Unbounded operators are a crucial concept in functional analysis, mapping subsets of Banach spaces. Unlike bounded operators, their domains aren't the entire space. Examples include differentiation and multiplication operators on specific function spaces.
Closed operators are characterized by the Closed Graph Theorem. They're essential in studying partial differential equations and quantum mechanics. Closable operators, a related concept, have graphs that can be extended to closed operators.
Closed and Closable Operators
Definition of unbounded operators
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An unbounded operator T maps a subset D(T) of a Banach space X to another Banach space Y
The domain D(T) is a linear subspace of X but not necessarily the entire space (unlike bounded operators)
The graph of T is the set of ordered pairs (x,Tx) where x∈D(T) (G(T)={(x,Tx):x∈D(T)})
Examples of unbounded operators include differentiation operator on C1([a,b]) and multiplication operator on L2(R)
Characterization of closed operators
Closed Graph Theorem states an unbounded operator T:D(T)⊂X→Y between Banach spaces is closed iff for any sequence (xn)⊂D(T) with xn→x and Txn→y, we have x∈D(T) and Tx=y
Proof:
(⇒) Assume T is closed. Let (xn)⊂D(T) with xn→x and Txn→y. Since G(T) is closed and (xn,Txn)∈G(T) for all n, (x,y)∈G(T), so x∈D(T) and Tx=y
(⇐) Assume the condition holds. Let (xn,Txn) be a sequence in G(T) converging to (x,y). Then xn→x and Txn→y. By the condition, x∈D(T) and Tx=y, so (x,y)∈G(T). Thus, G(T) is closed, and T is closed
Examples of closed operators include the Laplace operator on H2(Rn) and the multiplication operator on L2(R) with a bounded measurable function
Identification of closed vs closable operators
To check if an unbounded operator T is closed, consider a sequence (xn)⊂D(T) with xn→x and Txn→y. If x∈D(T) and Tx=y for all such sequences, then T is closed
To check if T is closable, consider the closure of its graph G(T). If G(T) is the graph of an operator, then T is closable, and the associated operator is the closure of T
Equivalently, T is closable iff for any sequence (xn)⊂D(T) with xn→0 and Txn→y, we have y=0
Examples of closable operators include the differentiation operator on C([a,b]) and the multiplication operator on L2(R) with an unbounded measurable function
Construction of operator closures
The closure of a closable operator T:D(T)⊂X→Y, denoted T, is defined by:
D(T)={x∈X:∃(xn)⊂D(T) such that xn→x and Txn converges in Y}
For x∈D(T), Tx=limn→∞Txn, where (xn) is a sequence as in the definition of D(T)
Properties of the closure:
T is a closed extension of T
T is the smallest closed extension of T, i.e., if S is any other closed extension of T, then T⊂S
The graph of T is the closure of the graph of T, i.e., G(T)=G(T)
Examples of closures include the closure of the differentiation operator on C([a,b]) is the weak derivative operator on W1,p([a,b]) for 1≤p<∞