Let $H$ be a Hilbert space, $P : H \rightarrow H$ a positive-definite (bounded) operator and $K : H \rightarrow H$ a compact (not necessarily self-adjoint) operator.
Let $T = P + K$. In particular, $T$ is Fredholm of index zero.
For each negative real number $c$, let $S_c$ denote the set of solutions $x$ of the equation
$$T(x) = cx.$$
How can I show that $S = \cup_{c < 0} S_c$ is finite?
I know about the Fredholm alternative, but I don't know how to apply it to this situation, since I need the operator $T$ to be of the form $\lambda\mathrm{Id} + K$, and the usual tricks (changing basis, composing with inverses, etc.) don't seem to quite work. If there is such a trick, I'd appreciate a more detailed explanation of why it works. Thanks!