1. ## Self-adjoint transformation

V is an inner product space over C, and T: V -> V is a self-adjoint transformation. I need to prove that T+iaI is invertible for every real, non-zero a.

I know that a transformation is not invertible iff it has an eigenvalue 0, but how can i use this?

Thanks for any help.

2. ## Re: Self-adjoint transformation

Assume that $T+iaI$ has eigenvalue 0. Derive a contradiction with the fact that eigenvalues of $T$ are real.