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.

**Definition 1.1**

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 .

Proof.

The following are the basic properties about closed operators.

**Proposition 1.4**

(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

**Definition 1.5**

A linear operator is said to be densely defined if is dense in .

Consider the following “adjoint” .

**Definition 1.6**

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

.

**Theorem 1.7**

defined above must be closed. Also, imples .

Proof.

**Theorem 1.8**

(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.

Proof.

**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.

**Remark 1.10**

(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 .

Examples.