Graph of a linear operator

The graph of $T:X\to Y$ is the linear subspace $\Gamma (T)=\{(x,Tx):x\in X\}$ of $X\times Y$.

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 resolvent formalism studies the spectrum through the resolvent of an operator and its dependence on a complex spectral parameter. This connects operator theory with complex analysis.

For a closed linear operator $A$, the resolvent is the bounded inverse $(zI-A{)}^{-1}$ when $z$ lies in the resolvent set of an operator. This definition includes unbounded $A$. It specializes to the resolvent of an element when $A$ is an element of a Banach algebra.

For a Hermitian matrix $X$ and nonreal $z$, this is the negative of the usual resolvent of an operator convention $(zI-X{)}^{-1}$. Its normalized trace is the Stieltjes transform of a measure of the empirical spectral measure with kernel $(x-z{)}^{-1}$. Naming the convention prevents sign errors in self-consistency identities.

For a zero-diagonal real symmetric matrix, put $q_i=x_i^T(X^{(i)}-zI{)}^{-1}{x}_i$ and $g_N=N^{-1}\mathrm{Tr}(X-zI{)}^{-1}$. The Schur complement formula for a diagonal resolvent entry gives $g_N=-N^{-1}{\sum }_i(z+q_i{)}^{-1}$. Subtracting the comparison value $-(z+g_N{)}^{-1}$ gives ${\epsilon }_N=N^{-1}{\sum }_i(q_i-g_N)/[(z+g_N)(z+q_i)]$. Upper-half-plane positivity and the principal minor resolvent trace bound control this defect by the average of $|q_i-g_N^{(i)}|$ plus a trace correction.

The eigenvalue interlacing of a Hermitian matrix and a principal minor bounds the difference of their resolvent traces by a constant times $|\mathrm{Im}z{|}^{-1}$. When both traces are normalized by the original dimension $N$, the bound acquires $N^{-1}$. One proof writes the difference using the interlacing counting functions, whose difference is at most one, and bounds the integral of $|t-z{|}^{-2}$ by $\pi /|\mathrm{Im}z|$. This gives the admissible absolute constant $c=\pi$.

By an orthonormal eigenbasis, the imaginary part of the Stieltjes matrix resolvent has eigenvalues $\eta /|\lambda -z{|}^2>0$. Thus its normalized trace has positive imaginary part, and its quadratic form at any vector has nonnegative imaginary part. This yields $|z+g|\ge \eta$ and $|z+q|\ge \eta$ for the normalized trace $g$ and a quadratic form $q$ used in Schur-complement estimates.

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 operator $R(z,A)=(zI-A{)}^{-1}$.

A linear map between Banach spaces is continuous if and only if its graph is closed.

A linear map $T:E\to F$ between Fréchet spaces is continuous if its graph of a linear operator is closed in $E\times F$. The closed graph is itself a Fréchet space. Its first-coordinate projection is a continuous linear bijection onto $E$, so the open mapping theorem for Fréchet spaces makes the inverse continuous. Compose that inverse with the second-coordinate projection to obtain $T$. Completeness of both original spaces is part of the theorem.

For a closed graph, the projection $\Gamma (T)\to X$ is a bounded bijection between Banach spaces. Its bounded inverse followed by projection to $Y$ is $T$.

If bounded $S:X\to Y$ is injective and $\mathrm{im}T\subseteq \mathrm{im}S$, then $R=S^{-1}T$ has closed graph: $x_n\to x$ and $R{x}_n\to z$ imply $Sz=Tx=S(Rx)$ and hence $z=Rx$.

On incomplete $c_{00}$ with the ${\ell }^2$ norm, let $S(x_n)=(x_n/n)$ and let $T$ be inclusion into ${\ell }^2$. Their images agree, but $S^{-1}T(e_n)=n{e}_n$ is unbounded.

Let $V$ be a real Banach space and $T:V\to V^{\ast }$ be linear with $T_v(v)\ge 0$ for every $v$. If $v_n\to 0$ and $T_{v_n}\to f$ in $V^{\ast }$, positivity of $T_{v_n+tw}(v_n+tw)$ and passage to the limit givefor every real $t$ and every $w$. Taking small $t$ of either sign forces $f(w)=0$. Thus the graph of $T$ is closed, and the closed graph theorem makes $T$ continuous.