| The corona problem was raised when mathematicians studied the homomorphism of Hinfinity(D) and maximal ideals in 1960's. It has two equivalent formulations. We refer to the following one as the corona problem. If fin i=1⊂Hinfinity (D), when does there exist functions gin i=1⊂Hinfinity (D) such that i=1n figi = 1 in the unit disk?;In 1962, Carleson determined when a finitely generated ideal in Hinfinity(D) is actually all of Hinfinity(D), by giving a function theory condition on the generators. Namely, I(f 1, f2, f3, ..., fn) = Hinfinity( D) if and only if there exists a epsilon > 0, so that i=1nf iz 2>32 for all z ∈ D. Actually, more was shown, as abound for the size of solutions was given. More precisely, if fin i=1⊂Hinfinity (D) such that 1 ≥ i=1nf iz 2>32>0 for all z ∈ D, then there exists a B(epsilon, n) < infinity and gin i=1⊂Hinfinity (D) with i=1n figi = 1 and i=1ngi z&vbm0;2 ≤ B(epsilon, n) for all z ∈ D. Carleson's proof of what has come to be known as the corona theorem was based on an intricate geometrical construction, of combinatorial character. His proof has been improved by the work of T. Wolff, done in the spring of 1979.;In this paper, we extend the vector valued corona theorem on a reproducing kernel Hilbert space to the matrix case. We proved that if the vector valued corona theorem on a reproducing kernel Hilbert space holds, then so does the corresponding matrix version of corona theorem on some algebras of functions on reproducing kernel Hilbert spaces. |
