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Vector-valued holomorphic and harmonic functions

100%
Arendt W.
Concrete Operators
|
2016
|
tom 3
|
nr 1
68-76
EN
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 with values in a Banach space.
2
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Forms, functional calculus, cosine functions and perturbation

63%
Arendt W., Batty C.
Banach Center Publications
|
2007
|
tom 75
|
nr 1
17-38
EN
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 $H^∞$-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.
3
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Decomposing and twisting bisectorial operators

63%
Arendt W., Zamboni A.
Studia Mathematica
|
2010
|
tom 197
|
nr 3
205-227
EN
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 we can express the solution of the evolution equation on the line by an integral operator involving two semigroups associated with A.
4
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Fourier multipliers for Hölder continuous functions and maximal regularity

51%
Arendt W., Batty C., Bu S.
Studia Mathematica
|
2004
|
tom 160
|
nr 1
23-51
EN
Two operator-valued Fourier multiplier theorems for Hölder spaces are proved, one periodic, the other on the line. In contrast to the $L^{p}$-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.
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