From Corollary 3.5 in [Berkani, M; Sarih, M.; Studia Math. 148 (2001), 251– 257] we know that if S, T are commuting B-Fredholm operators acting on a Banach space X, then ST is a B-Fredholm operator. In this note we show that in general we do not have ind(ST) = ind(S) + ind(T), contrarily to what has been announced in Theorem 3.2 in [Berkani, M; Proc. Amer. Math. Soc. 130 (2002), 1717–1723]. However, if there exist U, V ∈ L(X) such that S, T, U, V are commuting and US + V T = I, then ind(ST) = ind(S) + ind(T), where ind stands for the index of a B-Fredholm operator.
A necessary and sufficient condition for the boundedness of a solution of the third problem for the Laplace equation is given. As an application a similar result is given for the third problem for the Poisson equation on domains with Lipschitz boundary.
Let K ⊂ ℝm (m ≥ 2) be a compact set; assume that each ball centered on the boundary B of K meets K in a set of positive Lebesgue measure. Let C(1) 0 be the class of all continuously differentiable real-valued functions with compact support in m and denote by σm the area of the unit sphere in m. With each ϕ ∈ C(1) 0 we associate the function WKϕ(z) = 1⁄ σm ∫ Rm\K grad ϕ(x) · z − x |z − x| m dx of the variable z ∈ K (which is continuous in K and harmonic in K \ B). WKϕ depends only on the restriction ϕ|B of ϕ to the boundary B of K. This gives rise to a linear operator WK acting from the space C(1)(B) = {ϕ|B; ϕ ∈ C(1) 0 } to the space C(B) of all continuous functions on B. The operator TK sending each f ∈ C(1)(B) to TKf = 2WKf − f ∈ C(B) is called the Neumann operator of the arithmetical mean; it plays a significant role in connection with boundary value problems for harmonic functions. If p is a norm on C(B) ⊃ C(1)(B) inducing the topology of uniform convergence and G is the space of all compact linear operators acting on C(B), then the associated p-essential norm of TK is given by ωpTK = inf Q∈G sup {p[(TK − Q)f]; f ∈ C(1)(B), p(f) ≤ 1} . In the present paper estimates (from above and from below) of ωpTK are obtained resulting in precise evaluation of ωpTK in geometric terms connected only wit K.