Extremal compressions of closed operators

Abstract:

Let $X$ be a Banach space, $A$ a closed linear operator on $X$, and ${\lambda _1}, \ldots ,{\lambda _n}$ isolated eigenvalues of $A$ of finite multiplicity. If $P$ is a projection on $X$ such that ${\lambda _1}, \ldots ,{\lambda _n}$ belong to the resolvent of the compression of $A$ on the range of $P$ it is easy to see that \[ \dim N\left ( P \right ) \geq \max \left \{ {\dim N\left ( {{\lambda _i} - A} \right ):1 \leq i \leq n} \right \}.\] It is shown that there exist such projections where we have equality in this inequality.