Journal of Operator Theory

Volume 46, Issue 2, Fall 2001 pp. 265-280.

Relative angular derivatives

Authors: Jonathan E. Shapiro
Author institution: Mathematics Department, California Polytechnic State University, San Luis Obispo, CA 93407, Ad.2, USA

Summary: We generalize the notion of the angular derivative of a holomorphic self-map $b$, of the unit disk, by replacing the usual difference quotient $\frac{b(z)-b(z_{0})}{z-z_{0}}$ with a difference quotient relative to an inner function $u$, $\frac{1-b(z)}{1-u(z)}$. We relate properties of this generalized difference quotient to properties of the Aleksandrov measures associated with the functions $b$ and $u$. Six conditions are shown to be equivalent to each other, and these are used to define the notion of a relative angular derivative. We see that this generalized derivative can be used to reproduce some known results about ordinary angular derivatives, and the generalization is shown to obey a form of the product rule.

Keywords: Angular derivative, Hardy space, Aleksandrov measure, de Branges-Rovnyak space

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