This article introduces several new relations among related Hilbert space
operators. In particular, we prove some L\"{o}wener partial orderings among T,β£Tβ£,RT,IT,β£Tβ£+β£Tββ£ and many other related forms, as a
new discussion in this field; where RT and IT are the
real and imaginary parts of the operator T. Our approach will be based on
proving the positivity of some new matrix operators, where several new forms
for positive matrix operators will be presented as a key tool in obtaining the
other ordering results. As an application, we present some results treating
numerical radius inequalities in a way that extends some known results in this
direction, in addition to some results about the singular values