Singular values and Schmidt pairs of composition operators on the Hardy space. (English) Zbl 1066.47026

Clifford, John H.; Dabkowski, Michael G.

J. Math. Anal. Appl. 305, No. 1, 183-196 (2005).

The main result of the present paper solves the functional equation \[ G(z)f(\varphi(z))=\lambda f(z), \] where \(G\) is a given bounded analytic function in the unit disk \(D\) and \(\varphi\) is a given analytic self-map of \(D\). The unknowns of the equation are \(f\), which is to be analytic in \(D\), and \(\lambda\), which is to be a complex number. As a consequence, the singular values of the compact composition operator \(C_\varphi\) on the classical Hardy space \(H^2\) are computed, where \(\varphi(z)=az+b\) with \(| a| +| b| <1\). The corresponding problem for weighted Bergman spaces is discussed as well.

Reviewer: Kehe Zhu (Albany)

MSC:

47B33	Linear composition operators
30H05	Spaces of bounded analytic functions of one complex variable
39B32	Functional equations for complex functions

Keywords:

composition operator; singular value; Schröder’s equation; Schmidt pair; weighted composition operator

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