Closed Operator - Extension-related

Extension-related

By definition, an operator T is an extension of an operator S if Γ (S) ⊆ Γ (T). An equivalent direct definition: for every x in the domain of S, x belongs to the domain of T and Sx = Tx.

Note that an everywhere defined extension exists for every operator, which is a purely algebraic fact explained at Discontinuous linear map#General existence theorem and based on the axiom of choice. If the given operator is not bounded then the extension is a discontinuous linear map. It is of little use since it cannot preserve important properties of the given operator (see below), and usually is highly non-unique.

An operator T is called closable if it satisfies the following equivalent conditions:

• T has a closed extension;
• the closure of the graph of T is the graph of some operator;
• for every sequence (x_n) of points from the domain of T such that x_n converge to 0 and also Tx_n converge to some y it holds that y = 0.

Not all operators are closable.

A closable operator T has the least closed extension called the closure of T. The closure of the graph of T is equal to the graph of

Other, non-minimal closed extensions may exist.

A densely defined operator T is closable if and only if T∗ is densely defined. In this case and

If S is densely defined and T is an extension of S then S∗ is an extension of T∗.

Every symmetric operator is closable.

A symmetric operator is called maximal symmetric if it has no symmetric extensions, except for itself.

Every self-adjoint operator is maximal symmetric. The converse is wrong.

An operator is called essentially self-adjoint if its closure is self-adjoint.

An operator is essentially self-adjoint if and only if it has one and only one self-adjoint extension.

An operator may have more than one self-adjoint extension, and even a continuum of them.

A densely defined, symmetric operator T is essentially self-adjoint if and only if both operators T – i, T + i have dense range.

Let T be a densely defined operator. Denoting the relation "T is an extension of S" by S ⊂ T (a conventional abbreviation for Γ(S) ⊆ Γ(T)) one has the following.

• If T is symmetric then T ⊂ T∗∗ ⊂ T∗.
• If T is closed and symmetric then T = T∗∗ ⊂ T∗.
• If T is self-adjoint then T = T∗∗ = T∗.
• If T is essentially self-adjoint then T ⊂ T∗∗ = T∗.