## Abstract

Let P_{t} (a ≤ t ≤ b) be a function whose values are projections in a Banach space. The paper is devoted to bounded linear operators A admitting the representation A = ∫ _{a} ^{b} ϕ(t)dP_{t} + V, where ϕ(t) is a scalar function and V is a compact quasi-nilpotent operator such that P_{t}V P_{t} = V P_{t} (a ≤ t ≤ b). We obtain norm estimates for the resolvent of A and a bound for the spectral variation of A. In addition, the representation for the resolvents of the considered operators is established via multiplicative operator integrals. That representation can be considered as a generalization of the representation for the resolvent of a normal operator in a Hilbert space. It is also shown that the considered operators are Kreiss-bounded. Applications to integral operators in L^{p} are also discussed. In particular, bounds for the upper and lower spectral radius of integral operators are derived.

Original language | English |
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Pages (from-to) | 113-139 |

Number of pages | 27 |

Journal | Advances in Operator Theory |

Volume | 4 |

Issue number | 1 |

DOIs | |

State | Published - 1 Jan 2019 |

## Keywords

- Banach space
- Integral operator
- Invariant chain of projections
- Resolvent
- Spectral variation