On the topological boundary of semi-Fredholm operators

Abstract

We prove several distance formulas from a fixed operator in B(H) to some classes of operators connected with the semi-Fredholm ones. Here H is a separable Hilbert space. In particular, Fredholm and upper and lower semi-Fredholm operators have the same boundary in B(H). Let H be a separable Hilbert space and let B(H) be the algebra of all bounded linear oprators on H. For an operator T E B(H), we will denote by T*, R(T), N(T) and a(T) its adjoint, range, kernel and spectrum, respectively. Let K(H) be the ideal of compact operators and C(H) = B(H)/K(H) be the Calkin algebra. Denote by ir: B(H) -> C(H) the canonical projection. Endowed with the essential norm I TIle = I Ir(T) I 1, C(H) is a C*-algebra. The index of an operator T E B(H) will be denoted by ind(T) and is defined by ind(T) = dim N(T) dim N(T*), with the convention oo -oo0. We introduce the following notation for several classes of operators * F+ = {T E B(H): R(T) is closed, dim N(T) < oo} is the set of all upper semi-Fredholm operators. * F_ = {T E B(H): R(T) is closed, dim N(T*) < oo} is the set of all lower semi-Fredholm operators. * F? = F+ U F_ is the set of semi-Fredholm operators. * F = F+ n F_ is the set of Fredholm operators. * Int = {T E B(H): ind(T) = n} with n E Z = Z U {-oo, +oo}. * F= F? n In, with n G 2, the connected component of index n in F?. For a set X in B(H), we will denote by intX, X and OX the interior, closure and (topological) boundary, respectively. For a linear operator T E B(H), we will denote by ae(T) = { C: T--AI V F} the essential spectrum of T. Let me (T) = inf{cre (ITI)} (cf. [1]), where ITI = (T*T)1/2, and Me (T) = max{me (T); me (T*)} Using Theorem 1.1 of [4] and Theorem 3.1 of [7], we easily obtain the following result: Received by the editors April 30, 1996 and, in revised form, October 14, 1996. 1991 Mathematics Subject Classification. Primary 47A53; Secondary 47A55.