Why we care about domains of unbounded operators: The Hellinger-Toeplitz theorem

Here is a sufficient condition for a linear operator to be bounded:

Theorem (Hellinger-Toeplitz theorem)
Let A be an everywhere defined linear operator on a Hilbert space X with (x,Ay) = (Ax,y) for all x and y in X. Then A is bounded.

This theorem states that an everywhere defined linear operator on a Hilbert space H that is symmetric everywhere on H is always bounded. The Hellinger-Toeplitz theorem implies that an unbounded symmetric operator cannot be defined on all of H! It tells you that when dealing with unbounded operators it is very important to specify the domain on which the operator is defined.

The Hellinger-Toeplitz theorem is an immediate consequence of the closed graph theorem. The graph of a linear operator is defined as follows:

Definition (Graph)
Let A:X \to Y be a linear map from one normed linear space into another. Its graph \Gamma(A) is the set

    \begin{displaymath} \Gamma(A):=\{ \langle x,y \rangle | \langle x,y \rangle \in X \times Y, y = A x \}. \end{displaymath}

The closed graph theorem states:

Theorem (Closed graph theorem)
Let A: X \to Y be a linear operator mapping from one Banach space into another. Then A is bounded if and only if its graph \Gamma(A) is closed.

One can show that for an everywhere defined linear operator the symmetry implies that the graph is closed and hence, the closed graph theorem gives that the operator is bounded.

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