A note on operator norm inequalities

Richard J. Fleming, Sivaram K. Narayan, Sing Cheong Ong

Research output: Contribution to journalArticlepeer-review

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: {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.

Original languageEnglish
Pages (from-to)722-729
Number of pages8
JournalIntegral Equations and Operator Theory
Volume15
Issue number5
DOIs
StatePublished - Sep 1992

Keywords

  • MSC: Primary, 47A30, Secondary, 47A50

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