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Some aspects of the minimal, Möbius-invariant space of analytic functions on the unit disc. (English) Zbl 0553.46021

Interpolation spaces and allied topics in analysis, Proc. Conf., Lund/Swed. 1983, Lect. Notes Math. 1070, 24-44 (1984).
[For the entire collection see Zbl 0534.00013.]
A complete semi-normed space X of analytic functions on the unit disc U is Möbius-invariant if the Möbius group Aut(U) acts isometrically on X by composition in a strongly continuous way. The authors study Möbius-invariant spaces with the emphasis on the minimal Möbius- invariant space \({\mathcal M}\), consisting of all analytic functions f on U which admit a representation \(f(z)=\sum_{j}\lambda_ j(z-a_ j)/(1- \bar a_ jz),\) where \(| a_ j| \leq 1\) and \(\sum_{j}| \lambda_ j| <\infty\), with \(\| f\| =\inf \sum_{j}| \lambda_ j|.\) It is shown that the action of Aut(U) on \({\mathcal M}\) is strongly continuous. From this one gets that \(f\in {\mathcal M}\) if and only if \(\int_{U}| f''(z)| dxdy<\infty.\) It follows that \({\mathcal M}\) is an algebra. Also, with respect to the Möbius-invariant pairing \((f,g)=\int_{U}f'(z)\overline{g'(z)}dxdy.\) The dual of \({\mathcal M}\) is the Bloch space: \[ {\mathcal B}=\{f;\quad f\quad analytic\quad on\quad U,\quad (1-| z|^ 2)| f'(z)| =O(1)\}\quad as\quad | z| \to 1. \] This yields the following surprising fact: if \(f\in {\mathcal M}\) is not constant, then any \(g\in {\mathcal M}\) can be represented as \(g(z)=\sum_{j}\lambda_ jf(\mu_ j\cdot (z-a_ j)/(1-\bar a_ jz))\) where \(| a_ j| <| \mu_ j| =1\) and \(\sum_{j}| \lambda_ j| <\infty.\) The extreme points of the unit ball are found to be the functions \(z\to \lambda (z-a)/(1-\bar az),\) \(| \lambda | =1\), \(| a| \leq 1\), and the isometries of \({\mathcal M}\) are just the trivial ones \(f\to \lambda \cdot f\circ \phi,\), \(| \lambda | =1\), \(\phi\in Aut(U)\). The authors ask some general questions concerning Möbius-invariant spaces. In the appendix it is proved that besides the Dirichlet space, no Möbius-invariant space has a 1- symmetric basis.

MSC:

46E15 Banach spaces of continuous, differentiable or analytic functions
46J15 Banach algebras of differentiable or analytic functions, \(H^p\)-spaces

Citations:

Zbl 0534.00013