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 = -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...

Let {Tt}t&gt;0 be the semigroup of linear operators generated by a Schrödinger operator -A = Δ - V, where V is a nonnegative potential that belongs to a certain reverse Hölder class. We define a Hardy space HA1 by means of a maximal function associated with the semigroup {Tt}t&gt;0. Atomic and Riesz transforms characterizations of HA1 are shown.

For a Schrödinger operator A = -Δ + V, where V is a nonnegative polynomial, we define a Hardy $H_A^1$ space associated with A. An atomic characterization of $H_A^1$ is shown.

Let ${K_t}_{t>0}$ be the semigroup of linear operators generated by a Schrödinger operator -L = Δ - V with V ≥ 0. We say that f belongs to $H{¹}_L$ if $\left|\right|su{p}_{t>0}|K_t{f\left(x\right)\left|\right||}_{L¹\left(dx\right)}<\infty$. We state conditions on V and $K_t$ which allow us to give an atomic characterization of the space $H{¹}_L$.

We consider the fractional Laplacian $-{(-\Delta )}^{\alpha /2}$ on an open subset in ${ℝ}^d$ with zero exterior condition. We establish sharp two-sided estimates for the heat kernel of such a Dirichlet fractional Laplacian in $C^{1,1}$ open sets. This heat kernel is also the transition density of a rotationally symmetric $\alpha$-stable process killed upon leaving a $C^{1,1}$ open set. Our results are the first sharp twosided estimates for the Dirichlet heat kernel of a non-local operator on open sets.

A Hille-Yosida Theorem is proved on convenient vector spaces, a class, which contains all sequentially complete locally convex spaces. The approach is governed by convenient analysis and the credo that many reasonable questions concerning strongly continuous semigroups can be proved on the subspace of smooth vectors. Examples from literature are reconsidered by these simpler methods and some applications to the theory of infinite dimensional heat equations are given.

Motivated by a great deal of interest recently in operators that may not be densely defined, we deal with regularized semigroups and integrated semigroups that are either not exponentially bounded or not defined on [0,∞) and generated by closed operators which may not be densely defined. Some characterizations and related examples are presented. Our results are extensions of the corresponding results produced by other authors.

A class of Banach spaces of compact operators in Hilbert spaces is introduced, and the holomorphic automorphism groups of the unit balls of these spaces are investigated.