On Certain Integral Operator Inequalities in Normed Spaces

Abstract

A lot of researches have been carried out on inner product type integral transformers (IPTIT) with regard to various aspects including spectra, numerical ranges and operator inequalities. Consider $M$ and $N$ to be weakly $\mu$-measurable operator valued (OV) functions such that $M,N:\Omega\rightarrow B(X)$ for any $Q\in \mathcal{B}(H).$ If $M$ and $N$ are integrable with respect to Gel'fand axiom, then we obtain a linear transformation arising from the inner product space as $Q\mapsto \int_{\Omega}M_{t}QN_{t}\partial(t).$ There exists an open problem regarding IPTIT while studying inequalities for IPTIT with spectra limited to the unit disc in complex domains. It has been pointed out that the inequalities, and in particular Cauchy-Schwarz (CS) and Cauchy-Buniakowski-Schwarz (CBS) inequalities, can only be attained for these IPTIT if only one of the operator $M$ or $N$ is normal. Therefore, in this note we solve this problem by obtaining CBS-inequalities for IPTIT in Banach spaces.