Hellinger–Toeplitz theorem

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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 if and only if

$\langle A x | y \rangle = \langle x | A y\rangle$

for all x, y in the domain of A. Note that symmetric everywhere defined operators are necessarily self-adjoint, so this theorem can also be stated that an everywhere defined self-adjoint operator is bounded.

This theorem is a corollary of the closed graph theorem, as self-adjoint operators are closed. The theorem is named after 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)

$[Hf](x) = - \frac12 \frac{\mbox{d}^2}{\mbox{d}x^2} f(x) + \frac12 x^2 f(x).$