Rayleigh Quotient and Surjectivity of Nonlinear Operators in Hilbert space

Raffaele Chiappinelli*

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. 

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Published on: Sep 28, 2024 Pages: 277-278

Full Text PDF Full Text HTML DOI: 10.17352/amp.000132
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