Analysis on geometric structures of operators in definite or indefinite inner product spaces

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For selfadjoint operators/Hermitian matrices, we can define the order structure. For a real-valued function f we know the functional calculus by f. If this operation preserves the preceding order, then we say that f is operator/matrix monotone. It is well-known that this is equivalent to the positive-semidefiniteness of the corresponding Loewner matrices to f. Rajendra Bhatia and I studied the conditional positive/negative definiteness of Loewner matrices to have their characterization with applications. Fumio Hiai and I gave more precise arguments to have their generalizations.