On the continuity of the spectrum in certain Banach algebras (Q1592831)

Some classes of linear operators \(T_0\in L(H)\) on a Hilbert spaces \(H\) (mainly Toeplitz, Wiener-Hopf and singular integral) such that the spectrum \(\sigma (T,L(H))\) is continuous at the points \(T_0\) are considered. Here \(\sigma (x,{\mathcal A})\) is the spectrum of the element \(x\) in algebra \({\mathcal A}\). In addition, some subalgebras \({\mathcal A}\) of the algebras \(\widetilde{\mathcal A}\) for which the spectrum \(\sigma (x,\widetilde{\mathcal A})\) becomes continuous at a point \(x_0\) when restricted to the subalgebra \({\mathcal A}\). The main results of the paper are the following. The spectrum \(\sigma (x,\widetilde{\mathcal A})\) is continuous in Banach algebras \(\widetilde{\mathcal A}\) with polynomial identities which are a generalization of commutative algebras. Some conditions which provide the continuity of the restriction of \(\sigma (x,\widetilde{\mathcal A})\) to an algebra \({\mathcal A}\) with polynomial identity are obtained. A local principle for the continuity of the spectrum is proposed. Some conditions of continuity of the spectrum \(\sigma (T,L(H))\) at the point \(T_0\) where \(T_0\) is a Toeplitz operator or singular integral operator are obtained. Continuity of the spectrum in algebras \(M_n(\widetilde{\mathcal A})\) of \(n\times n\) matrices with entries from \(\widetilde{\mathcal A}\) is studied. The paper contains many useful examples.