Different metrics in the problem of ideals for the algebra 𝐻^{∞}

Abstract

Starting with the 1970s, significant attention has been paid to estimates of solutions of equations arising in the corona theorem and the associated so-called ideal problem. Naturally, the question emerged about the metrics in which such estimates should be sought. In the case of the corona theorem, the answer to this question is known: it is practically always possible to pass from estimates in one reasonable metric to estimates in any other. In this article, something similar is proved for the ideal problem. It should be noted that in the case of the ideal problem, formulations themselves with different metrics are not always obvious. The article concludes with several applications to the operator corona theorem and ideal problem. The proofs of the main results heavily rely on fixed point theorems.