Integr Equat Oper Th 0378-620X/92/050722-0851.50+0.20/0 Vol. 15 (1992) (c) 1992 Birkh~user Verlag, Basel A NOTE ON OPERATOR NORM INEQUALITIES Richard J. Fleming, Sivaram K. Narayan and Sing-Cheong Ong 1 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: II[qo(P)]-lT[tp(P)]ll < 12 max{llTII, IIp-1TPII} for any bounded operator T on H, where q~ is a continuous, concave, nonnegative, nondecreasing function on [0, IIPII]. This inequality is extended to the class of normal operators with dense range to obtain the inequality II[tp(N)]-lT[tp(N)]ll < 12c 2 max{llTII, IIN-ITNII} where tp is a complex valued function in a class of functions called vase-like, and c is a constant which is associated with q~ 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 tp(N), where q0is vase-like. Similar results are obtained for scalar-type spectral operators on a Hilbert space, INTRODUCTION It is well known that the range of a bounded operator T on a Hilbert space H is the same as the range of the positive operator (YF*) ~/2 [1]. Thus, most of the work on operator ranges deal with ranges of positive operators. This has led to a proof of operator norm inequalities [3] which arose from interpolation norms [6]. To be precise, if P is a positive operator on H with dense range and if tp is a continuous, concave, nonnegative, nondecreas- ing function on [0, IIPIE] (resp. [0, IIPII2]) then for any bounded operator T, w 12 max {IITII, lip -1 T vii} (resp. Ilia(P=)] r max {][T[[, lIP-1T vii} ), where the norm of an unbounded operator is taken to be ~. It is natural at this point to consider instead of a positive operator something more 1 This author gratefully acknowledges the support of Central Michigan University in the form of a Research Professorship.