For a compact normal operator, the space can be written as the sum of generalized eigenspaces. So every element can be written as a linear combination of the eigenvectors, one from each eigenspace. But an eigenspace may have more than one linearly independant eigenvector so does this say 2 (linearly independent) eigenvectors from the same eigenspace cannot be involved in the same linear combination?
Thanks
Let $K$ be the operator in question. Its spectral decomposition is $$K=∑_{n}λ_{n}P_{n},$$ where $P_{n}$ is the eigenprojector associated with the eigenvalue $λ_{n}$. It is finite dimensional and we can write $$P_{n}=∑_{j_{n}}P_{j_{n}}$$ with the $P_{j_{n}}$ a finite set of one-dimensional orthogonal projectors. Hence $$K=∑_{n}∑_{j_{n}}λ_{n}P_{j_{n}}$$