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Journal Articles Operator Theory: Advances and Applications Year : 2010

Contractivity of the $H^\infty$-calculus and Blaschke products

Abstract

It is well known that a densely defined operator $A$ on a Hilbert space is accretive if and only if A has a contractive $H^\infty$-calculus for any angle bigger than $\pi/2$. A third equivalent condition is that $(A − w)(A + w)^{−1} \leq 1$ for all $\Re w \geq 0$. In the Banach space setting, accretivity does not imply the boundedness of the H ∞-calculus any more. However, we show in this note that the last condition is still equivalent to the contractivity of the $H^\infty$ calculus in all Banach spaces. Furthermore, we give a sufficient condition for the contractivity of the $H^\infty$-calculus on $\C_+$, thereby extending a Hilbert space result of Sz.-Nagy and Foiaş to the Banach space setting.
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Dates and versions

hal-03123922 , version 1 (28-01-2021)

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  • HAL Id : hal-03123922 , version 1

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Christoph Kriegler, Lutz Weis. Contractivity of the $H^\infty$-calculus and Blaschke products. Operator Theory: Advances and Applications, 2010. ⟨hal-03123922⟩

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