Bounded linear operators are the key ingredient studied in basic functional analysis. However, some important linear operators in analysis and physics are not bounded. Here I give an elementary introduction to unbounded operators, which may interest the readers especially analysts, hopefully.
1 CLOSED OPERATORS
Unless specified, always let and be two Banach spaces. Endow the norm to the product space . Denote and to be their continuous duals. Also let be a Hilbert space.
Given a linear operator , which its domain is a subspace of and its range space is a subspace of . Define the graph of to be
, which is a subspace of .
is said to be closed if is closed in .
Definition 1.2 (Extension and closability)
Given linear. If , then we call to be an extension of , denoted by .
A linear operator is said to be closable if there exists an extension such that . is the closure of , denoted by .
Not all linear operators are closable, we give a condition to determine whether is the graph of some linear operator.
Proposition 1.3 Given linear. The following are equivalent.
(1) is closable.
(2) If , then .
The following are the basic properties about closed operators.
(1) If is an injective closed operator, then is also closed.
(2) is closed implies the null space is closed in .
(3) If is closable and is closed, then . This means the closure of is its smallest closed extension.
(4) If is closed and , then is bounded.
Proof. (1)(2)(3) follow by direct check. (4) follows by closed graph theorem.
By Propostion 1.4(4), what we interest in closed unbounded operators are those with . What we like to study is
A linear operator is said to be densely defined if is dense in .
Consider the following “adjoint” .
Given a densely defined linear . Denote
Define by . is called the adjoint operator of , and to be the domain of .
is densely defined guarantees is uniquely determined, and it can be easily checked that is also linear.
If is a Hilbert space and , then can rewrite
defined above must be closed. Also, imples .
(1) is a densely defined linear operator with . Then is closable if and only if is densely defined. In either case .
(2) If is a densely defined linear operator and is reflexive, then the conclusion as above still holds, but in either case , where and are canonical embeddings.
Definition 1.9 (Symmetry, self-adjointness, essentially self-adjointness)
Given densely defined linear operator from to . is said to be symmetric if . is said to be self-adjoint if . is said to be essentially self-adjoint if is closable and is self-adjoint.
(1) is symmetric if and only if for all , .
(2) Symmetric operators are closable. Indeed, if is symmetric, then and hence . This implies is densely defined. Thus is closable by Theorem 1.8(1).
(3) If , symmetry is equivalent to self-adjointness. In general, is self-adjoint if and only if is symmetry and .
(4) If is self-adjoint, is symmetric and , then . Indeed, since (Theorem 1.7 is used), . This means, self-adjoint operator is the maximal symmetric extension of itself.
If is essentially self-adjoint, then is its unique closed self-adjoint extension, because if is another one, then implying .