So I have the following operator: $A(x_1,x_2,x_3,...) = (x_2,x_4,x_6)$ and I need to find its spectrum and its decomposition so I started by calculating the point spectrum: $$ Ax = \lambda *x \implies (x_2,x_4, x_6,...) = (\lambda x_1, \lambda x_2,...) $$
So from here we can take any $\lambda \neq 0$ and get that it's an eigenvalue with the vectors of the form: $$ (x_1,\lambda x_1, x_3, \lambda^2 x_1, x_5, \lambda x_3,...)$$
Is this correct, does it mean that my residual and continuous spectrum are empty? Please don't provide me with a solution just point to my mistakes.