graphon shift operator eigenvalues

the eigenvalues of $T_W$ lie in $[-1,1]$.

This follows from

• $0\le W\le 1$ and
• $||W|{|}_2\le 1$
  (The reasoning is analogous to matrix norms are bounded below by the spectral radius)

the only accumulation point for the eigenvalues of $T_W$ is 0.

This follows directly from the second part of the spectral theorem for self-adjoint compact operators on Hilbert spaces and because $||T_W|{|}_{HS}\le 1$.

If we then order the eigenvalues as $\{{\lambda }_j{\}}_{j\in \mathbb{Z}\setminus \{0\}}$ so that

Then, for any $c\ne 0$

This is equivalent to the convergence/accumulation of eigenvalues about 0 result above presented in a possibly more convenient way.