Unbounded operators and their domains

An operator A on H is a linear map from a linear subspace D(A) into H. It is densely defined when that subspace is dense in the Hilbert norm. On ℓ2, define (Ax)_n=n x_n with domain consisting of sequences for which the sum of n²|x_n|² is finite. Finite-support sequences lie in this domain, making it dense. The rule is unbounded because the unit input e_n has output norm n. Each permitted input nevertheless has a well-defined output in H.

The graph consists of pairs (x,Ax). A is closed when convergence of x_j to x and Ax_j to y forces x into D(A) with Ax=y. Closedness is a compatibility between input and output limits, not a norm bound. The diagonal example is closed and unbounded on a proper domain. Equipping D(A) with the graph norm makes it complete when A is closed. An everywhere-defined closed linear operator between Banach spaces would instead be bounded by the closed graph theorem.

For densely defined A, a vector y belongs to D(A*) when the functional x↦inner-product(y,Ax), initially defined on D(A), is bounded with respect to the H norm of x. Riesz representation then determines A*y uniquely. Density ensures uniqueness of the representing vector. The adjoint's domain can be larger than D(A), and need not be all of H. This is the crucial difference from the adjoint of a bounded everywhere-defined operator.

A densely defined operator is symmetric if inner-product(y,Ax)=inner-product(Ay,x) for all x and y in D(A). Equivalently A is contained in A*: it has the same action on its domain, but the adjoint may admit more inputs. Self-adjoint means equality of both actions and domains. A symmetric operator has real expectation values on its domain, yet that alone does not grant the spectral theorem or a unique unitary evolution. Self-adjoint extension theory addresses which additional domains can supply those properties.

On L2(0,1), take A=−i d/dx on the Sobolev domain H1_0(0,1), whose endpoint traces vanish. Integration by parts makes A symmetric because the boundary term vanishes. Its adjoint acts by the same differential expression on H1(0,1), with no endpoint condition. The domains differ, so this symmetric A is not self-adjoint. This example uses Sobolev weak derivatives and their endpoint traces, not an assumption that all L2 functions have ordinary endpoint values.

A different realization, with H1 functions satisfying f(1)=exp(iθ)f(0) for a fixed real θ, is self-adjoint. Matching the boundary term against every allowed test function imposes the same condition on the adjoint domain. Its allowed wave numbers are θ+2πk for integer k. The expression −i d/dx did not change, but the domain did. A boundary prescription therefore changes the mathematical observable, rather than merely changing how an existing operator is printed.