  Motivated by the need of processing functional-valued data, or more general,
operatorvalued data, we introduce the notion of the operator reproducing kernel
Hilbert space (ORKHS). This space admits a unique operator reproducing kernel
which reproduces a family of continuous linear operators on the space. The
theory of ORKHSs and the associated operator reproducing kernels are
established. A special class of ORKHSs, known as the perfect ORKHSs, are
studied, which reproduce the family of the standard point-evaluation operators
and at the same time another different family of continuous linear operators.
The perfect ORKHSs are characterized in terms of features, especially for those
with respect to integral operators. In particular, several specific examples of
the perfect ORKHSs are presented. We apply the theory of ORKHSs to sampling and
regularized learning, where operator-valued data are considered. Specifically,
a general complete reconstruction formula from linear operators values is
established in the framework of ORKHSs. The average sampling and the
reconstruction of vector-valued functions are considered in specific ORKHSs. We
also investigate in the ORKHSs setting the regularized learning schemes, which
learn a target element from operator-valued data. The desired representer
theorems of the learning problems are established to demonstrate the key roles
played by the ORKHSs and the operator reproducing kernels in machine learning
from operator-valued data. We finally point out that the continuity of linear
operators, used to obtain the operator-valued data, on an ORKHS is necessary
for the stability of the numerical reconstruction algorithm using the resulting
data.
