Cauchy–Schwarz Operator and Norm Inequalities for Inner Product Type Transformers in Norm Ideals of Compact Operators, with Applications

Abstract

In this survey paper we present operator and norm inequalities of Cauchy–Schwarz type:

$$\displaystyle \begin{aligned} \bigg \vert {\!\bigg \vert {\:\!\int _{\Omega }\! A_t X B_t \,d\mu (t)\:\!}\bigg \vert \!}\bigg \vert _{\Psi } \!\!\:\!\le \bigg \vert {\!\bigg \vert {\:\!\!\bigg ({\int _{\Omega } \! A_t^* A_t \,d\mu (t)\!\;\!\!}\bigg )^{\:\!{\!\:\! 1/2}} \!\;\!\! X\!\:\! \bigg ({\int _{\Omega } \! B_t B_t^*\,d\mu (t)\!\;\!\!}\bigg )^{\:\!{\!\:\! 1/2}}\:\!}\bigg \vert \!}\bigg \vert _{\Psi }\!, \end{aligned} $$

for strongly square integrable operator families ${\{A_t\}_{t\in \Omega }},{\{B_t^*\}_{t\in \Omega }}$ and symmetrically norming functions Ψ, such that the associated unitarily invariant norm is nuclear, Q^∗ or arbitrary, under some additional commutativity conditions. The applications of this and complementary inequalities for Q and Schatten–von Neumann norms to Aczél–Bellman, Grüss–Landau, arithmetic–geometric, Young, Minkowski, Heinz, Zhan, Heron, and generalized derivation norm inequalities are also presented.