The Asymptotic Behaviour of the Reduced Minimum Modulus of a Fredholm Operator

Let 7(S) denote the reduced minimum modulus of a linear operator S acting in a complex Banach space X, and let / denote the identity on X. In this paper it is shown that for a (not necessarily bounded) Fredholm operator T acting in X, the limit lim y^fy/rt exists and is equal to the supremum of all positive numbers S such that the dimension of the null space and the codimension of the range of T — XI are constant on o < |a| < s. Introduction. Throughout this paper T will be a linear operator with domain D{T) and range RÎT) in the complex Banach space X. By definition the reduced minimum modulus y\T) of T is the supremum of all real numbers y such that \\Tx\\>yd(x, N(T)\ x e D{T) (cf. [6, p. 231] and [3, Definition IV.1.31). Here d{x, N(T)) denotes the distance of x to the null space NvT) of T. Observe that we do not require Nix) to be closed in X. In [3] y(T) is called the minimum modulus of T, but in the present paper this term is reserved for the object studied in [2]. Let tz(T) denote the dimension of Nix) and d(T) the codimension of R(T) in X. We call T a Fredholm operator if T is a closed linear operator with w(T) and d(T) both finite. If T is a closed linear operator with closed range such that at least one of the numbers n{T) and d(T) is finite, then T is said to be a semi-Fredholm operator. Since the range of a Fredholm operator is closed (cf. [5, Lemma 332]), any Fredholm operator is semi-Fredholm. In §1 we show that for a semi-Fredholm operator T (1) lim yt,Tn)Un n— oo exists. Further we compute the limit (1) for a few examples. Received by the editors January 29, 1974. AMS (MOS) subject classifications (1970). Primary 47B30, 47A55; Secondary 47A10.