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A sufficient condition is given under which the sum, product and indeed any polynomial combination of a well-bounded operator and a commuting real scalar-type spectral operator is well-bounded. This generalizes a result of Gillespie for Hilbert space operators. It is shown in particular that if $X$ is a UMD space, then the sum of finitely many commuting real scalar-type spectral operators acting on $X$ is a well-bounded operator (a result which fails on general reflexive Banach spaces). [ABSTRACT FROM AUTHOR]

Here, it is observe that ΛBV(p) (σ, 픹), the class of functions of bounded p-Λ- variation from a non-empty compact subset σ of ℝ into a commutative unital Banach algebra 픹, is a commutative unital Banach algebra. Moreover, (Λ¹ , ..., ΛN)*BV(p)(∏Ni=1σi, 픹), the class of N-variables functions of bounded p-((Λ¹ , ..., ΛN)*-variation from ∏Ni=1σi into 픹, is a Banach space, where σi are non-empty compact subsets of ℝ, for all i = 1 to N. [ABSTRACT FROM AUTHOR]

Abstract  We investigate R-bounded representations $$\Psi: L^{1}\left( G\right) \rightarrow {\mathcal{L}}\left( X\right) $$ , where X is a Banach space and G is a lca group. Observing that Ψ induces a (strongly continuous) group homomorphism $$U:G\rightarrow {\mathcal{L}}\left( X\right) $$ , we are then able to analyze certain classical homomorphisms U (e.g. translations in Lp (G)) from the viewpoint of R-boundedness and the theory of scalar-type spectral operators. [ABSTRACT FROM AUTHOR]

Let A be the generator of an analytic C0-semigroup on a Banach space X. We associate a closed operator ${\cal A}_{1}$ with A defined on Rad(X) and show that when X is a UMD-space, the Cauchy problem associated with A has maximal regularity if and only if the operator ${\cal A}_{1}{\rm g}$ generates an analytic C0-semigroup on Rad(X). This allows us to exploit known results on analytic C0-semigroups to study maximal regularity. Our results show that ${\cal R}$-boundedness is a local property for semigroups: an analytic C0-semigroup T of negative type is ${\cal R}$-bounded if and only if it is ${\cal R}$-bounded at z = 0. As applications, we give a perturbation result for positive semigroups. Finally, we show the following: when X is a UMD-space, T is an analytic C0-semigroup of negative type, then for every $f\in L^{p}(\RR_{+}; X)$, the mild solution of the corresponding inhomogeneous Cauchy problem with initial value 0 belongs to $W^{\theta,p}(\RR_{+}; X)$ for every $0<\theta < 1$. [ABSTRACT FROM PUBLISHER]