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 $Ax=y$.

A densely defined closed operator is sectorial of angle $\theta$ if its spectrum lies in $\{z:|\arg z|\le \theta \}\cup \{0\}$ and, for every larger angle $\varphi <\pi$, its resolvent outside that larger sector satisfies $\Vert (z-A{)}^{-1}\Vert \le C_{\varphi }/|z|$. If $\theta <\pi /2$, then $-A$ generates an analytic semigroup. A real shift of the generator allows exponential growth without changing the short-time smoothing estimates.

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.

Let $X$ be a normed space, $Y$ a Banach space, and $T:X\to Y$ a linear map. If its kernel is complete in the original norm and $S:Y\to X$ is a bounded right inverse, then the graph norm $\Vert x{\Vert }_T=\Vert x{\Vert }_X+\Vert Tx{\Vert }_Y$ is complete, even if $X$ with its original norm is incomplete or $T$ is unbounded.

For a Cauchy sequence $(x_n)$ in the graph norm, $y_n=T{x}_n$ converges to $y\in Y$. The elements $w_n=x_n-S{y}_n$ lie in the kernel and satisfy $\Vert w_n-w_m{\Vert }_X\le (1+\Vert S\Vert )\Vert x_n-x_m{\Vert }_T$. Hence $w_n\to w\in \ker T$. The element $x=w+Sy$ satisfies $Tx=y$ and

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.

For an everywhere-defined linear map $T:E\to F$ between Banach spaces, its separating space measures the possible limiting defect in continuity at zero. It is a closed linear subspace: linear combinations combine witnessing sequences, and diagonal choices with $\Vert x_j\Vert <1/j$ and $\Vert T{x}_j-y_j\Vert <1/j$ handle a convergent sequence $y_j$ of defects. The closed graph theorem gives $T$ continuous exactly when $S(T)=\{0\}$, since a limit $(x_n,T{x}_n)\to (x,y)$ has $y-Tx\in S(T)$.

Let $S:E\to Y$ be linear between Banach spaces, let $T_n:E\to E$ be bounded, and let $U_n:Y\to Z_n$ be bounded. If $U_{n}S{P}_m$ is continuous whenever $m>n$, then $U_{n}S{P}_n$ is continuous for all sufficiently large $n$. After rescaling the bounded operators, assume their norms are at most one. If the conclusion failed, discontinuity would allow successively tiny vectors $x_j$ whose diagonal images dominate all previous contributions. Choose their norms also to control the continuous maps $U_{n_i}S{P}_{n_i+1}$ on subsequent tails. The convergent sum $x={\sum }_j{P}_{n_j}{x}_j$ would then satisfy $\Vert U_{n_j}Sx\Vert >j$ despite $\Vert U_{n_j}Sx\Vert \le \Vert Sx\Vert$, a contradiction. The tail is factored through $P_{n_i+1}$ before applying $S$, so no continuity of $S$ is assumed.

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.