A note on operator norm inequalities

Abstract

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: ‖[ϕ(P)]⁻¹T[ϕ(P)]‖<-12 max {‖T‖, ‖P⁻¹TP‖} for any bounded operator T on H, where φ is a continuous, concave, nonnegative, nondecreasing function on [0, ‖P‖]. This inequality is extended to the class of normal operators with dense range to obtain the inequality ‖[φ(N)]⁻¹T[φ(N)]‖<-12c² max {tT‖, ‖N⁻¹TN‖} 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.