A structure theorem on non-homogeneous linear equations in Hilbert spaces

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