Open Mapping and Closed Graph Theorems - Applications

The Hellinger-Toeplitz Theorem demonstrates a surprising automatic continuity result for symmetric operators on Hilbert spaces, showing that algebraic symmetry implies boundedness.

TheoremHellinger-Toeplitz Theorem

Let $H$ be a Hilbert space and $T : H \to H$ a linear operator defined on all of $H$ . If $T$ is symmetric (i.e., $\langle Tx, y \rangle = \langle x, Ty \rangle$ for all $x, y \in H$ ), then $T$ is bounded.

This theorem is remarkable because it shows that an unbounded symmetric operator cannot be defined on the entire Hilbert space. This has profound implications for quantum mechanics, where unbounded self-adjoint operators (like momentum and position) must have restricted domains.

Proof

We show that $T$ has a closed graph, then apply the Closed Graph Theorem.

Suppose $x_n \to x$ and $Tx_n \to y$ in $H$ . For any $z \in H$ : $\langle y, z \rangle = \lim_{n \to \infty} \langle Tx_n, z \rangle = \lim_{n \to \infty} \langle x_n, Tz \rangle = \langle x, Tz \rangle = \langle Tx, z \rangle$

Since this holds for all $z \in H$ , we have $y = Tx$ . Thus $\Gamma(T)$ is closed.

By the Closed Graph Theorem, $T$ is bounded.

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ExampleConsequences

Quantum Mechanics: The momentum operator $p = -i\hbar \frac{d}{dx}$ is symmetric but unbounded, so it cannot be defined on all of $L^2(\mathbb{R})$ . Its domain is $H^1(\mathbb{R}) \subsetneq L^2(\mathbb{R})$
Multiplication Operators: If $M_\phi : L^2 \to L^2$ defined by $(M_\phi f)(x) = \phi(x) f(x)$ is symmetric and everywhere defined, then $\phi \in L^\infty$
Differential Operators: The Laplacian $\Delta : H^2(\Omega) \to L^2(\Omega)$ is symmetric but unbounded. For it to be everywhere defined and symmetric, it would need to be bounded—but it isn't
Restriction of Domain: To study unbounded symmetric operators, one must carefully specify their domains

TheoremClosed Range Theorem

Let $X, Y$ be Banach spaces and $T \in \mathcal{B}(X, Y)$ . Then the range of $T$ is closed if and only if the range of $T^*$ is closed.

Moreover, if the range is closed, then: $\text{Range}(T) = (\ker(T^*))^\perp \quad \text{and} \quad \text{Range}(T^*) = (\ker(T))^\perp$

Remark

The Hellinger-Toeplitz Theorem explains why functional analysis distinguishes carefully between bounded and unbounded operators. Many physically important operators are unbounded, requiring sophisticated domain theory and spectral analysis beyond the scope of bounded operator theory.

These automatic continuity results show the power of the Closed Graph Theorem and Open Mapping Theorem in establishing boundedness without explicit norm estimates.