Study On Some Properties Of Operator-Valued Reproducing Kernels

Since the reproducing kernel Hilbert space can ensure the stability of the sampling process,it has become an ideal space of functions for processing point-evaluation data.The well-known Riesz representation theorem of continuous linear functionals in Hilbert spaces establishes a bijective correspondence between a reproducing kernel Hilbert space and its reproducing kernel.This provides a solid foundation of mathematics for developing kernel-based methods for processing point-evaluation data.Large amount of vector-valued data emerging in practical applications and the great success of kernel-based methods for processing scalar-valued data promote the investigation of the theory of vector-valued reproducing kernel Hilbert spaces and operator-valued reproducing kernels.The main content of this paper includes two aspects.On the one hand,we shall induce and summarize systematically the theory of scalar-valued reproducing kernel Hilbert spaces and reproducing kernels.These results include: the notion of reproducing kernels and reproducing kernel Hilbert spaces and the bijective correspondence between them;the feature representation of reproducing kernels;the basic properties and operational properties of reproducing kernels.On the other hand,we shall establish the isometric isomorphism between a vector-valued reproducing kernel Hilbert space and a scalar-valued reproducing kernel Hilbert space.By making use of the isometric isomorphism we shall build bijective correspondence between a vectorvalued reproducing kernel Hilbert spaces and its operator-valued reproducing kernel and then study some important properties of operator-valued reproducing kernels.We need to point out that the theory of operator-valued reproducing kernels and vector-valued reproducing kernel Hilbert spaces have been established and studied systematically in the literature.The proofs can be done in the similar manner to the scalar case.However,in this paper,we shall use the isometric isomorphism as a bridge between scalar-valued reproducing kernel Hilbert spaces and vector-valued reproducing kernel Hilbert spaces.Then by isometric isomorphism,the theory of scalar-valued reproducing kernel Hilbert spaces will be transformed to the vector-valued case.By virtue of this point of view,we can easily obtain the bijective correspondence between a vector-valued reproducing kernel Hilbert spaces and its operatorvalued reproducing kernel and the important properties of operator-valued reproducing kernels.