Invertibility and Fredholmness of linear combinations of quadratic, k-potent and nilpotent operators

Abstract

Recently, the invertibility of linear combinations of two idempotents has been studied by several authors. Let P and Q be idempotents in a Banach algebra. It was shown that the invertibility of P + Q is equivalent to that of aP + bQ for nonzero a, b with a + b = 0 . In this note, we obtain a similar result for square zero operators and those operators satisfying x2 = dx for some scalar d . More generally, we show that if P,Q satisfy a quadratic polynomial (x − c)(x − d) then the linear combination aP + bQ − c(a + b) being invertible or Fredholm (and the index) is independent of the choice of the nonzero scalars a, b . However, this is not the case when P and Q are involutions, unitaries, partial isometries, k -potents ( k 3 ) and other nilpotents, as counterexamples are provided. Mathematics subject classification (2000): 15A03, 47A53, 47C05.