Let be an infinite-dimensional complex Banach space and a continuous/bounded linear operator. Suppose that for every nonzero , the "orbit space" is finite-dimensional, say has dimension .

I would like to prove or disprove the following conjecture:

Conjecture 1:has infinitely many linearly independent eigenvectors.

Conjecture 1 easily follows from the following:

Conjecture 2:There is a sequence of nonzero vectors in such that for all .

Proof that Conjecture 2 implies Conjecture 1:First we claim that for each nonzero , has an eigenvector in . For proof, notice that since is -invariant then the restriction of to is a linear operator on a finite-dimensional vector space. Being defined on a nonzero but finite-dimensional vector space, this means has eigenvalues. Thus we can find an eigenvector under , and the claim is proved. In particular, for each we can find an eigenvector . Next we claim that the resulting sequence is linearly independent. For suppose otherwise, towards a contradiction. Then we can find , not all zero, such that . Without loss of generality suppose . Then . However , contradicting the hypothesis that .

So if anyone can help me prove either conjecture, that would be great. Thanks!