The generalized regular points and narrow spectrum points of bounded linear operators on Hilbert spaces (Q627570)

Let \(H\) be a Hilbert space, \(A\) be a bounded linear operator and \(\sigma(A)\) be its spectrum. A point \(\mu\in\mathbb{C}\) is called a generalized regular point if the range \(R(A-\mu I)\) of \(A-\mu I\) is closed and its orthogonal complement intersects only trivially with \(R(A-\lambda I)\) for any \(\lambda\) taken from some vicinity of \(\mu\). The set of such points is denoted by \(\rho_g(A)\) and its complement by \(\sigma_W(A)\) (a narrow spectrum). The authors state several properties of \(\sigma_W(A)\). They prove, in particular, that \(\sigma_W(A)\) is nonempty and closed and that the boundary points of \(\sigma(A)\) belong to \(\sigma_W(A)\). They also prove that, if \(\sigma(A)\setminus\sigma_W(A)\neq \emptyset\), then there exists a nontrivial invariant subspace for \(A\).