Hilbert Schmidt on Hilbert probability space

Now use twenty minutes to explain why we consider Hilbert Schmidt operator on Hilbert probability space.

We consider an operator F: H\rightarrow H where H is a Hilbert space with probability measure \mu. Then when F is Hilbert Schmidt, let’s denote z=\sum a_i e_i where F^*F e_i=\lambda_i e_i, then we have

\int\|Fz\|^2_H\mu(dz)=\int <F^*Fz,z>_H\mu(dz)=\int <\sum a_i\lambda_i e_i, \sum a_i e_i>_H \mu(a_1,a_2,\dots)=\int \lambda_i<\sum a_i\lambda_i e_i, \sum a_i e_i>_H \mu(a_1,a_2,\dots)