Vol. 53, No. 2, 1974

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Equiconvergence of derivations

Anthony G. O’Farrell

Vol. 53 (1974), No. 2, 539–554

This paper is a study of bounded point derivations on the classical Banach algebras of analytic functions of a complex variable. The results are positive in character. The higher-order Gleason metrics dp of R(X) are introduced and conditions are studied under which convergence takes place with respect to these metrics. In particular, if R(X) admits a pth-order bounded point derivation at a point x ∂X and X satisfies a cone condition at x, then dp(y,x) tends to 0 as y tends to x along the midline of the cone. Similar results hold for the other classical function algebras. In the case of the algebra H(U), for open U C, the analogous results hold only for regular derivations (a regular p-th-order derivation maps zp to a nonzero complex number). The points of the maximal ideal space of H(U) at which regular bounded point derivations exist are characterized in terms of analytic capacity, following Hallstrom.

Mathematical Subject Classification 2000
Primary: 46J15
Received: 12 April 1973
Published: 1 August 1974
Anthony G. O’Farrell