11.2 Closed and closable operators

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

• 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 $x \in D(T)$ ($G(T) = \{(x, Tx) : x \in D(T)\}$)
• Examples of unbounded operators include differentiation operator on $C^1([a,b])$ and multiplication operator on $L^2(\mathbb{R})$

Characterization of closed operators

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

Identification of closed vs closable operators

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

Construction of operator closures

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