Operator acting in orthogonal subspace

[seek]

Homework Statement

Consider a normal operator A

If R^{perpendicular}_a1 is the orthogonal complement to the subspace of eigenvectors of A with eigenvalue a1, show that if y exists in all R^{perpendicular}_a1 then Ay exists in all R^{perpendicular}_a1

The Attempt at a Solution

This could be answered very simply, if I knew that all normal operators were linear. Were that the case, A would simply be mapping y onto the subspace in which it already existed. Am I right?

*Edit:
Wikipedia tells me that a normal operator is indeed a linear operator. I think I can do this now.

 

[lippyka]

Let y be a vector in Rperpendiculara1. We know that for any normal operator A, it satisfies the equation A*A* = A*A. Thus, A*y must also be in Rperpendiculara1, since it is an eigenvector of A with eigenvalue a1. Therefore, Ay exists in all Rperpendiculara1.