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On the Möbius invariant function spaces. (English) Zbl 0757.46050

Let \(D\) be the unit disc in \(\mathbb{C}\) and let \(C\) be the Möbius group, i.e. the set of all one-to-one analytic functions of \(D\) into itself. Denote by \(\mathbb{M}\) the complete normed algebra of analytic functions of type \(f(z)=\sum_{k=1}^ \infty \lambda_ k {{a_ k-a}\over{1-a_ k z}}\), \(| a_ k|<1\), \(|\lambda_ k|=1\), \(\sum_{k=1}^ \infty |\lambda_ k|<\infty\), equipped with the norm \(\| f\|_{\mathbb{M}}=\inf\sum_{k=1}^ \infty |\lambda_ k|\), where \(\{\lambda_ k\}\) runs over all representations of \(f(z)\) of the above type. The author expands the investigations on Möbius invariant function spaces \(X\) and their duals \(X^*\) by J. Arazy, S. Fisher and J. Peetre from J. Reine Angew. Math. 363, 110-145 (1985; Zbl 0566.30042).
From the paper: In the first part it is proved that the polynomials are dense in \(X^*\); \(\mathbb{M}\) is continuously contained in \(X^*\); the Möbius group \(C\) spans both \(X\) and \(X^*\). The second part concerns the ideal theory of \(\mathbb{M}\). The maximal ideals of \(\mathbb{M}\) are determined. Some other properties important from the point of view of ideal theory are obtained as well.

MSC:

46J15 Banach algebras of differentiable or analytic functions, \(H^p\)-spaces
30H05 Spaces of bounded analytic functions of one complex variable
40E20 Tauberian constants

Citations:

Zbl 0566.30042
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