Relative angular derivatives. (English) Zbl 1002.46021

Shapiro, Jonathan E.

J. Oper. Theory 46, No. 2, 265-280 (2001).

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 \({{b(z)-b(z_0)}\over{z-z_0}}\) with a difference quotient relative to an inner function \(u\), \({{1-b(z)}\over{1-u(z)}}\). We relate properties of this generalized difference quotient to the 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.

MSC:

46E22	Hilbert spaces with reproducing kernels (= (proper) functional Hilbert spaces, including de Branges-Rovnyak and other structured spaces)
46E30	Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)

Keywords:

angular derivative; Hardy space; Aleksandrov measure; de Branges-Rovniak space; holomorphic self-map; inner function; generalized difference quotient; product rule