Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

If $H$ is a Hilbert space with norm $\| . \|$, and $A$ is an operator, we call it a Hilbert Schmidt Operator if $$\sum_{n=1}^\infty \|Ax_n\|^2<\infty$$ for some orthonormal basis $\{x_i\}.$

Consider $L^2(X,\mu)$. How could one prove that every Hilbert Schmidt Operator on this space is given by $$(Af)(x)=\int_X k(x,y)f(y)dy$$ for some $$k(x,y)\in L^2(X\times X, \mu \times \mu).$$

I am not really sure where to start, but I imagine this would be in many textbooks/online notes? Does this fact have a particular name? Ideally I would love it if someone could show why it is true, but a reference is useful as well.

Thanks for any help!

share|cite|improve this question
up vote 6 down vote accepted

How many details do you want?

Let $$A x_n = \sum_k \alpha_{n,k} x_k.$$

Show $\sum |\alpha_{n,k}|^2 < \infty$.

Now define $$k = \sum_{n,k} \alpha_{n,k} (x_n \otimes \overline{x}_k)$$

Then show that this works.

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.