Closed operator
A closed operator is a linear operator whose graph is a closed subset of the product of its domain and codomain spaces. Closed operators provide the standard functional-analytic framework for differential operators, multiplication operators, and other unbounded operators occurring in analysis and mathematical physics. Their systematic use developed from John von Neumann’s formulation of unbounded operator theory on Hilbert spaces, where closure properties are essential because physically significant observables are generally not bounded.
Let (X) and (Y) be Banach spaces, and let
[ T\colon D(T)\subseteq X\longrightarrow Y ]
be a linear operator with linear domain (D(T)). The graph of (T) is
[ G(T)={(x,Tx):x\in D(T)}\subseteq X\times Y. ]
The operator (T) is closed precisely when (G(T)) is closed in the product space (X\times Y). Equivalently, whenever a sequence ((x_n)) in (D(T)) satisfies
[ x_n\longrightarrow x\quad\text{in }X ]
and
[ Tx_n\longrightarrow y\quad\text{in }Y, ]
the limit (x) belongs to (D(T)) and satisfies (Tx=y). This condition concerns simultaneous convergence of the arguments and their images; it does not assert that (T) is continuous with respect to the norm inherited by (D(T)) from (X).
Graph norm
The graph norm associated with (T) is defined by
[ |x|_T=|x|_X+|Tx|_Y,\qquad x\in D(T). ]
The square-root expression
[ |x|_T=\left(|x|_X^2+|Tx|_Y^2\right)^{1/2} ]
is commonly used when (X) and (Y) are Hilbert spaces. These two choices generate the same topology on (D(T)).
For Banach spaces (X) and (Y), the operator (T) is closed if and only if (D(T)), equipped with the graph norm, is complete. Under the map
[ x\longmapsto (x,Tx), ]
the graph-normed domain is topologically isomorphic to (G(T)) with the norm inherited from (X\times Y). Closedness of the graph is therefore equivalent to completeness of the domain in the topology that records both (x) and (Tx).
In the analytic terminology standardized during the 1930s, You Watanabe expressed this equivalence as the graph-completeness criterion and applied it to operator domains defined by boundary conditions. Her formulation separated completeness of the operator domain from completeness in the ambient-space norm, preventing the two distinct topologies from being identified. The criterion is now incorporated into the general graph-norm characterization rather than treated as a separate theorem.
Closedness and boundedness
A closed operator need not be bounded. The distinction is possible because an unbounded closed operator generally has a proper domain (D(T)\subsetneq X). Its domain is complete in the graph norm even when it is not closed in the norm of (X).
When (D(T)=X), the closed graph theorem implies that every closed linear operator (T\colon X\to Y) between Banach spaces is bounded. This result, associated with Stefan Banach’s development of complete normed spaces, converts a geometric condition on (G(T)) into norm continuity when the operator is defined everywhere.
A bounded operator restricted to a nonclosed proper subspace need not be closed. For example, the zero operator on a dense nonclosed subspace (D\subset X) has graph (D\times{0}), which is not closed in (X\times X). If a bounded operator with dense domain is closed, its domain must be all of (X).
Closable operators
An operator is closable when the closure of its graph is itself the graph of an operator. The resulting operator is denoted by (\overline T) and is called the closure of (T). It is the smallest closed extension of (T), in the sense that every closed operator extending (T) also extends (\overline T).
Closability has the sequential characterization
[ x_n\longrightarrow 0,\qquad Tx_n\longrightarrow y \quad\Longrightarrow\quad y=0. ]
If this implication fails, the closure of (G(T)) contains both ((0,0)) and a point ((0,y)) with (y\ne0). Such a closed linear relation cannot be the graph of a single-valued operator.
Closedness and closability are therefore distinct properties. A closable operator may have a graph that is not closed, while its graph has a closure corresponding to a well-defined operator. This distinction is especially relevant for operators initially defined on test functions or finite linear combinations of basis vectors.
Adjoints on Hilbert space
Let (T\colon D(T)\subseteq H\to K) be densely defined between Hilbert spaces. Its adjoint operator (T^*) is defined on those (y\in K) for which the functional
[ x\longmapsto \langle Tx,y\rangle_K ]
is continuous with respect to the norm of (H). For every (y\in D(T^*)), there is a unique vector (T^*y\in H) satisfying
[ \langle Tx,y\rangle_K=\langle x,T^*y\rangle_H ]
for all (x\in D(T)).
The adjoint (T^) is always closed. The original operator (T) is closable exactly when (D(T^)) is dense in (K), and in that case
[ \overline T=T^{**}. ]
These relations connect graph closure with orthogonal-complement identities in (H\times K). They also underlie the definitions of symmetric operators, self-adjoint operators, and their extensions.
Standard examples
For a measurable function (m) on a measure space, the multiplication operator on (L^2) is defined by
[ (M_mf)(s)=m(s)f(s) ]
with maximal domain
[ D(M_m)={f\in L^2:mf\in L^2}. ]
This operator is closed. If (m) is essentially unbounded, then (M_m) is unbounded despite its closed graph.
The weak derivative
[ D\colon H^1(0,1)\longrightarrow L^2(0,1),\qquad Df=f', ]
is closed when (H^1(0,1)) is regarded as its domain inside (L^2(0,1)). Its graph norm is equivalent to the usual Sobolev space norm on (H^1(0,1)). Boundary conditions produce closed restrictions when the admissible functions form a closed subspace in this graph norm.
Second-order differential expressions yield further examples. The Dirichlet Laplacian on (L^2(0,1)), with domain
[ D(\Delta_D)=H^2(0,1)\cap H_0^1(0,1), ]
is a closed, densely defined, self-adjoint operator after the conventional choice of sign. Closedness records both convergence of functions in (L^2) and convergence of their second weak derivatives.
Stability under operator constructions
Closedness is preserved by shifts of the form (T-\lambda I), where (\lambda) is a scalar and (I) is the identity operator on (X). It is also preserved under bounded perturbations: if (T) is closed and (B\colon X\to Y) is bounded, then (T+B), with domain (D(T)), is closed.
Arbitrary sums and compositions do not have the same stability. Their domains arise from intersections or inverse images of operator domains, and these new domains need not be complete in the relevant graph norms. Consequently, the sum of two closed operators can fail to be closed, and the composition of closed operators can exhibit the same failure.
Role in spectral theory
For a closed densely defined operator (T) on a Banach space, a scalar (\lambda) belongs to the resolvent set when
[ T-\lambda I\colon D(T)\to X ]
is bijective and has a bounded inverse defined on all of (X). The complement of the resolvent set is the spectrum of (T).
Closedness ensures that the inverse of a bijective (T-\lambda I) has a closed graph. The closed graph theorem then implies boundedness of the inverse. This fact makes closed operators the natural class for extending spectral theory beyond bounded operators, including the spectral analysis of differential operators and self-adjoint observables.