On invertibility of combinations of \(k\)-potent operators (Q417602)

A bounded linear operator \(P\) on a complex Hilbert space \(H\) is called \(k\)-potent if \(P^k=P\), where \(k \geq 2\). Let \(P_1,P_2\) be non-zero \(k\)-potents on \(H\). The authors study certain properties of operators of the form \(c_1P_1+c_2P_2-c_3P_1^sP_2^{k-1-s}\), where \(c_1,c_2\) and \(c_3\) are complex numbers, \(k\) is a positive integer, \(k \geq 2\) and \(s \leq k-1\). Questions concerning the invertibility and generalized invertibility of these operators are also studied.