Confusion between spectral radius of matrix and spectral radius of the operator

Question

The adjacency matrix $A(G)$ of an infinite undirected graph $G$ is considered as a bounded self-adjoint linear operator $A$ on the Hilbert Space $l^2(G)$ (last section of https://en.wikipedia.org/wiki/Spectral_radius). Then the spectral radius of the $A(G)$ would be the spectral radius of the $A$ (I presume).

The spectral radius of the matrix $A(G)$ by Rayleigh quotient formula $R(A(G),x)=\frac{x^*A(G)x}{x^*x}=\frac{<x,A(G)x>}{<x,x>}$ is $R(A(G),v_{max})=\lambda_{max}$, that is, $\rho(A(G))=\sup_{x\neq0}\frac{<x,A(G)x>}{<x,x>}$.

And the spectral radius of the linear operator by Gelfand's formula is $\rho(A)=\lim_{k\to\infty}\|A^k\|^{\frac1k}$ which converts into $\rho(A)=\|A\|$ for self-adjoint operators because they are normal. The operator norm for the p-norm vectors of $l^2 (G)$ is $\|A\|=\|A\|_2=\sup_{x\neq0}\frac{\|Ax\|_2}{\|x\|_2}$(https://en.wikipedia.org/wiki/Matrix_norm). Which means $\rho(A)=\sup_{x\neq0}\frac{<Ax,Ax>^\frac12}{<x,x>^\frac12}$.

I am confused: why $\rho(A(G))$ and $\rho(A)$ are not equal? I am surely making a mistake or missing something important. Please help.

Answer 1

Your two expressions for the spectral radius are not identical, but their values on (finite) symmetric matrices are equal.

One comment. For an infinite graph of bounded degree, it's perfectly reasonable to view its adjacency matrix as an operator on $\ell^2$, but it's not clear to me that this is even an established convention, let alone a rule.

Answer 2

As, $R(A(G),x)=\frac{<x,A(G)x>}{<x,x>}$,

$R(A^2(G),x)=\frac{<x,A^2(G)x>}{<x,x>}=\frac{<A(G)x,A(G)x>}{<x,x>}=\frac{\|A(G)x\|_2^2}{\|x\|_2^2}=\|A(G)\|_2^2$ for $x=v_{max}$.

Also, $\rho(A^2)=\rho(A)^2=\|A\|^2_2$.

So, $R(A^2(G),x)=\rho(A^2)$ for $x=v_{max}$.

If $\mu$ is the spectral radius of $A^2$, then $\sqrt{\mu}$ is the spectral radius of $A$.(similar argument holds for A(G)).

Which means that the both spectral radii are equal.