On the numerical radii of certain operator forms

The main goal of this paper is to present new bounds for the norms and numerical radii of certain Hilbert space operators, which involve the sums of operators' products. The obtained result will be utilized to obtain some equivalent conditions regarding certain operator identities, to find upper bounds for the sum of two operators, to obtain certain refinements of celebrated numerical radius bounds, and to present a new reverse for the triangle inequality when positive operators are treated.