Closed linear operator

A linear operator $A:D(A)\subseteq X\to Y$ is closed when its graph is closed in $X\times Y$. Equivalently, if $x_n\to x$ and $A{x}_n\to y$, then $x\in D(A)$ and $Ax=y$.

The graph norm of $A:D(A)\subseteq X\to Y$ is $\Vert x{\Vert }_A=\Vert x{\Vert }_X+\Vert Ax{\Vert }_Y$. If $X$ and $Y$ are Banach spaces, then $A$ is a closed linear operator exactly when its domain is complete in the graph norm.

The resolvent set $\rho (A)$ consists of the complex numbers $z$ for which $zI-A:D(A)\to X$ is bijective and has a bounded inverse on $X$. This inverse is the resolvent of an element $R(z,A)=(zI-A{)}^{-1}$.