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 G associated with the squared-modulus of the reproducing kernel of H. We discuss the properties of this coisometry, and 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 operators are the Riesz representations of integral functionals on G, drawing a direct connection between the approximation of trace-class integral operators with PSD kernels and the approximation of integral functionals on RKHSs associated with squared-modulus kernels.

Mots clés: reproducing kernel Hilbert spaces; integral operators; low-rank approximation; kernel embedding of measures; kernel quadrature

Domaines: Mathématiques [math]

Bertrand Gauthier. Isometric representation of integral operators with positive-semidefinite kernels. 2023. ⟨hal-03848105v2⟩