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 ($1\<p\<\infty$) 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...

The existence of solutions of Hammerstein equations in the space of bounded and continuous functions is proved. It is obtained by the Schauder fixed point theorem using a compactness theorem. The result is applied to Wiener-Hopf equations and to ODE's.

The paper studies a construction of nontrivial solution for a class of Hammerstein–Nemytskii type nonlinear integral equations on half-line with noncompact Hammerstein integral operator, which belongs to space $L_1(0,+\infty )\cap L_{\infty }(0,+\infty )$. This class of equations is the natural generalization of Wiener-Hopf type conservative integral equations. Examples are given to illustrate the results. For one type of considering equations continuity and uniqueness of the solution is established.

It is an open question whether Nehari's theorem on the circle group has an analogue on the infinite-dimensional torus. In this note it is shown that if the analogue holds, then some interesting inequalities follow for certain trigonometric polynomials on the torus. We think these inequalities are false but are not able to prove that.