Positive multiplication preserves dissipativity in commutative \(C^*\)-algebras (Q5952854)

An operator \(T:(\Omega \subseteq X)\to X\), \(X\) being a Banach space, is called dissipative, and \(-T\) accretive, if \(\|u-v-\lambda(Tu-Tv)\|\geq\|u-v\|\) for all \(\lambda >0\), \(u,v\in \Omega \), and strongly dissipative if there exists \(c>0\) such that \(\|u-v-\lambda(Tu-Tv)\|\geq(1+\lambda c)\|u-v\|\). Let \({\mathcal A}\) be a commutative \(C^{\ast}\)-algebra and let \({\mathcal A}^{+}\) be the cone of positive elements of \({\mathcal A}\). Here positive means self-adjoint with a real nonnegative spectrum and ``strictly'' positive refers to a positive spectrum. The following theorem is shown: Let \(T:(\Omega \subseteq{\mathcal A}) \to{\mathcal A}\) be a dissipative [accretive] operator. Then for every \(a\in{\mathcal A}^{+}\), the operator \(aT(\cdot)\) is still dissipative [accretive]. If \(T\) is strongly dissipative [accretive] with constant \(c\), and \(a\) is a strictly positive, then \(aT(\cdot)\) is strongly dissipative [accretive] itself, with constant \(c\min \sigma (a)\), where \(\sigma(a)\) denotes the spectrum of \(a\). A simple counterexample, which shows that this result is not valid in general commutative and involutory Banach algebra is also given.