Estimates of solutions of the $H^p$ and BMOA corona problem

Abstract

We prove new sharper estimates of solutions to the H p -corona problem in strictly pseudoconvex domains; in particular we show that the constant is independent of the number of generators. We also obtain sharper estimates for solutions to the BMOA corona problem. The proofs also lead to new results about the Taylor spectrum of analytic Toeplitz operators on H p and BMOA.

DIVISION PROBLEM IN GENERALIZED GROWTH SPACES ON THE UNIT BALL IN ℂn

“…Formulas for explicit solutions of such division problems were studied by many authors in various situations and norms (see [1], [2], [3], [4], [5], [11], [12], [14], [15], [16], [17], [18]). In particular, the H p -corona problem asks for the condition on holomorphic n-tuples G = ( …”

The essential spectrum of holomorphic Toeplitz operators on Hpspaces

“…This idea was used, though not formalized, in [3] and goes back to Wolff's proof of the corona problem. To see this, let n = 1 and p = ∞.…”

“…Moreover, the right essential spectrum σ ress (T g ) is the set of w such that the homology of (1.1) at l = 0 is infinite-dimensional. In [3] the following was proved:…”

“…[3]. The equality σ r (T g , X) = g(D) has been proved for a large number of other function spaces X in strictly pseudoconvex domains such as various Besov spaces ( [7] and [8]) and Q p spaces ( [13] and [4]).…”

Operator Theory and Function Theory in Drury–Arveson Space and Its Quotients

“…Over the years a lot of effort was put into proving an analogue of this celebrated theorem in several variables, and some results were obtained [9,85,88,116,117,120]; see also the recent survey [55]. However, the most natural several variables analogues of Theorem 12.4.1, which are precisely the same statement in the theorem but with the disc D replaced by either the unit ball B d or the polydisc D d , remain to this day out of reach.…”