graphon shift operator eigenvalues

[[concept]]

For a graphon $W$ and graphon shift operator $T_W$:

Note 1

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

Proof

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)

Note 2

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

Proof

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

Note 3

If we then order the eigenvalues as $\{{\lambda }_j{\}}_{j\in \mathbb{Z}\setminus \{0\}}$ so that ${\lambda }_{-1}\le {\lambda }_{-2}\le \cdots \le 0\le \cdots \le {\lambda }_2\le {\lambda }_1$ Then, for any $c\ne 0$ $\left|\{{\lambda }_i:|{\lambda }_i|\ge c\}\right|=n_c\le \infty$

Proof

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

Mentions

Mentions

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