1
$\begingroup$

Consider the 1-torus $\mathbb{T}$. Let $k$ be a smooth function on $\mathbb{T}^2$ and $K$ be the integral operator on $L^2(\mathbb{T})$ with kernel $k$. One can show that $K$ is of trace class, hence $|K|^{1/2}$ is a Hilbert Schmidt operator=integral operator. But what is the kernel of $|K|^{1/2}$?

$\endgroup$
  • 2
    $\begingroup$ Can you do the matrix case? Given $n \times n$ complex matrix $K = [k_{ij}]$, what are the entries of the matrix $|K|^{1/2}$? $\endgroup$ – Gerald Edgar Nov 21 '10 at 23:22
  • $\begingroup$ Diagonalize $K*K=UDU*$ and $|K|^{1/2}=UD^{1/4}U*$ $\endgroup$ – m07kl Nov 22 '10 at 17:30
4
$\begingroup$

It seems to me that you are looking for a formula for the kernel of $|K|^{1/2}$. But, as Gerald, mentioned, such a formula (in the case where the space $\mathbb{T}$ is replaced by a finite set) would give you a formula for the entries of the square root of an arbitrary positive matrix. And I don't think such a thing exists (or, at least, I don't think it is known).

| cite | improve this answer | |
$\endgroup$
  • 2
    $\begingroup$ I agree, although I have neither a proof nor a reference. $\endgroup$ – Denis Serre Nov 22 '10 at 7:38

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

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