不等式的保留”《不平等与应用杂志》第 2 卷，387-395（1998 年）。