On the boundedness and continuity of the spectral factorization mapping (Q2719154)

The spectral factorization mapping \(\phi: B_{\text{pos}}\to B_+\) is considered on a decomposing Banach algebra \(B\) (the paper provides a large number of examples of such algebras), given by NEWLINE\[NEWLINE\phi(f)(z):= \exp\Biggl\{(1/4\pi) \int^{2\pi}_0 ((e^{it}+ z)/(e^{it}- z))\log f(e^{it}) dt\Biggr\},NEWLINE\]NEWLINE so that \(\phi(f) \overline{\phi(f)}= f\), \(f\in B_{\text{pos}}\). It is proved that \(\phi\) is locally Lipschitz continuous. For every \(f\in B_{\text{pos}}\), there exist \(\rho\), \(c>0\) such that NEWLINE\[NEWLINE\|\phi(f_1)- \phi(f_2)\|_B\leq C\|f_1- f_2\|_BNEWLINE\]NEWLINE and \(\|\phi(f_1)^{-1}- \phi(f_2)^{-1}\|_B\leq C\|f_1- f_2\|_B\) whenever \(\|f_i- f\|_B<\rho\), \(i= 1,2\). In spite of this, \(\phi\) fails to be bounded on all decomposing Banach algebras in consideration.