Let $H$ be a Hilbert space and $L$ is a self- adjoint operator with a discrete spectrum $\{\lambda_{j}\}$.
I would ask about this inequality because I don't understand it$$\displaystyle{\|(L- \lambda I)^{-1}L\| \leq \sup_{j}\frac{\lambda_{j}}{|\lambda - \lambda_{j}|}}\,\mbox{for} \, \lambda \notin \sigma(L).$$
And what are the conditions that guarantee to have similar estimation in a Banach space?
In the conditions you have $L=\sum_j\lambda_jP_j$ for pairwise orthogonal projections $\{P_j\}$ which add to the identity. Then for $\lambda\not\in\sigma(L)$ $$ (L-\lambda I)^{-1}L=\sum_j\dfrac{\lambda_j}{\lambda_j-\lambda}\,P_j. $$ The norm of such operator (a diagonal operator) is simply the supremum over the absolute values of the diagonal entries, which proves your estimate (with $|\lambda_j|$ in the numerator).
I don't see an easy way to characterize this property in a general Banach algebra.
Edit: the result is actually true (again with equality) for an arbitrary normal operator $L$. Indeed, you can always represent a normal operator $L$ (via the Gelfand Transform) as the function $z\mapsto z$ in $C(\sigma(L))$. Then $$ \|(L-\lambda I)^{-1}L\|=\sup\left\{\frac{z}{z-\lambda}:\ z\in\sigma(L)\right\}. $$