A STRUCTURE THEOREM ON NON-HOMOGENEOUS LINEAR EQUATIONS IN HILBERT SPACES

Abstract

A very particular by-product of the result announced in the title reads as follows:  Let $(X,<.,.>)$be a real Hilbert space, $T : X \to X$ a compact and symmetric linear operator, and $z \in X$ such that the equation $T(x) - \|T \|x = z$ has no solution in $X$. For each $r > 0$, set$\gamma(r) = \sup_{x \in Sr} J(x),$ where $J(x) = \langle T(x)-2z, x \rangle$ and $S_r = {x \in X : \|x\|2 = r}$. Then, the function $\gamma$ is $C^1,$ increasing and strictly concave in$]0,+\infty[,$ with $\gamma(]0,+\infty[) =]\|T\|,+1[;$ moreover, for each $r > 0$, the problem of maximizing $J$ over $S_r$ is wellposed, and one has $$T(\hat{x}_r) - \gamma(r)\hat{x}_r = z$$where $\hat{x}_r$ is the only global maximum of $J_{|Sr} .$