## 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)]^{-1}T[φ{symbol}(P)]{norm of matrix}<-12 max {{norm of matrix}T{norm of matrix}, {norm of matrix}P^{-1}TP{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)]^{-1}T[φ(N)]{norm of matrix}<-12c^{2} max {tT{norm of matrix}, {norm of matrix}N^{-1}TN{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 language | English |
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Pages (from-to) | 722-729 |

Number of pages | 8 |

Journal | Integral Equations and Operator Theory |

Volume | 15 |

Issue number | 5 |

DOIs | |

State | Published - Sep 1992 |

## Keywords

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