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11.2 Closed and closable operators

3 min readLast Updated on July 22, 2024

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 TT maps a subset D(T)D(T) of a Banach space XX to another Banach space YY
  • The domain D(T)D(T) is a linear subspace of XX but not necessarily the entire space (unlike bounded operators)
  • The graph of TT is the set of ordered pairs (x,Tx)(x, Tx) where xD(T)x \in D(T) (G(T)={(x,Tx):xD(T)}G(T) = \{(x, Tx) : x \in D(T)\})
  • Examples of unbounded operators include differentiation operator on C1([a,b])C^1([a,b]) and multiplication operator on L2(R)L^2(\mathbb{R})

Characterization of closed operators

  • Closed Graph Theorem states an unbounded operator T:D(T)XYT: D(T) \subset X \to Y between Banach spaces is closed iff for any sequence (xn)D(T)(x_n) \subset D(T) with xnxx_n \to x and TxnyTx_n \to y, we have xD(T)x \in D(T) and Tx=yTx = y
  • Proof:
    1. (\Rightarrow) Assume TT is closed. Let (xn)D(T)(x_n) \subset D(T) with xnxx_n \to x and TxnyTx_n \to y. Since G(T)G(T) is closed and (xn,Txn)G(T)(x_n, Tx_n) \in G(T) for all nn, (x,y)G(T)(x, y) \in G(T), so xD(T)x \in D(T) and Tx=yTx = y
    2. (\Leftarrow) Assume the condition holds. Let (xn,Txn)(x_n, Tx_n) be a sequence in G(T)G(T) converging to (x,y)(x, y). Then xnxx_n \to x and TxnyTx_n \to y. By the condition, xD(T)x \in D(T) and Tx=yTx = y, so (x,y)G(T)(x, y) \in G(T). Thus, G(T)G(T) is closed, and TT is closed
  • Examples of closed operators include the Laplace operator on H2(Rn)H^2(\mathbb{R}^n) and the multiplication operator on L2(R)L^2(\mathbb{R}) with a bounded measurable function

Identification of closed vs closable operators

  • To check if an unbounded operator TT is closed, consider a sequence (xn)D(T)(x_n) \subset D(T) with xnxx_n \to x and TxnyTx_n \to y. If xD(T)x \in D(T) and Tx=yTx = y for all such sequences, then TT is closed
  • To check if TT is closable, consider the closure of its graph G(T)\overline{G(T)}. If G(T)\overline{G(T)} is the graph of an operator, then TT is closable, and the associated operator is the closure of TT
  • Equivalently, TT is closable iff for any sequence (xn)D(T)(x_n) \subset D(T) with xn0x_n \to 0 and TxnyTx_n \to y, we have y=0y = 0
  • Examples of closable operators include the differentiation operator on C([a,b])C([a,b]) and the multiplication operator on L2(R)L^2(\mathbb{R}) with an unbounded measurable function

Construction of operator closures

  • The closure of a closable operator T:D(T)XYT: D(T) \subset X \to Y, denoted T\overline{T}, is defined by:
    • D(T)={xX:(xn)D(T) such that xnx and Txn converges in Y}D(\overline{T}) = \{x \in X : \exists (x_n) \subset D(T) \text{ such that } x_n \to x \text{ and } Tx_n \text{ converges in } Y\}
    • For xD(T)x \in D(\overline{T}), Tx=limnTxn\overline{T}x = \lim_{n \to \infty} Tx_n, where (xn)(x_n) is a sequence as in the definition of D(T)D(\overline{T})
  • Properties of the closure:
    • T\overline{T} is a closed extension of TT
    • T\overline{T} is the smallest closed extension of TT, i.e., if SS is any other closed extension of TT, then TS\overline{T} \subset S
    • The graph of T\overline{T} is the closure of the graph of TT, i.e., G(T)=G(T)G(\overline{T}) = \overline{G(T)}
  • Examples of closures include the closure of the differentiation operator on C([a,b])C([a,b]) is the weak derivative operator on W1,p([a,b])W^{1,p}([a,b]) for 1p<1 \leq p < \infty


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AP® and SAT® are trademarks registered by the College Board, which is not affiliated with, and does not endorse this website.
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