Find the spectrum σ(A).

Question

Let A: l² → l² be given by A(x₁,x₂,...)=($\frac{1}{2}$x₁,$\frac{2}{3}$x₂,...,$\frac{n}{n+1}$x_n,...). Find the spectrum $\sigma$(A) and the spectral radius.

Answer 1

$A$ is a real multiplication operator, so it is self-adjoint; observe that for $x,y\in\ell^2(\mathbb R)$ $$ \langle Ax,y\rangle = \sum_{j=1}^\infty \left(\frac j{j+1}x_j\right)y_j = \sum_{j=1}^\infty x_j\left(\frac j{j+1}y_j\right) = \langle x,Ay\rangle, $$and hence $A=A^\star$. Self-adjoint operators have real spectrum, and indeed $A-\lambda I$ is not invertible precisely when $\lambda = \frac j{j+1} =:\lambda_j$. It is clear that the eigenspace corresponding to $\lambda_j$ is the span of $\{e_j\}$, with $(e_j)_{j\in\mathbb N}$ being the canonical orthonormal basis of $\ell^2(\mathbb R)$. So the spectrum of $A$ is $$\sigma(A) = \bigcup_{j=1}^\infty \left\{\frac j{j+1} \right\}, $$ and hence the spectral radius is $\rho(A)=1$.