We show that the growth rates of solutions of the abstract differential equations ẋ(t) = Ax(t), $ẋ\left(t\right)=A^{-1}x\left(t\right)$, and the difference equation $x_d(n+1)=(A+I){(A-I)}^{-1}{x}_d\left(n\right)$ are closely related. Assuming that A generates an exponentially stable semigroup, we show that on a general Banach space the lowest growth rate of the semigroup ${\left(e^{A^{-1}t}\right)}_{t\ge 0}$ is O(∜t), and for $\left((A+I){(A-I)}^{-1}\right)ⁿ$ it is O(∜n). The similarity in growth holds for all Banach spaces. In particular, for Hilbert spaces the best estimates are O(log(t)) and O(log(n)), respectively. Furthermore, we give conditions...

It will be proved that if $N$ is a bounded nilpotent operator on a Banach space $X$ of order $k+1$, where $k\ge 1$ is an integer, then the $\gamma$-th order Cesàro mean $C_t^{\gamma }:=\gamma t^{-\gamma }{\int }_0^t{(t-s)}^{\gamma -1}T\left(s\right)ds$ and Abel mean $A_{\lambda }:=\lambda {\int }_0^{\infty }{e}^{-\lambda s}T\left(s\right)ds$ of the uniformly continuous semigroup ${\left(T\left(t\right)\right)}_{t\ge 0}$ of bounded linear operators on $X$ generated by $iaI+N$, where $0\ne a\in ℝ$, satisfy that (a) $\parallel C_t^{\gamma }\parallel \sim t^{k-\gamma }(t\to \infty )$ for all $0<\gamma \le k+1$; (b) $\parallel C_t^{\gamma }\parallel \sim t^{-1}(t\to \infty )$ for all $\gamma \ge k+1$; (c) $\parallel A_{\lambda }\parallel \sim \lambda (\lambda \downarrow 0)$. A similar result will be also proved for the uniformly continuous cosine function ${\left(C\left(t\right)\right)}_{t\ge 0}$ of bounded linear operators on $X$ generated by ${(iaI+N)}^2$.

We characterise the boundedness of the $H^{\infty }$ calculus of a sectorial operator in terms of dilation theorems. We show e. g. that if $-A$ generates a bounded analytic $C_0$ semigroup $\left(T_t\right)$ on a UMD space, then the $H^{\infty }$ calculus of $A$ is bounded if and only if $\left(T_t\right)$ has a dilation to a bounded group on $L^2([0,1],X)$. This generalises a Hilbert space result of C.LeMerdy. If $X$ is an $L^p$ space we can choose another $L^p$ space in place of $L^2([0,1],X)$.

Let $A$ be the $L^p$ realization ($1\&lt;p\&lt;\infty$) of a differential operator $P(D_x,D_t)$ on ${ℝ}^n\times {ℝ}^{+}$ with general boundary conditions $B_k(D_x,D_t)u(x,0)=0$ ($1\le k\le m$). Here $P$ is a homogeneous polynomial of order $2m$ in $n+1$ complex variables that satisfies a suitable ellipticity condition, and for $1\le k\le m$$B_k$ is a homogeneous polynomial of order...

We give a concise exposition of the basic theory of $H^{\infty }$ functional calculus for N-tuples of sectorial or bisectorial operators, with respect to operator-valued functions; moreover we restate and prove in our setting a result of N. Kalton and L. Weis about the boundedness of the operator $f(T₁,...,T_N)$ when f is an R-bounded operator-valued holomorphic function.

Let A be a linear closed densely defined operator in a complex Banach space X. If A is of type ω (i.e. the spectrum of A is contained in a sector of angle 2ω, symmetric around the real positive axis, and $\parallel \lambda {(\lambda I-A)}^{-1}\parallel$ is bounded outside every larger sector) and has a bounded inverse, then A has a bounded $H^{\infty }$ functional calculus in the real interpolation spaces between X and the domain of the operator itself.

Let A be a linear closed one-to-one operator in a complex Banach space X, having dense domain and dense range. If A is of type ω (i.e.the spectrum of A is contained in a sector of angle 2ω, symmetric about the real positive axis, and ${\left|\right|\lambda (\lambda I-A)}^{-1}\left|\right|$ is bounded outside every larger sector), then A has a bounded $H^{\infty }$ functional calculus in the real interpolation spaces between X and the intersection of the domain and the range of the operator itself.

Let A = -Δ + V be a Schrödinger operator on ${ℝ}^d$, d ≥ 3, where V is a nonnegative potential satisfying the reverse Hölder inequality with an exponent q > d/2. We say that f is an element of $H_A^p$ if the maximal function $su{p}_{t>0}\left|T_{t}f\left(x\right)\right|$ belongs to $L^p\left({ℝ}^d\right)$, where ${T_t}_{t>0}$ is the semigroup generated by -A. It is proved that for d/(d+1) < p ≤ 1 the space $H_A^p$ admits a special atomic decomposition.

Let (X) be the algebra of all bounded operators on a Banach space X, and let θ: G → (X) be a strongly continuous representation of a locally compact and second countable abelian group G on X. Set σ¹(θ(g)): = λ/|λ| | λ ∈ σ(θ(g)), where σ(θ(g)) is the spectrum of θ(g), and let ${\Sigma }_{\theta }$ be the set of all g ∈ G such that σ¹(θ(g)) does not contain any regular polygon of (by a regular polygon we mean the image under a rotation of a closed subgroup of the unit circle different from 1). We prove that θ is uniformly...