How to find the point spectrum $\sigma_p(A)$ of $A$

Question

How to find the point spectrum $\sigma_p(A)$ of $A$

Consider the Hilbert space $H=l_2$ over the complex field and $A:H\rightarrow H.$

$A(x)=(x_1,0,0,\frac{1}{2}x_4,0,0,0,0,\frac{1}{3}x_9,0,...,0,\frac{1}{n}x_{n^2},0,...)$

Could you please help.

Answer 1

The point spectrum is the set of all eigenvalues, you can think of it as the set of $\lambda$ such that $A-\lambda I$ is not invertible,I.e is not injective. In this case it is the set of values $\{\lambda:\lambda=(1/n)x_{n^2}:n\in\Bbb N\}$