# Boundness and adjoint of an integration operator

Let's define the operator $$K$$ as $$\begin{eqnarray*} K :&H&\longrightarrow &\;\:Z&\\ &y\left( x;\sigma \right) &\mapsto &\!\!\!\!\int_{0}^{1}C\left( \sigma \right) y\left( x;\sigma \right) d\sigma& \end{eqnarray*}\:$$ where $$H$$, $$Z$$ are Hilbert spaces, $$C\left( \sigma \right)$$ is a linear bounded operator from $$H$$ to $$Z$$, $$x$$ and $$\sigma$$ are real parameters. My question is: how can I prove that $$K$$ is bounded, and what does its adjoint operator look like?

• I don't think the question can be answered in the generality in which you've posed it, which relies at least on being able to interpret elements of $H$ as functions of a real parameter $\sigma$ (as well as possibly of some other unspecified parameter $x$). – LSpice May 6 at 16:59

I may assume that $$H=Z=L^2(\mathbb R)$$ and the mapping $$K$$ to be given by a distribution kernel $$k(s,t)$$ via a formula $$Ku(s)=\int k(s,t) u(t) dt,$$ meaning that for $$u,v\in C^\infty_c(\mathbb R)$$, we have $$\langle Ku, v\rangle_{\mathscr D'(\mathbb R),\mathscr D(\mathbb R)}=\langle k, v\otimes u\rangle_{\mathscr D'(\mathbb R^2),\mathscr D(\mathbb R^2)}$$. Now your question could be reformulated as follows: are there general criteria for extending $$K$$ to a bounded operator on $$L^2(\mathbb R)$$? The answer is yes, some general criteria are available.
1. Schur's criterion, for a locally integrable kernel, $$\sup_s\int \vert{k(s,t)}\vert dt,\sup_t\int \vert{k(s,t)}\vert ds\quad \text{ both finite.}$$
2. Singular integrals criteria, which can be applied for instance to $$k=pv(1/(s-t))$$, which is a Fourier multiplier by a bounded function.