Hellinger-Toeplitz theorem

In functional analysis, a branch of mathematics, the Hellinger-Toeplitz theorem states that an everywhere defined symmetric operator on a Hilbert space is bounded. An operator A is symmetric iff

for all x, y in the domain of A. Note that symmetric everywhere defined operators are necessarily self-adjoint, that is A = A*. Unbounded symmetric operators (particularly self-adjoint ones) are plentiful and in fact are extremely important in applications to physics.

This theorem is a corollary of the closed graph theorem. Indeed, self-adjoint operators are closed operators. It is named for Ernst David Hellinger and Otto Toeplitz.

The Hellinger-Toeplitz theorem leads to some technical difficulties in the mathematical formulation of quantum mechanics. Observables in quantum mechanics correspond to self-adjoint operators on some Hilbert space, but some observables (like energy) are unbounded. Such operators cannot be everywhere defined (but they may be defined on a dense subset). Take for instance the quantum harmonic oscillator. Here the Hilbert space is L²(R), the space of square integrable functions on R, and the energy operator H is defined by (assuming the units are chosen such that ℏ = m = ω = 1)

This operator is self-adjoint and unbounded (its eigenvalues are 1/2, 3/2, 5/2, ...), so it cannot be defined on the whole of L²(R).