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

$0 \leq W \leq 1$ and
$| | W | |_{2} \leq 1$
(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 $| | T_{W} | |_{H S} \leq 1$ .

Note 3

If we then order the eigenvalues as ${λ_{j}}_{j \in Z ∖ {0}}$ so that

λ_{- 1} \leq λ_{- 2} \leq \dots \leq 0 \leq \dots \leq λ_{2} \leq λ_{1}

Then, for any $c \neq 0$

| {λ_{i} : | λ_{i} | \geq c} | = n_{c} \leq \infty

Proof

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

Mentions

File
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we can write a graphon in the basis of its shift operator

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