Consider the Hilbert space $l^2$ and the Left and Right-shift operator

\begin{align*} L(x_1,x_2,\cdots) &= (x_2,x_3,\cdots)\\ R(x_1,x_2,\cdots) &= (0,x_1,x_2,\cdots ) \end{align*}

I'm trying to find out the continuous spectrum of $R+L$, as I've found out the point specturum and residual spectrum are both empty using the fact $R+L$ is self-adjoint. Now, I am trying to figure out when does $(\lambda I-(R+L))^{-1}$ become unbounded so that I can distinguish the continuous spectrum and the resolvent set. Thanks!


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