Various metrics in the problem of ideals for the algebra $H^\infty$

Starting with the 1970th, considerable attention has been paid to estimates of solutions of the equations arising in the corona theorem and in the so-called ideal problem. Naturally, also the question about metrics suitable for these estimated also arose. In the case of the corona theorem, the answer to this question is known: mostly, it is possible to pass from estimates in a particular reasonable metric to those in any other. Something similar is proved in this paper about the ideal problem. It should ne noted that, in this case, the proper claims themselves about different metrics are not so easy to find.
In conclusion, some applications to the operator corona problem and operator analog of the ideal problem are presented. The proof of the main results involve a fixed point theorem in a crucial way.