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Isometries in $$H^ 2$$, generating functions and extremal problems. (English) Zbl 0572.47003
If $$\phi$$ is a Möbius function consider the multiplication operator M($$\phi)$$ on $$H^ 2$$; M($$\phi)$$ is an isometry. A detailed description of this isometry is given. The results are used to justify some formal operations with generating functions. As an application, a concrete matrix representation is given for the operator which realizes the maximum of $$| A^ n|$$ as A ranges over all contractions on n- dimensional Hilbert space such that the spectral radius of A does not exceed a given number $$r<1$$.

##### MSC:
 47A20 Dilations, extensions, compressions of linear operators 47A10 Spectrum, resolvent 47B37 Linear operators on special spaces (weighted shifts, operators on sequence spaces, etc.)
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