Maximality properties for isometric interpolating sequences and sequences of trivial points in the spectrum of H ∞

Let ( x n ) be an isometric interpolating sequence or a sequence of trivial points in the spectrum of H ∞ . It is shown that either every cluster point of that sequence has a maximal support set or there exists y ∈ M ( H ∞ + C ) such that the support of x n is contained in the support of y for infinitely many n . Similar results for Gleason parts are obtained, too. We also investigate the H ∞ ‐convex hulls of countable unions of support sets and show that whenever supp x ⊂ supp y and x /∈ $ \overline {P(y)} $ , then the H ∞ ‐convex hull of supp x does not meet $ \overline {P(y)} $ . (© 2005 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)