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 \(r<1\).


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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