# 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

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