graphon shift operator

[[concept]]

The graphon shift operator

Let $W$ be a graphon and $x$ a graphon signal. The graphon shift operator is the integral linear operator $T_{W} : L^{2} ([0, 1]) \to L^{2} ([0, 1])$ which maps $x \mapsto T_{W} x$

T_{W} x = \int_{0}^{1} W (u, \cdot) x (u) d x

Note

A graphon shift operator is a Hilbert-Schmidt integral operator (continuous and compact) because graphons are bounded

⟹ W \in L^{\infty} ⟹ W \in L^{2}

And has Hilbert-Schmidt norm

| | T_{W} | |_{H S}^{2} = | | W | |_{2}^{2} = \int_{0}^{1} \int_{0}^{1} W^{2} (u, v) d u d v

Review

#flashcards/math/dsg

A graphon shift operator is a {ass||Hilbert-Schmidt integral operator||type}(ie it is {hah||continuous and compact}) because {hha||graphons are bounded||why type}

Graphon shift operators are also {as||self-adjoint||special property} because graphons are {ha||symmetric||why}.

Mentions

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2025-04-02 lecture 17	2025-06-05
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graphon shift operators are self-adjoint	2025-05-30
we can write a graphon in the basis of its shift operator	2025-05-30
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Created 2025-03-31 Last Modified 2025-06-05