On the primarity of \(H^{\infty}\)-spaces
Let \(D^ m\) denote the open unit polydisc in \({\mathbb{C}}^ m\) \((m=1,2,...)\) where \({\mathbb{C}}\) is the complex plane. It was proved by \textit{P. Wojtaszczyk} [Duke Math. J. 46, 635-644 (1979; Zbl 0423.46016)] that \(H^ p\) \((D^ m)\) is isomorphic to the \(\ell_ p\)-direct sum \((\sum^{\infty}_{n=1}H^ p(D))_ p\) for \(1\leq p\leq \infty\). The author and A. Pełczyński [in preparation] have applied this result to prove that \(H^{\infty}(D^ m)\) is isomorphic to \((\sum^{\infty}_{N=0}L_ N^{\infty}(D^ m))_{\infty},\) where \(L_ N^{\infty}(D^ m)\) is the space of polynomials on \(D^ m\) of degree \(\leq N\) with \(L^{\infty}\)-norm. By using this and establishing a result about factorization of the identity operator on \(L_ n^{\infty}(D^ m)\), the author proves the following Theorem. If \(H^{\infty}(D^ m)\) decomposes as the direct sum of two spaces X,Y, then either X or Y is isomorphic to \(H^{\infty}(D^ m).\) This gives an affirmative answer to a question raised by \textit{P. Casazza} [''Complemented subspaces of A,H' and \(H^ r\)''. Proc. Linear Operators Function Theory, Vol. ll, Leningrad 1978, 20-22].
scientific article; zbMATH DE number 3877803