### A characterization for the spectral capacity of a finite system of operators

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We give a new constructive proof of the composition rule for Taylor's functional calculus for commuting operators on a Banach space.

For a holomorphic function ψ defined on a sector we give a condition implying the identity ${(X,\left({A}^{\alpha}\right))}_{\theta ,p}=x\in X|{t}^{-\theta Re\alpha}\psi \left(tA\right)\in L{\u204e}^{p}((0,\infty );X)$ where A is a sectorial operator on a Banach space X. This yields all common descriptions of the real interpolation spaces for sectorial operators and allows easy proofs of the moment inequalities and reiteration results for fractional powers.

If -A is the generator of an equibounded ${C}_{0}$-semigroup and 0 < Re α < m (m integer), its fractional power ${A}^{\alpha}$ can be described in terms of the semigroup, through a formula that is only valid if a certain function ${K}_{\alpha ,m}$ is nonzero. This paper is devoted to the study of the zeros of ${K}_{\alpha ,m}$.

We introduce a notion of analytic generator for groups of unbounded operators, on Banach modules, arising from Esterle’s quasimultiplier theory. Characterizations of analytic generators are given in terms of the existence of certain functional calculi. This extends recent results about C₀ groups of bounded operators. The theory is applicable to sectorial operators, representations of ${H}^{\infty}$, and integrated groups.

We show that some unital complex commutative LF-algebra of ${\mathcal{C}}^{\left(\infty \right)}$$\mathbb{N}$-tempered functions on ${\mathbb{R}}^{+}$ (M. Hemdaoui, 2017) equipped with its natural convex vector bornology is useful for functional calculus.

The Fourier expansion in eigenfunctions of a positive operator is studied with the help of abstract functions of this operator. The rate of convergence is estimated in terms of its eigenvalues, especially for uniform and absolute convergence. Some particular results are obtained for elliptic operators and hyperbolic equations.

For 1 ≤ q < ∞, let ${}_{q}\left(\right)$ denote the Banach algebra consisting of the bounded complex-valued functions on the unit circle having uniformly bounded q-variation on the dyadic arcs. We describe a broad class ℐ of UMD spaces such that whenever X ∈ ℐ, the sequence space ℓ²(ℤ,X) admits the classes ${}_{q}\left(\right)$ as Fourier multipliers, for an appropriate range of values of q > 1 (the range of q depending on X). This multiplier result expands the vector-valued Marcinkiewicz Multiplier Theorem in the direction q >...