Abstract
The concept of spectrum of a finite graph discussed in the previous chapter is not extended directly to an infinite graph. An alternative approach is brought by the idea of spectral distribution on the basis of quantum probability. In short, quantum probability provides algebraic axiomatization of the traditional (Kolmogorovian) probability theory, and is useful for statistical questions in non-commutative analysis.
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- 1.
The terms “positive semidefinite,” “positive definite,” “non-negative definite,” “positive,” “non-negative,” “strictly positive,” etc. are mixed up in literatures. Our wording is after Bhatia [16] and Horn-Johnson [80].
- 2.
For a linear operator \(T:\mathscr {H}_0\rightarrow \mathscr {H}\) let \(\mathop {\mathrm {Dom}\,}\nolimits (T^*)\) be the space of all \(\xi \in \mathscr {H}\) such that the linear function \(\mathscr {H}_0\ni \eta \mapsto \langle \xi ,T\eta \rangle \) is continuous. If \(\xi \in \mathop {\mathrm {Dom}\,}\nolimits (T^*)\), by Riesz theorem there exists a unique \(\zeta \in \mathscr {H}\) such that \(\langle \xi ,T\eta \rangle =\langle \zeta ,\eta \rangle \) for all \(\eta \in \mathscr {H}_0\). A linear operator \(T^*\) defined by \(\zeta =T^*\xi \), \(\xi \in \mathop {\mathrm {Dom}\,}\nolimits (T^*)\), is called the adjoint operator of T. From the definition we do not know relation between \(\mathscr {H}_0\) and \(\mathop {\mathrm {Dom}\,}\nolimits (T^*)\).
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Obata, N. (2017). Spectral Distributions of Graphs. In: Spectral Analysis of Growing Graphs. SpringerBriefs in Mathematical Physics, vol 20. Springer, Singapore. https://doi.org/10.1007/978-981-10-3506-7_3
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DOI: https://doi.org/10.1007/978-981-10-3506-7_3
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