A CORONA THEOREM FOR CERTAIN SUBALGEBRAS OF H∞(D) II

Abstract. We prove corona theorems (with estimates on solutions) for var-ious uniformly closed subalgebras of H∞(D), including those algebras that have the form j /∈K j<kp BjC + Bkp+1H∞(D) where B is an inner function and K is a finite subset of N. In this paper we prove corona theorems (with estimates on solutions) for var-ious uniformly closed subalgebras of H∞(D), including those considered in [RT, MSW]. The algebras of the form C + BH∞(D), for B an inner function, were studied in [S, Ra, and DPRS] in connection with Nevanlinna-Pick interpolation problems. That our algebras can be more easily handled for corona problems, rather than Nevanlinna-Pick interpolation problems, stems from the fact that the Koszul complex satisfies nice algebraic relations. We include a new proof of the relevant relations. Unfortunately, a nice result of Scheinberg [S] shows that there exist closed uni-tal subalgebras of H∞(D) for which the corona theorem fails. All algebras we consider are multiplier algebras for reproducing kernel Hilbert spaces. However, a characterization of Mittal and Paulsen [MP] shows that none of the Schein-berg algebras can be multiplier algebras for a reproducing kernel Hilbert space. Moreover, it is known that the corona theorem can fail for multiplier algebras of reproducing kernel Hilbert spaces [Tr2], but it does not seem to be known if the corona theorem can fail for multiplier algebras of reproducing kernel Hilbert spaces of analytic functions on the unit disk