Minlos' theorem

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In mathematics, Minlos' theorem is a theorem in functional analysis. It states that a bounded linear operator between two Hilbert spaces is γ-radonifying if and only if it is Hilbert-Schmidt. The result is also important in the study of stochastic processes and the Malliavin calculus, since results concerning probability measures on infinite-dimensional spaces are of central importance in these fields.

[edit] Statement of the theorem

Let G and H be two Hilbert spaces and let T : G → H be a bounded operator from G to H. Recall that T is said to be γ-radonifying if the push forward of the canonical Gaussian cylinder set measure on G is a bona fide measure on H. Recall also that T is said to be Hilbert-Schmidt if there is an orthonormal basis { e_i | i ∈ I } of G such that

Then Minlos' theorem is that T is γ-radonifying if and only if it is Hilbert-Schmidt.

[edit] Remarks

It is a further result that the canonical Gaussian cylinder set measure on an infinite-dimensional Hilbert space can never be a bona fide measure; equivalently, the identity function on such a space cannot be γ-radonifying. Therefore, the push forward measure T_*(γ) will be supported only on a finite-dimensional subspace of H, and so will not in general be strictly positive.

[edit] References