Pseudo-integral operators

Abstract:

Let $(X, \mathcal {a}, m)$ be a standard finite measure space. A bounded operator T on ${L^2}(X)$ is called a pseudo-integral operator if $(Tf)(x) = \int {f(y) \mu (x, dy)}$, where, for every x, $\mu (x, \cdot )$ is a bounded Borel measure on X. Main results: 1. A bounded operator T on ${L^2}$ is a pseudo-integral operator with a positive kernel if and only if T maps positive functions to positive functions. 2. On nonatomic measure spaces every operator unitarily equivalent to T is a pseudo-integral operator if and only if T is the sum of a scalar and a Hilbert-Schmidt operator. 3. The class of pseudo-integral operators with absolutely bounded kernels form a selfadjoint (nonclosed) algebra, and the class of integral operators with absolutely bounded kernels is a two-sided ideal. 4. An operator T satisfies $(Tf)(x) = \int {f(y) \mu (x, dy)}$ for $f \in {L^\infty }$ if and only if there exists a positive measurable (almost-everywhere finite) function $\Omega$ such that $\left | {(Tf)(x)} \right | \leqslant {\left \| f \right \|_\infty }\Omega (x)$.