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

在国家助学金(C)资助下，共发表论文5篇，受理论文3篇，提交论文1篇。它们的内容如下：1.对于非负算子(或矩阵)A，B，对于算子非调函数f，我们有llf(A)f(B)ll*f(Roo&lt；&gt；llABll)^2特别地，对于对数函数乘积的范数，我们有lllog(1 A)log(1+B)ll*log(1+Roo&lt；&gt；llABll)^2更多地推广到实数幂的情形，因此可以称之为Minkowski-型不等式。进一步，我们研究并证明了类似的Minkowski型不等式对于行列式也是成立的。数学不等式和应用第一卷(2)(1998)279-284.2。Furuta推广了Heinz-Kato不等式。推广如下：对于算子单调函数f(T)，g(T)&gt；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)。数学。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，我们弄清楚了已有定理的证明，得到了新的Korovkin集。函数f称为算子单调函数，如果对算子A，B0*f(A)*f(B)当0*A*B时，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是强单调函数。

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.