spectrum of two bounded linear operators

Question

Suppose that L and B are bounded linear operator on H, assume $0\in \rho(L) \cap \rho(L+B)$ and that $L^{-1}$ is compact. Prove that L+B also has a compact inverse.

Answer 1

Recall that the compact operators form a two-sided ideal in $B(X)$ for any normed vector space $X$.

Writing $T=L^{-1}LT$ shows that every bounded operator is therefore compact. In particular your $(L+B)^{-1}$.

Remark: actually, the existence of a compact invertible operator in $B(X)$ is equivalent to $X$ being finite-dimensional. Indeed, this is equivalent to the identity being compact, hence to the closed unit ball being compact. If $X$ has finite dimension, it follows from the equivalence of norms that the closed unit ball is compact. And if the closed unit ball is compact, it is a consequence of Riesz's lemma that $X$ has finite dimension.