ISSN: 2689-7636

Short Communication
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Department of Information Engineering and Mathematical Sciences, University of Siena, 53100 Siena, Italy

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This short work does not contain any detailed discussion or proof, but merely a few statements and comments concerning some properties of nonlinear (= not necessarily linear) operators acting in a Hilbert space; it aims at inviting people interested in the subject to further study the matter. References for proofs of the results are given throughout. Thus let *H* be a real Hilbert space with scalar product denoted
$\langle \mathrm{.,.}\rangle $
and corresponding norm
$\parallel x\parallel \mathrm{=}\sqrt{\langle x\mathrm{,}x\rangle}$
. If *F* is any map of *H* into itself, it makes sense to define its *Rayleigh quotient* by the formula

$\frac{\langle F\mathrm{(}x\mathrm{),}x\rangle}{\parallel x{\parallel}^{2}}\text{}\mathrm{(}x\in H\mathrm{,}x\ne \mathrm{0).}\text{(1)}$

If we suppose in addition that *F* is continuous and such that

$\parallel F\mathrm{(}x\mathrm{)}\parallel \le A\parallel x\parallel \text{(2)}$

for some *A*≥0 and all *x∈H*, then its Rayleigh quotient is a bounded continuous real function, and we look in particular at the numbers

$m\mathrm{(}F\mathrm{)=}\underset{x\ne 0}{{\displaystyle \mathrm{inf}}}\frac{\langle F\mathrm{(}x\mathrm{),}x\rangle}{\parallel x{\parallel}^{2}}\mathrm{,}\text{}M\mathrm{(}F\mathrm{)=}\underset{x\ne 0}{{\displaystyle \mathrm{sup}}}\frac{\langle F\mathrm{(}x\mathrm{),}x\rangle}{\parallel x{\parallel}^{2}}\text{(3)}$

which are quite useful in the study of the spectral properties of *F*. Indeed it is immediate that if *λ* is an eigenvalue of *F* (meaning that
$F\mathrm{(}x\mathrm{)}-\lambda x\mathrm{=0}$
for some *x*≠0), then
$m\mathrm{(}F\mathrm{)}\le \lambda \le M\mathrm{(}F\mathrm{)}$
. Moreover in the special case that *F=T*, a bounded linear operator, then the whole spectrum *σ(T)* of *T* satisfies the inclusion

$\sigma \mathrm{(}T\mathrm{)}\subset \mathrm{[}m\mathrm{(}T\mathrm{),}M\mathrm{(}T\mathrm{)]}\text{(4)}$

as follows for instance using the Lax-Milgram Lemma (see, e.g., [1]). More can be said if *T* is in addition self-adjoint and/or compact; and quite surprisingly, similar interesting properties can be drawn also when *T* is replaced by a nonlinear operator *F* acting in *H*. For instance, if *F* is Lipschitz continuous and satisfies the condition

${m}_{0}\mathrm{(}F\mathrm{)}\equiv \underset{x\ne y}{{\displaystyle \mathrm{inf}}}\frac{\langle F\mathrm{(}x\mathrm{)}-F\mathrm{(}y\mathrm{),}x-y\rangle}{\parallel x-y{\parallel}^{2}}\mathrm{>0,}\text{(5)}$

then *F* is a *Lipeomorphism* (in the language of [2]), in the sense that it is a Lipschitz homeomorphism of *H* onto itself with Lipschitz inverse *F ^{-1}*: as for surjectivity, this follows easily from the Minty-Browder Theorem (see, e.g., [1]).

Something can be said also in case *F*, rather than being Lipschitzian, satisfies the weaker condition (2): for if we put
$b\mathrm{(}F\mathrm{)=}\underset{x\ne 0}{{\displaystyle \mathrm{inf}}}\frac{\parallel F\mathrm{(}x\mathrm{)}\parallel}{\parallel x\parallel}$
and

${\Sigma}_{b}\mathrm{(}F\mathrm{)=\{}\lambda \in \mathbb{R}\mathrm{:}b\mathrm{(}F-\lambda I\mathrm{)=0\}}\text{(6)}$

(a sort of ``approximate point spectrum" of F), then we have
${\Sigma}_{b}\mathrm{(}F\mathrm{)}\subset \mathrm{[}m\mathrm{(}F\mathrm{),}M\mathrm{(}F\mathrm{)]}$
. Finally, some surjectivity properties of *F* can be derived through the numbers *m(F), M(F)* at least in the case *F* is a *gradient* operator, i.e., is such that

$\langle F\mathrm{(}x\mathrm{),}y\rangle \mathrm{=}{f}^{\prime}\mathrm{(}x\mathrm{)}y\text{}\mathrm{(}x\mathrm{,}y\in H\mathrm{)}\text{(7)}$

for some differentiable functional
$f\mathrm{:}H\to \mathbb{R}$
; here f′(*x*) denotes the (Fréchet) derivative of *f* at the point *x∈H*. Indeed using the Ekeland Variational Principle (see, e.g., [3]) we can show that for such an operator, the conditions

*m*(*F*)>0 and *ω(F)*>0 (8)

where

$\omega \mathrm{(}F\mathrm{)=}\mathrm{inf}\mathrm{\{}\frac{\alpha \mathrm{(}F\mathrm{(}A\mathrm{))}}{\alpha \mathrm{(}A\mathrm{)}}\mathrm{:}A\subset E\mathrm{,}A\text{\hspace{0.05em}}\text{bounded}\text{\hspace{0.05em}}\mathrm{,}\alpha \mathrm{(}A\mathrm{)>0\}}\text{(9)}$

and *α(A)* denotes the measure of non-compactness of the bounded set *A*⊂*H*, ensure that *F* is surjective. This implies in particular the surjectivity of *F-λl* when

$\lambda \notin \mathrm{[}m\mathrm{(}F\mathrm{),}M\mathrm{(}F\mathrm{)]}\cup {\sigma}_{\omega}\mathrm{(}F\mathrm{),}\text{(10)}$

where ${\sigma}_{\omega}\mathrm{(}F\mathrm{)}\equiv \mathrm{\{}\lambda \in \mathbb{R}\mathrm{:}\omega \mathrm{(}F-\lambda I\mathrm{)=0\}}$ .

Proofs of these statements can be found in [4-6], while we refer to [2] for a general introduction to the subject and also as a reference for further study.

*Communication was held at the Conference on* Topological Methods for Nonlinear Analysis and Dynamical Systems *(Firenze, 27-28 September 2024) organized in honor of the retirement of Professor Patrizia Pera*.

- Brezis H. Functional Analysis, Sobolev Spaces and Partial Differential Equations. New York: Springer; 2011. Available from: https://link.springer.com/book/10.1007/978-0-387-70914-7
- Appell J, De Pascale E, Vignoli A. Nonlinear Spectral Theory. Berlin: de Gruyter; 2004. Available from: https://doi.org/10.1515/9783110199260
- de Figueiredo DG. Lectures on the Ekeland Variational Principle with Applications and Detours. Bombay: Tata Institute of Fundamental Research; 1989. Available from: https://mathweb.tifr.res.in/sites/default/files/publications/ln/tifr81.pdf
- Chiappinelli R. Surjectivity of coercive gradient operators in Hilbert space and nonlinear spectral theory. Ann Funct Anal. 2019;10(2):170-179. Available from: http://dx.doi.org/10.1215/20088752-2018-0003
- Chiappinelli R, Edmunds DE. Measure of noncompactness, surjectivity of gradient operators and an application to the p-Laplacian. J Math Anal Appl. 2019;471:712-727. Available from: https://doi.org/10.1016/j.jmaa.2018.11.010
- Chiappinelli R, Edmunds DE. Remarks on surjectivity of gradient operators. Mathematics. 2020;8:1538. Available from: https://doi.org/10.3390/math8091538

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