On the well-posedness of the heat equation on unbounded domains.
An Order Theoretical Characterization of the Fourier Transform.
Factorization by Lattice Homomorphisms.
Kato's equality and spectral decomposititon for positive C0-Groups.
Resolvent positive operators and inhomogeneous boundary conditions
Vector-valued holomorphic and harmonic functions
Holomorphic and harmonic functions with values in a Banach space are investigated. Following an approach given in a joint article with Nikolski [4] it is shown that for bounded functions with values in a Banach space it suffices that the composition with functionals in a separating subspace of the dual space be holomorphic to deduce holomorphy. Another result is Vitali’s convergence theorem for holomorphic functions. The main novelty in the article is to prove analogous results for harmonic functions...
Decomposing and twisting bisectorial operators
Bisectorial operators play an important role since exactly these operators lead to a well-posed equation u'(t) = Au(t) on the entire line. The simplest example of a bisectorial operator A is obtained by taking the direct sum of an invertible generator of a bounded holomorphic semigroup and the negative of such an operator. Our main result shows that each bisectorial operator A is of this form, if we allow a more general notion of direct sum defined by an unbounded closed projection. As a consequence...
Inegalités de Kato et semi-groups sous-Markoviens.
Order Isomorphisms of Fourier-Stieltjes Algebras.
Integral representations of resolvents and semigroups.
Fourier multipliers for Hölder continuous functions and maximal regularity
Two operator-valued Fourier multiplier theorems for Hölder spaces are proved, one periodic, the other on the line. In contrast to the -situation they hold for arbitrary Banach spaces. As a consequence, maximal regularity in the sense of Hölder can be characterized by simple resolvent estimates of the underlying operator.
Forms, functional calculus, cosine functions and perturbation
In this article we describe properties of unbounded operators related to evolutionary problems. It is a survey article which also contains several new results. For instance we give a characterization of cosine functions in terms of mild well-posedness of the Cauchy problem of order 2, and we show that the property of having a bounded -calculus is stable under rank-1 perturbations whereas the property of being associated with a closed form and the property of generating a cosine function are not....
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