The isometries of 𝐻¹_{}ℋ

Cambern, Michael; Jarosz, Krzysztof

doi:10.1090/S0002-9939-1989-0979225-8

The isometries of $H^ 1_ \mathcal {H}$
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by Michael Cambern and Krzysztof Jarosz PDF

Proc. Amer. Math. Soc. 107 (1989), 205-214 Request permission

Abstract:

Let $\mathcal {H}$ be a finite-dimensional complex Hilbert space. In this article we characterize the linear isometries of the Banach space $H_\mathcal {H}^1$ onto itself. We show that $T$ is such an isometry iff $T$ is of the form $TF(z) = UF(\psi (z))\psi ’(z)$, for $F \in H_\mathcal {H}^1$ and $z$ in the unit disc, where $\psi$ is a conformal map of the disc onto itself, and $U$ is a unitary operator on $\mathcal {H}$.

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