Advanced Search

In analogy to the analyticity condition $\parallel A{e}^{tA}\parallel \le C{t}^{-1}$, t > 0, for a continuous time semigroup ${\left(e^{tA}\right)}_{t\ge 0}$, a bounded operator T is called analytic if the discrete time semigroup ${\left(T^n\right)}_{n\in \mathbb{N}}$ satisfies $\parallel (T-I)T^n\parallel \le C{n}^{-1}$, 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 $\parallel R({\lambda }_0,T)-R({\lambda }_0,S)\parallel$ is small enough for some ${\lambda }_0\in \varrho \left(T\right)\cap \varrho \left(S\right)$, then the type $\omega$ of the semigroup $\left(e^{t(S-I)}\right)$ also controls the analyticity of S in the sense that $\parallel (S-I)S^n\parallel \le C(\omega +n^{-1})e^{\omega n}$, n ∈ ℕ. As an application we generalize...

We consider the maximal regularity problem for the discrete time evolution equation $u_{n+1}-Tuₙ=fₙ$ for all n ∈ ℕ₀, u₀ = 0, where T is a bounded operator on a UMD space X. We characterize the discrete maximal regularity of T by two types of conditions: firstly by R-boundedness properties of the discrete time semigroup ${\left(Tⁿ\right)}_{n\in \mathbb{N}₀}$ and of the resolvent R(λ,T), secondly by the maximal regularity of the continuous time evolution equation u’(t) - Au(t) = f(t) for all t > 0, u(0) = 0, where A:= T - I. By recent results of...

Hörmander’s famous Fourier multiplier theorem ensures the $L_p$-boundedness of $F(-{\Delta }_{ℝ}D)$ whenever $F\in \mathscr{H}\left(s\right)$ for some $s\>\frac{D}{2}$, where we denote by $\mathscr{H}\left(s\right)$ the set of functions satisfying the Hörmander condition for $s$ derivatives. Spectral multiplier theorems are extensions of this result to more general operators $A\ge 0$ and yield the $L_p$-boundedness of $F\left(A\right)$ provided $F\in \mathscr{H}\left(s\right)$ for some $s$ sufficiently large. The harmonic oscillator $A=-{\Delta }_{ℝ}+x^2$ shows that in general $s\>\frac{D}{2}$ is not sufficient even if $A$ has a heat kernel satisfying gaussian estimates. In this paper,...