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On an integral transform by R. S. Phillips

Sten Bjon (2010)

Open Mathematics

The properties of a transformation f ↦ f ˜ h by R.S. Phillips, which transforms an exponentially bounded C 0-semigroup of operators T(t) to a Yosida approximation depending on h, are studied. The set of exponentially bounded, continuous functions f: [0, ∞[→ E with values in a sequentially complete L c-embedded space E is closed under the transformation. It is shown that ( f ˜ h ) k ˜ = f ˜ h + k for certain complex h and k, and that f ( t ) = lim h → 0 + f ˜ h ( t ) , where the limit is uniform in t on compact subsets of the positive real line. If f is Hölder-continuous...

On analytic semigroups and cosine functions in Banach spaces

V. Keyantuo, P. Vieten (1998)

Studia Mathematica

If A generates a bounded cosine function on a Banach space X then the negative square root B of A generates a holomorphic semigroup, and this semigroup is the conjugate potential transform of the cosine function. This connection is studied in detail, and it is used for a characterization of cosine function generators in terms of growth conditions on the semigroup generated by B. The characterization relies on new results on the inversion of the vector-valued conjugate potential transform.

On coerciveness in Besov spaces for abstract parabolic equations of higher order

Yoshitaka Yamamoto (1999)

Studia Mathematica

We are concerned with a relation between parabolicity and coerciveness in Besov spaces for a higher order linear evolution equation in a Banach space. As proved in a preceding work, a higher order linear evolution equation enjoys coerciveness in Besov spaces under a certain parabolicity condition adopted and studied by several authors. We show that for a higher order linear evolution equation coerciveness in Besov spaces forces the parabolicity of the equation. We thus conclude that parabolicity...

On ergodicity for operators with bounded resolvent in Banach spaces

Kirsti Mattila (2011)

Studia Mathematica

We prove results on ergodicity, i.e. on the property that the space is a direct sum of the kernel of an operator and the closure of its range, for closed linear operators A such that | | α ( α - A ) - 1 | | is uniformly bounded for all α > 0. We consider operators on Banach spaces which have the property that the space is complemented in its second dual space by a projection P. Results on ergodicity are obtained under a norm condition ||I - 2P|| ||I - Q|| < 2 where Q is a projection depending on the operator A....

On optimal L p regularity in evolution equations

Alessandra Lunardi (1999)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni

Using interpolation techniques we prove an optimal regularity theorem for the convolution u ⁢ t = ∫ 0 t T ⁢ t - s ⁢ f ⁢ s ⁢ d ⁢ s , where T ⁢ t is a strongly continuous semigroup in general Banach space. In the case of abstract parabolic problems – that is, when T ⁢ t is an analytic semigroup – it lets us recover in a unified way previous regularity results. It may be applied also to some non analytic semigroups, such as the realization of the Ornstein-Uhlenbeck semigroup in L p ⁢ R n , 1 < p < ∞ , in which case it yields new optimal regularity results in fractional...

Currently displaying 261 – 280 of 451