Operator-valued positive definite kernels and differentiable universality

We present a characterization for a positive definite operator-valued kernel to be universal or C-0-universal, and apply these characterizations to a family of operator-valued kernels that are shown to be well behaved. Later, we obtain a characterization for an operator-valued differentiable kernel to be C-q-universal and C-0(q)-universal. In order to obtain such characterization and examples, we generalize some well-known results concerning the structure of differentiable kernels to the operator-valued context. On the examples is given an emphasis on the radial kernels on Euclidean spaces.