A Characterization of Finite Blaschke Products with Degree n

Abstract

In this paper, the authors give a characterization of finite Blaschke products with degree n. The main results are: (1) An n-dimensional complex vector can be the first n Taylor coefficients of a finite Blaschke product with degree no more than n − 1 if and only if the vector induces a lower triangular Toeplitz matrix with norm 1; (2) an n-dimensional complex vector can be the first n Taylor coefficients of an inner function if and only if the vector induces a lower triangular Toeplitz matrix with norm no more than 1. Möbius transformations acting on contraction matrices play an important role in the proofs.

Keywords

Finite Blaschke products / Toeplitz matrices / Contractions / Inner functions / Taylor coefficients

References

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