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We introduce the notion of Khatri-Rao product for operator matrices acting on the direct sum of Hilbert spaces. This notion generalizes the tensor product and Hadamard product of operators and the Khatri-Rao product of matrices. We investigate algebraic properties, positivity, and monotonicity of the Khatri-Rao product. Moreover, there is a unital positive linear map taking Tracy-Singh products to Khatri-Rao products via an isometry

Khatri-Rao Products for Operator Matrices Acting on the Direct Sum of Hilbert Spaces