Some properties of Glisson distances in the upper half-plane

Abstract

The author compares the Gleason distance with the distance of Euclid in the unit disk in the upper half plane. The concept of “the Gleason distance” was formulated in the work of H.S. Bear [1] The Gleason distance is defined as follows (see [1]): d = sup |f(z2)-f(z1)|, f(Z)εB1(K) where B 1 (K) is the unit ball in the space of bounded analytic in K functions. The author of the article proves that in the circle K the distances of Gleason and Euclid are equal only when the points are opposite. He found necessary and sufficient conditions, when the distances are equal for the two given points which are symmetrical about the ordinate axis.