Rayleigh Quotient and Surjectivity of Nonlinear Operators in Hilbert space - Chiappinelli, Raffaele (Annals of Mathematics and Physics, 2024)
We consider continuous operators acting in a real Hilbert space and indicate conditions ensuring their continuous invertibility and/or surjectivity. In the case of bounded linear operators, these facts are well-known from basic Functional Analysis. The objective of this work is to indicate how similar properties can be proved also when the operators are not necessarily linear, using as a main tool their Rayleigh quotient and especially its lower and upper bound. In particular, we focus our attention on gradient operators and show a quantitative criterion that ensures their surjectivity through the positivity of an additional constant related to the measure of noncompactness.