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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

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

for all x, y in the domain of A. Note that symmetric everywhere defined operators are necessarily self-adjoint, 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)

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

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).

03-10-2013 05:06:04