Isometric representation of integral operators with positive-semidefinite kernels
Abstract
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.
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