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Perturbation of analytic operators and temporal regularity of discrete heat kernels

Sönke Blunck (2000)

Colloquium Mathematicae

In analogy to the analyticity condition ∥ A e t A ∥ ≤ C t - 1 , t > 0, for a continuous time semigroup ( e t A ) t ≥ 0 , a bounded operator T is called analytic if the discrete time semigroup ( T n ) n ∈ ℕ satisfies ∥ ( T - I ) T n ∥ ≤ C n - 1 , n ∈ ℕ. We generalize O. Nevanlinna’s characterization of powerbounded and analytic operators T to the following perturbation result: if S is a perturbation of T such that ∥ R ( λ 0 , T ) - R ( λ 0 , S ) ∥ is small enough for some λ 0 ∈ ϱ ( T ) ∩ ϱ ( S ) , then the type ω of the semigroup ( e t ( S - I ) ) also controls the analyticity of S in the sense that ∥ ( S - I ) S n ∥ ≤ C ( ω + n - 1 ) e ω n , n ∈ ℕ. As an application we generalize...

Perturbation of the spectrum οf an essentially selfadjoint operator

Andrzej Pokrzywa (1993)

Applicationes Mathematicae

The aim of this paper is to find estimates of the Hausdorff distance between the spectra of two nonselfadjoint operators. The operators considered are assumed to have their imaginary parts in some normed ideal of compact operators. In the case of the classical Schatten ideals the estimates are given explicitly.

Perturbation theorems for local integrated semigroups

Chung-Cheng Kuo (2010)

Studia Mathematica

We apply the contraction mapping theorem to establish some bounded and unbounded perturbation theorems concerning nondegenerate local α-times integrated semigroups. Some unbounded perturbation results of Wang et al. [Studia Math. 170 (2005)] are also generalized. We also establish some growth properties of perturbations of local α-times integrated semigroups.

Perturbation theory relative to a Banach algebra of operators

Bruce Barnes (1993)

Studia Mathematica

Let ℬ be a Banach algebra of bounded linear operators on a Banach space X. Let S be a closed linear operator in X, and let R be a linear operator in X. In this paper the spectral and Fredholm theory relative to ℬ of the perturbed operator S + R is developed. In particular, the situation where R is S-inessential relative to ℬ is studied. Several examples are given to illustrate the usefulness of these concepts.

Perturbations of bi-continuous semigroups

Bálint Farkas (2004)

Studia Mathematica

The notion of bi-continuous semigroups has recently been introduced to handle semigroups on Banach spaces that are only strongly continuous for a topology coarser than the norm-topology. In this paper, as a continuation of the systematic treatment of such semigroups started in [20-22], we provide a bounded perturbation theorem, which turns out to be quite general in view of various examples.

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