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I am asking perhaps a stupid question. How can I characterize all positive semi-definite operators in $\mathcal{B}(L^2(X,\lambda))$, where $\lambda$ is the Lebesgue measure. For a start, let us consider $X=[0,1]$.

The way first I thought was by using some (positive) kernel $K(x,y)$ such that any such operator $T$ can be represented by that kernel as $Tf(x)=\int_X K(x,y)f(y)dy$. I suspect, I am making something wrong. Moreover they may not represent all positive semi-definite operators (if if this approach is true). So I want to know, how to represent such objects, and their properties etc. Advanced thanks for any help, suggestions, references. Edit it if you need to and also change the tags if you think it is required.

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up vote 2 down vote accepted

You can represent Hilbert-Schmidt operators on $L^2$ using a kernel, but not all bounded operators.

A good way to approach self-adjoint operators on a Hilbert space is using the Spectral Theorem.

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