Kato decomposition of linear pencils
T. Kato  found an important property of semi-Fredholm pencils, now called the Kato decomposition. M. A. Kaashoek  introduced operators having the property $P(S: k)$ as a generalization of semi-Fredholm operators. In this work, we study this class of operators. We show that it is characterized by a Kato-type decomposition. Other properties are also proved.