This paper develops Kato-type inequalities for bounded operators on Hilbert spaces by using the Moore–Penrose inverse of closed range operators. The results extend classical forms of Kato’s inequality to products involving a closed range operator and additional bounded operator factors, and they provide flexible estimates controlled by an interpolation parameter. Consequences are obtained for the operator norm and for the numerical radius, including power-type numerical radius bounds. Several special cases and applications are presented, showing that the Moore–Penrose inverse gives a unified framework for deriving refined inequalities for Hilbert space operators.
Altwaijry et al. (Tue,) studied this question.