The methods of arbitrarily high orders of accuracy for the solution of an abstract ordinary differential equation are studied. The right-hand side of the differential equation under investigation contains an unbounded operator which is an infinitesimal generator of a strongly continuous semigroup of operators. Necessary and sufficient conditions are found for a rational function to approximate the given semigroup with high accuracy.

Suppose ${G₁\left(t\right)}_{t\ge 0}$ and ${G₂\left(t\right)}_{t\ge 0}$ are two families of semigroups on a Banach space X (not necessarily of class C₀) such that for some initial datum u₀, G₁(t)u₀ tends towards an undesirable state u*. After remedying by means of an operator ρ we continue the evolution of the state by applying G₂(t) and after time 2t we retrieve a prosperous state u given by u = G₂(t)ρG₁(t)u₀. Here we are concerned with various properties of the semigroup (t): ρ → G₂(t)ρG₁(t). We define (X) to be the space of remedial operators...

Let $B\left(X\right)$ be the algebra of all bounded linear operators in a complex Banach space $X$. We consider operators $T_1,T_2\in B\left(X\right)$ satisfying the relation ${\sigma }_{T_1}\left(x\right)={\sigma }_{T_2}\left(x\right)$ for any vector $x\in X$, where ${\sigma }_T\left(x\right)$ denotes the local spectrum of $T\in B\left(X\right)$ at the point $x\in X$. We say then that $T_1$ and $T_2$ have the same local spectra. We prove that then, under some conditions, $T_1-T_2$ is a quasinilpotent operator, that is $\parallel {(T_1-T_2)}^n{\parallel }^{1/n}\to 0$ as $n\to \infty$. Without these conditions, we describe the operators with the same local spectra only in some particular cases.

In this paper, we shall give sufficient conditions for the ultimate boundedness of solutions for some system of third order non-linear ordinary differential equations of the form $$\stackrel{\text{t}o8pt...}{Xwidth0ptheight5.46pt}+F\left(\ddot{X}\right)+G\left(\dot{X}\right)+H\left(X\right)=P(t,X,\dot{X},\ddot{X})$$ where $X,F\left(\ddot{X}\right)$, $G\left(\dot{X}\right)$, $H\left(X\right)$, $P(t,X,\dot{X},\ddot{X})$ are real $n$-vectors with $F,G$, $H:{ℝ}^n\to {ℝ}^n$ and $P:ℝ\times {ℝ}^n\times {ℝ}^n\times {ℝ}^n\to {ℝ}^n$ continuous in their respective arguments. We do not necessarily require that $F\left(\ddot{X}\right),G\left(\dot{X}\right)$ and $H\left(X\right)$ are differentiable. Using the basic tools of a complete Lyapunov Function, earlier results are generalized.

Given an equibounded (₀)-semigroup of linear operators with generator A on a Banach space X, a functional calculus, due to L. Schwartz, is briefly sketched to explain fractional powers of A. Then the (modified) K-functional with respect to $(X,D\left({(-A)}^{\alpha }\right))$, α > 0, is characterized via the associated resolvent R(λ;A). Under the assumption that the resolvent satisfies a Nikolskii type inequality, ${\left|\right|\lambda R(\lambda ;A)f\left|\right|}_Y{\le c\phi (1/\lambda )\left|\right|f\left|\right|}_X$, for a suitable Banach space Y, an Ulyanov inequality is derived. This will be of interest if one has...

Let $𝔄$ be a $C^{*}$-algebra, $G$ a compact abelian group, $\tau$ an action of $G$ by $*$-automorphisms of $𝔄,{𝔄}^{\tau }$ the fixed point algebra of $\tau$ and ${𝔄}_F$ the dense sub-algebra of $G$-finite elements in $𝔄$. Further let $H$ be a linear operator from ${𝔄}_F$ into $𝔄$ which commutes with $\tau$ and vanishes on ${𝔄}^{\tau }$. We prove that $H$ is a complete dissipation if and only if $H$ is closable and its closure generates a $C_0$-semigroup of completely positive contractions. These complete dissipations are classified in terms of certain twisted negative...