I'm trying to read this paper http://arxiv.org/abs/1206.3325 , but I'm having a lot of difficulty making sense of two phrases. The setting is $L_2(\mathbb R^d)$.
i) He mentions that for a function $a$, the operator $a(-i\nabla)$ denotes multiplication by $a$ in momentum space. Can someone clarify what is meant by this?
ii) To a non-negative operator $\gamma$, he associates a function, the "diagonal" of $\gamma$, denoted by $\gamma(x,x)$. What precisely is this diagonal?
Any help is very much appreciated and thank you.
i) if $F(k)$ is the Fourier transform of $f(x)$, then $aF(k)$ is the Fourier transform of $a(-i\nabla)f(x)$.
ii) the operator $\gamma$ has kernel $\gamma(x,x')$, in the sense that $(\gamma f)(x)=\int \gamma(x,x')f(x')dx'$; this defines the diagonal of $\gamma$; in particular, the trace of $\gamma$ is the integral of the diagonal, ${\rm tr}\,\gamma=\int \gamma(x,x)dx$.