On the Kato Decomposition of Quasi–Fredholm and B–Fredholm Operators
نویسنده
چکیده
We construct a Kato-type decomposition of quasi-Fredholm operators on Banach spaces. This generalizes the corresponding result of Labrousse for Hilbert space operators. The result is then applied to B-Fredholm operators. Denote by B(X) the set of all bounded linear operators acting on a Banach space X. For T ∈ B(X) denote by N(T ) = {x ∈ X : Tx = 0} and R(T ) = TX its kernel and range, respectively. Let T ∈ B(X). For n ≥ 0 set αn(T ) = dimN(T)/N(T) and βn(T ) = dimR(T)/R(T). For n = 0 these numbers reduce to the well-known defect numbers α0(T ) = dimN(T ) and β0(T ) = codimR(T ). It is possible to show that αn(T ) = dim ( N(T ) ∩ R(T) ) , and similarly, βn(T ) = codim ( R(T ) + N(T) ) . This implies that the sequences αn(T ) and βn(T ) are nonincreasing. Further we define the ”difference sequence” kn(T ), see [4], by kn(T ) = dim ( R(T) ∩N(T ) ) / ( R(T) ∩N(T ) ) .
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