Operator-Adapted Reproducing Kernel Geometries for Structure-Preserving Representation Learning

The representation geometry determines which transformations appear to preserve, distort, or separate the represented objects. In most representation-learning settings, this geometry is fixed in advance or learned from data, while the action of an operator is subsequently studied within it. We consider the inverse problem by asking, given a linear operator, can one construct a reproducing kernel Hilbert space (RKHS) geometry in which its action is exactly norm preserving? We establish necessary and sufficient conditions for the existence of such an operator-adapted RKHS in finite-dimensional function spaces and characterize the complete family of reproducing kernels realizing this invariance. The resulting classification shows that existence is governed by the spectral structure of the operator, while the spectral multiplicities determine the degrees of freedom available in the invariant representation geometry. In particular, an operator that is not unitary under the original geometry may become unitary after an appropriate change of RKHS geometry, whereas operators with incompatible spectral structure admit no such positive-definite realization. These results provide a mathematical framework for constructing representations from the prescribed operator structure and suggest an alternative principle for structure-preserving representation learning where rather than modifying an operator to respect a predetermined geometry, one may construct the geometry to respect the operator.