On the completion problem for algebra \(H^{\infty}\) (Q2253108)

This paper in operator theory and in bounded holomorphic functions on a disc studies corona and operation completion problems in the manner of the author's previous papers on \(H^\infty\) of the disc. We quote one theorem. Theorem 2.3. (Oka-type theorem) A function \(F\) belongs to the space \(\mathcal{LI}_{\text{comp}}^0(X_1,X_2,Y)\) if and only if there exist functions \(H\) in the space \(C_{\text{comp}}(L(X_1\oplus Y,X_2))\) and \(G\) in the similar space \(C_{\text{comp}}(L(X_2,X_1\oplus Y))\) such that for all \(z\in{\mathbb C}\) with \(|z|<1\), the operator \(H(z)G(z)\) is the identity operator on \(X_2\), \(G(z)H(z)\) is the identity operator on \(X_1\oplus Y\), and \(H(z)|_{X_1}=F(z)\). Here the second and third spaces refer to continuous bounded functions on the unit disc with operator values whose ranges are relatively compact, and the first space refers to a set of components of the space of bounded holomorphic functions on the unit disc admitting a pointwise bounded family of left inverses with kernel type at zero equal to \(Y\); \(X_1,X_2,Y\) are Banach spaces.