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An operator connection is a binary operation assigned to each pair of positive operators satisfying monotonicity, continuity from above, and the transformer inequality. A normalized operator connection is called an operator mean. In this paper, we introduce and characterize the concepts of cancellability and regularity of operator connections with respect to operator monotone functions, Borel measures, and certain nonlinear operator equations. As applications, we investigate the existence and the uniqueness of solutions for operator equations involving various kind of operator means.

Cancellability and regularity of operator connections with applications to nonlinear operator equations involving means