285 research outputs found
The Matrix-Valued Corona Problem in the Disk and Polydisk
In this paper we consider the matrix-valued corona problem in the
disk and polydisk. The result for the disk is rather well known, and is usually
obtained from the classical Carleson Corona Theorem by linear algebra. Our
proof provides a streamlined way of obtaining this result and allows one to get
a better estimate on the norm of the solution. In particular, we were able to
improve the estimate found in the recent work of T. Trent. Note that, the
solution of the matrix corona problem in the disk can be easily
obtained from the corona problem either by factorization, or by the
Commutant Lifting Theorem. The corona problem in the polydisk was
originally solved by K.C. Lin. The solution used Koszul complexes and was
rather complicated because one had to consider higher order
-equations. Our proof is more transparent and it improves upon
Lin's result in several ways. First, we were able to show that the norm of the
solution is independent of the number of generators. Additionally, we
illustrate that the norm of the solution grows at most proportionally to the
dimension of the polydisk. Our approach is based on one that was originated by
M. Andersson. In the disk it essentially depends on Green's Theorem and duality
to obtain the estimate. In the polydisk we use Riesz projections to reduce the
problem to the disk case.Comment: 27 page
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