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.

A not necessarily continuous, linear or multiplicative function from an algebra A into itself is called a 2-local automorphism if agrees with an automorphism of A at each pair of points in A. In this paper, we study when a 2-local automorphism of a C-algebra, or a standard operator algebra on a locally convex space, is an automorphism.

Let and be a real uniformly smooth Banach space. Let be a nonempty closed convex subset of and be a strictly pseudocontractive mapping in the sense of FE Browder and WV Petryshyn [1]. Let be a bounded sequence in and be real sequences in satisfying some restrictions. Let be the bounded sequence in generated from any given by the Ishikawa iteration method with errors: , , . It is shown in this paper that if is compact or demicompact, then converges strongly to a fixed point of .