A note on operator norm inequalities |
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Authors: | Richard J. Fleming Sivaram K. Narayan Sing-Cheong Ong |
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Affiliation: | (1) Department of Mathematics, Central Michigan University, 48859 Mount Pleasant, MI, U.S.A. |
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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: [(P)]–1T[(P)]<-12 max {T, P–1TP} for any bounded operator T on H, where 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 [(N)]–1T[(N)]<-12c2 max {tT, N–1TN} where is a complex valued function in a class of functions called vase-like, and c is a constant which is associated with 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 (N), where is vase-like. Similar results are obtained for scalar-type spectral operators on a Hilbert space.This author gratefully acknowledges the support of Central Michigan University in the form of a Research Professorship. |
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Keywords: | Primary, 47 A 30 Secondary, 47 A 50 |
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