On the Boundedness of the Spectral Factorization Mapping on Decomposing Banach Algebras

Abstract

In [SIAM J. Control Optim., 40 (2001), pp. 88-106], Jacob and Partington studied the continuity and boundedness of the spectral factorization mapping on decomposing Banach algebras. Many function spaces considered in systems theory are decomposing Banach algebras. The most well known example is the Wiener algebra, the space of all absolutely convergent Fourier series. Jacob and Partington showed in the above paper that the spectral factorization is locally Lipschitz continuous on all decomposing algebras, but unbounded on the most important examples of decomposing algebras. Our paper gives an extension of this result and shows that the spectral factorization mapping is unbounded on every decomposing algebra.