2238 CHAPTER 65. STOCHASTIC INTEGRATION

Let U be a separable Hilbert space and let Q be a positive self adjoint operator. Thenconsider

J : Q1/2U →U1,

a one to one Hilbert Schmidt operator, where U1 is a separable real Hilbert space. First ofall, there is the obvious question whether there are any examples.

Lemma 65.4.3 Let A ∈L (U,U) be a bounded linear transformation defined on U a sep-arable real Hilbert space. There exists a one to one Hilbert Schmidt operator J : AU →U1where U1 is a separable real Hilbert space. In fact you can take U1 =U.

Proof: Let αk > 0 and ∑∞k=1 α2

k < ∞. Then let {gk}Lk=1 be an orthonormal basis for AU,

the inner product and norm given in Definition 65.4.1 above, and let

Jx≡L

∑k=1

(x,gk)AU αkgk.

Then it is clear that J ∈L (AU,U) . This is because,

||Jx||U ≤L

∑k=1|(x,gk)AU |αk ||gk||U

≤ CL

∑k=1|(x,gk)AU |αk

=1︷ ︸︸ ︷||gk||AU

≤ C

(L

∑k=1|(x,gk)AU |

2

)1/2( L

∑k=1

α2k

)1/2

= C

(L

∑k=1

α2k

)1/2

||x||AU

Also, from the definition, Jg j = α jg j. Say g j = A f j where f j ∈ U and 1 =∥∥g j∥∥

AU =∥∥ f j∥∥

U . Since A is continuous,∥∥g j∥∥

U =∥∥A f j

∥∥U ≤ ∥A∥

∥∥ f j∥∥

U = ∥A∥∥∥g j∥∥

AU = ∥A∥ ≡C1/2

ThusL

∑j=1

∣∣∣∣Jg j∣∣∣∣2

U =L

∑j=1

α2j∣∣∣∣g j∣∣∣∣2

U ≤CL

∑j=1

α2j < ∞

and so J is also a Hilbert Schmidt operator which maps AU to U . It is clear that J is one toone because each αk > 0. If AU is finite dimensional, L < ∞ and so the above sum is finite.

Definition 65.4.4 Let U1,U,H be real separable Hilbert spaces and let Q be a nonnegativeself adjoint operator, Q∈L (U,U) . Let Q1/2U be the Hilbert space described in Definition65.4.1. Let J be a one to one Hilbert Schmidt map from Q1/2U to U1.

U1J← Q1/2U Φ→ H