$H^\infty $ calculus and dilatations

“…However, one knows that on L p -spaces (1 < p < ∞) a bounded H ∞ (Σ ϕ )-calculus for some ϕ ∈ (0, π 2 ) directly implies R-analyticity [19,Remark 12.9c] and that the semigroups which dilate to a group whose generator is of scalar type are exactly those with such a calculus [14,Theorem 5.1]. So in this case one can always deduce R-analyticity directly without our methods.…”

Section: Some Remarks On Dilation Argumentsmentioning

confidence: 99%

Regularity of semigroups via the asymptotic behaviour at zero

Fackler

2013

Semigroup Forum

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An interesting result by T. Kato and A. Pazy says that a contractive semigroup (T (t)) t≥0 on a uniformly convex space X is holomorphic iff lim sup t↓0 T (t) − Id < 2. We study extensions of this result which are valid on arbitrary Banach spaces for semigroups which are not necessarily contractive. This allows us to prove a general extrapolation result for holomorphy of semigroups on interpolation spaces of exponent θ ∈ (0, 1). As an application we characterize boundedness of the generator of a cosine family on a UMD-space by a zero-two law. Moreover, our methods can be applied to R-sectoriality: We obtain a characterization of maximal regularity by the behaviour of the semigroup at zero and show extrapolation results.

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“…A dilation of a Hilbert space holomorphic semigroup (not necessarily contractive) to a Kreǐn space was constructed in [14]. In [8], one can find dilation results in the context of operators on UMD spaces.…”

Section: Theorem 24 the Spaces H Ctr Aθ And H Obsmentioning

confidence: 99%

H ∞-functional calculus and models of Nagy–Foiaş type for sectorial operators

2010

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We prove that a sectorial operator admits an H ∞ -functional calculus if and only if it has a functional model of Nagy-Foiaş type. Furthermore, we give a concrete formula for the characteristic function (in a generalized sense) of such an operator. More generally, this approach applies to any sectorial operator by passing to a different norm (the McIntosh square function norm). We also show that this quadratic norm is close to the original one, in the sense that there is only a logarithmic gap between them.

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$H^\infty $ calculus and dilatations

Cited by 30 publications

References 8 publications

A note on maximal estimates for stochastic convolutions

A note on maximal estimates for stochastic convolutions

Regularity of semigroups via the asymptotic behaviour at zero

H ∞-functional calculus and models of Nagy–Foiaş type for sectorial operators

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