I am working in the space of $l^2 (\alpha_1,\alpha_2,...)$ such that $\sum_{j=1}^\infty |\alpha_j|^2<\infty$. I need to classify the spectrum of the folloowing operator $$L|v> = (\alpha_2,\alpha_3,...)$$ where $|v> = (\alpha_1,\alpha_2,...)$.
I have realised that $||L||=1$ so the spectrum must be in $|z|\leq 1, z\in \mathbb{C}$, since $L$ is a linear bounded operator. Now, as far as I know, I should check whether $(L - \lambda I)^{-1}$ exists and if it is bounded or not but I do not know how to approach this.
If $|\lambda|<1$ then the vector $$ v=(1,\lambda,\lambda^2, \lambda^3, ...) $$ lies in your space and $$L(v)=\lambda v.$$ So $\lambda$ is an eigenvalue and hence the open disk is contained in the point spectrum. Moreover, you already know that the spectrum of $L$ is contained in the closed disk, so it is now enough to analyze complex numbers $\lambda$ with $|\lambda|=1$. Could these be eigenvalues?