Consider a compact operator $T$ on a Hilbert space with simple eigenvalue $\lambda$. Is it then true that left (the eigenvector of the adjoint with complex-conjugate eigenvalue) and right eigenvector are never orthogonal to each other?

If true, could this generalize in a way to degenerate eigenvalues? - I.e. there is a choice of eigenvectors such that this holds?

Consider a compact operator $T$ on a Hilbert space with algebraically simple eigenvalue $\lambda$. Is it then true that left (the eigenvector of the adjoint with complex-conjugate eigenvalue) and right eigenvector are never orthogonal to each other?

If true, could this generalize in a way to degenerate eigenvalues? - I.e. there is a choice of eigenvectors such that this holds?