AN OPERATOR EXTENSION OF THE PARALLELOGRAM LAW AND RELATED NORM INEQUALITIES

作者：Moslehian Mohammad Sal^*

来源：Mathematical Inequalities & Applications, 2011, 14(3): 717-725.

DOI：10.7153/mia-14-60

• Bochner integral
• Schatten p-norm
• norm inequality
• parallelogram law
• unitarily invariant norm
• convex function
• Bohr's inequality
• inner product space

We establish a general operator parallelogram law concerning a characterization of inner product spaces, get an operator extension of Bohr's inequality and present several norm inequalities. More precisely, let (sic) be a C*-algebra, T be a locally compact Hausdorff space equipped with a Radon measure mu and let (At)(t is an element of T) be a continuous field of operators in (sic) such that the function t bar right arrow A(t) is norm continuous on T and the function t bar right arrow parallel to A(t)parallel to is integrable. If alpha : T x T -> C is a measurable function such that <(alpha(t, s))over bar>alpha(s, t) = 1 for all t, s is an element of T, then we show that
integral(T)integral(T)vertical bar alpha(t,s)A(t) - alpha(s,t)A(s)vertical bar(2)d mu(t)d mu(s) + integral(T)integral(T)vertical bar alpha(t,s)B-t - alpha(s,t)B-s vertical bar(2)d mu(t)d mu(s)
-2 integral(T)integral(T)vertical bar alpha(t,s)A(t) - alpha(s,t)B-s vertical bar(2)d mu(t)d mu(s) - 2 vertical bar integral(T)(A(t) - B-t)d mu(t)vertical bar(2).

• 出版日期2011-7