Can a real operator on $H = L_2 (\mathbb{R})$ have infinitely many linearly independent eigenvector corresponding to a single eigenvalue?

Question

Let $\phi : L_2 (\mathbb{R}) \to L_2 (\mathbb{R})$ be a real operator ,meaning all of it eigenvalues are real, and assume the set of eigenvectors form a basis for $h = L_2 (\mathbb{R})$.

Can $\phi$ has infinitely many linearly independent eigenvector belonging to the same eigenvalue ?

Answer 1

Have you looked at the identity operator on $L^2(\Bbb{R})$ with eigenvalue 1?