Unbounded linear operators: domain, graph and extension

Throughout this page $H$ is a complex Hilbert space (Hilbert space) with inner product linear in the first variable and conjugate-linear in the second (Real and complex inner-product spaces and their induced length).

A (possibly unbounded) linear operator on $H$ is a linear map $T:D(T)\to H$ (Linear map between vector spaces over the same field) whose domain $D(T)\subseteq H$ is a linear subspace (Linear subspace of a vector space). The domain is part of the data: $T=S$ means $D(T)=D(S)$ and $Tx=Sx$ for all $x\in D(T)$. One writes $T\subseteq S$, and calls $S$ an extension of $T$, when $D(T)\subseteq D(S)$ and $Sx=Tx$ for every $x\in D(T)$.

The graph of $T$ is $\mathrm{\Gamma }(T):=\{(x,Tx):x\in D(T)\}\subseteq H\oplus H\mathrm{.}$ The direct sum $H\oplus H$ is read as the complex vector space of pairs with coordinatewise operations and the inner product $⟨(x,y),(u,v)⟩=⟨x,u⟩+⟨y,v⟩$, whose induced norm is $\mathrm{\parallel }(x,y)\mathrm{\parallel }=(\mathrm{\parallel }x{\mathrm{\parallel }}^2+\mathrm{\parallel }y{\mathrm{\parallel }}^2{)}^{1\mathrm{/}2};$ it is complete because $H$ is (Complete metric space: every Cauchy sequence converges in the space). The graph norm on $D(T)$ is $\mathrm{\parallel }x{\mathrm{\parallel }}_T:=(\mathrm{\parallel }x{\mathrm{\parallel }}^2+\mathrm{\parallel }Tx{\mathrm{\parallel }}^2{)}^{1\mathrm{/}2},$ so that $x\mapsto (x,Tx)$ is an isometric isomorphism of $(D(T),\mathrm{\parallel }\cdot {\mathrm{\parallel }}_T)$ onto $\mathrm{\Gamma }(T)$ with the norm restricted from $H\oplus H$.

$T$ is closed when $\mathrm{\Gamma }(T)$ is a closed subset of $H\oplus H$. A linear operator is determined by its graph, and the following elementary translations are used silently below: $\mathrm{\Gamma }(T)$ is a linear subspace of $H\oplus H$; $\mathrm{\Gamma }(T)\cap (\{0\}\oplus H)=\{(0,0)\}$; the image of $\mathrm{\Gamma }(T)$ under the first coordinate projection is $D(T)$; and $T\subseteq S$ if and only if $\mathrm{\Gamma }(T)\subseteq \mathrm{\Gamma }(S)$. In particular a closed operator is exactly one whose graph is a closed subspace of $H\oplus H$.

Notation. No boundedness of $T$ is assumed, and $T$ is frequently called unbounded to stress this; a bounded everywhere defined operator on $H$ is the special case $D(T)=H$ in which $\mathrm{\Gamma }(T)$ is a closed subspace by continuity. The letter $I$ denotes the identity operator on $H$ with domain $H$.

Completeness of $H\oplus H$. If $(x_n,y_n)$ is a Cauchy sequence in $H\oplus H$, then $\mathrm{\parallel }{x}_n-x_m\mathrm{\parallel }\le \mathrm{\parallel }(x_n-x_m,y_n-y_m)\mathrm{\parallel }$ and $\mathrm{\parallel }{y}_n-y_m\mathrm{\parallel }\le \mathrm{\parallel }(x_n-x_m,y_n-y_m)\mathrm{\parallel }$ show that $(x_n)$ and $(y_n)$ are Cauchy in $H$; let $x,y$ be their limits. Then $\mathrm{\parallel }(x_n,y_n)-(x,y){\mathrm{\parallel }}^2=\mathrm{\parallel }{x}_n-x{\mathrm{\parallel }}^2+\mathrm{\parallel }{y}_n-y{\mathrm{\parallel }}^2\to 0$, so $H\oplus H$ is complete. Consequently a sequence in $H\oplus H$ converges exactly when its two coordinate sequences converge in $H$.