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

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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.