Isometric representation of integral operators with positive-semidefinite kernels

Bertrand Gauthier (1)

1 School of Mathematics [Cardiff]

We describe a natural coisometry from the Hilbert space of all Hilbert-Schmidt operators on a separable reproducing kernel Hilbert space (RKHS) H, and onto the RKHS associated with the squared-modulus of the reproducing kernel of H. We discuss the properties of this coisometry, and in particular show that trace-class integral operators defined by general measures and the reproducing kernel of H always belong to its initial space. The images of such integral operators are the potentials of the underlying measures with respect to the squared-modulus kernel, drawing a direct connection between the approximation of integral operators with PSD kernels and the kernel embedding of measures in RKHSs associated with squared-modulus kernels.

Mots clés: reproducing kernel Hilbert spaces; integral operators; low-rank approximation; maximum mean discrepancy.

Domaines: Mathématiques [math]

Bertrand Gauthier. Isometric representation of integral operators with positive-semidefinite kernels. 2022. ⟨hal-03848105v1⟩