On the Punctured Neighbourhood Theorem

Let X, Y, Z be Banach spaces and X S(z) −→Y T (z) −→Z an analytically dependant sequence of operators satisfying T (z)S(z) = 0. We study properties of the function z 7→ dim Ker T (z)/ Im S(z). Let X, Y be complex Banach spaces. Denote by L(X,Y ) the set of all bounded linear operators from X to Y . If Y = X then we write for short L(X) = L(X, X). Recall the well-known punctured neighbourhood theorem: Theorem 1. Let T ∈ L(X) be a Fredholm operator. Then there exist ε > 0 and constants k1 ≤ dim Ker T , k2 ≤ codim Im T such that dim Ker(T − z) = k1 and codim Im(T − z) = k2 for all z, 0 < |z| < ε. In this paper we study a more general situation. Let X, Y, Z be Banach spaces, let U be an open subset of C, let S : U → L(X, Y ) and T : U → L(Y, Z) be analytic operator-valued functions satisfying T (z)S(z) = 0 for all z ∈ U . For z ∈ U write α(z) = dim Ker T (z)/ Im S(z). The aim of the paper is to study the behaviour of the function z 7→ α(z). The main result of the first section is the following generalization of Theorem 1 — if U ⊂ C, w ∈ U , Im T (w) is closed and α(w) < ∞ then α(z) = k is constant in a punctured neighbourhood of w. Clearly the classical punctured neighbourhood theorem follows easily from this generalization for sequences 0→ XT−z −→Y and XT−z −→Y → 0, respectively. In the second section we study the case n ≥ 2. This situation has been studied mainly in connection with the Koszul complex of an n-tuple of commuting operators.