Inheritance of quasinilpotency and Drazin invertibility between a matrix and its entry

An extensive matrix class over a complex Banach algebra is considered from the point of view of its quasinilpontency, nilpotency, as well as Drazin and generalized Drazin invertibilities. The matrices belonging to the class are obtained by alterations of a 2 × 2 anti-triangular matrix with one undetermined entry, and the general expression characterizing the class involves powers of the anti-triangular matrix. The main result of the paper determines those values of the powers for which the quasinilpontency of a matrix from the class is equivalent to the quasinilpontency of its undetermined entry. Similar result was established with respect to the nilpotency as well. The paper provides also related observations concerned with relationships between the Drazin and generalized Drazin invertibilities of the matrix and the entry. Though rooted in the work of Drazin published in the mid-twentieth century, the investigations carried out in the paper were inspired by and build upon recent results concerned with the Drazin invertibility.