A note on operator norm inequalities

If P is a positive operator on a Hilbert space H whose range is dense, then a theorem of Foias, Ong, and Rosenthal says that: {norm of matrix}[φ{symbol}(P)]^-1T[φ{symbol}(P)]{norm of matrix}<-12 max {{norm of matrix}T{norm of matrix}, {norm of matrix}P^-1TP{norm of matrix}} for any bounded operator T on H, where φ is a continuous, concave, nonnegative, nondecreasing function on [0, {norm of matrix}P{norm of matrix}]. This inequality is extended to the class of normal operators with dense range to obtain the inequality {norm of matrix}[φ(N)]^-1T[φ(N)]{norm of matrix}<-12c² max {tT{norm of matrix}, {norm of matrix}N^-1TN{norm of matrix}} where φ is a complex valued function in a class of functions called vase-like, and c is a constant which is associated with φ by the definition of vase-like. As a corollary, it is shown that the reflexive lattice of operator ranges generated by the range NH of a normal operator N consists of the ranges of all operators of the form φ(N), where φ is vase-like. Similar results are obtained for scalar-type spectral operators on a Hilbert space.