On the unicity of the Kato decomposition. (Q1408789)

The author defines and studies the Kato decomposition associated to a couple of operators \((T,S)\) acting on a Banach space \(X\) into another Banach space \(Y\). Using a Kaashoek property \(P(S, k)\), where \(k\) is a positive integer, he studies the unicity of the Kato decomposition under the assumption of its existence. In particular, he shows that the couple of operators \((T, S)\) has a nontrivial Kato decomposition if and only if \(0\) is a pole of order \( p> 0\) of the resolvent operator \((T + \lambda S)^ {-1}.\)