  We offer new results and new directions in the study of operator-valued
kernels and their factorizations. Our approach provides both more explicit
realizations and new results, as well as new applications. These include: (i)
an explicit covariance analysis for Hilbert space-valued Gaussian processes,
(ii) optimization results for quantum gates (from quantum information), (iii)
new results for positive operator-valued measures (POVMs), and (iv) a new
approach/result in inverse problems for quantum measurements.
