Semi-Fredholm operators with finite ascent or descent and perturbations
Semi-Fredholm operators with finite ascent or descent and perturbations
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DOI：
10.1090/s0002-9939-1995-1286004-6
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发表时间：
1995-12
期刊：
影响因子：
--
通讯作者：
V. Rakočević
中科院分区：
文献类型：
--
作者：
V. Rakočević
In this note we prove that the collection of upper (lower) semiFredholm operators with finite ascent (descent) is closed under commuting operator perturbations that belong to the perturbation class associated with the set of upper (lower) semi-Fredholm operators. Then, as a corollary we get the main result of S. Grabiner (Proc. Amer. Math. Soc. 71 (1978), 79-80). Let X be an infinite-dimensional complex Banach space and denote the set of bounded (compact) linear operators on X by B(X) (K(X)). For T in B(X) throughout this paper N(T) and R(T) will denote, respectively, the null space and the range space of T. Set N°°(T) = U^F"), R°°(T) = f]nR(Tn), a(T) = dimN(T) and ß(T) = dimX/R(T). Recall that an operator T e B(X) is semi-Fredholm if R(T) is closed and at least one of a(T) and ß(T) is finite. For such an operator we define an index i(T) by i(T) = a(T) ß(T). Let 0+(X) (-(X)) denote the set of upper (lower) semi-Fredholm operators, i.e., the set of semi-Fredholm operators with a(T) +(X), a(T) -(X), a(T) < oo and K e P(Q-(X)) =* d(T + K) < oo. Received by the editors March 24, 1994 and, in revised form, June 16, 1994. 1991 Mathematics Subject Classification. Primary 47A53, 47A55.
