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

TL;DR

This paper establishes a structure theorem for non-homogeneous linear equations in Hilbert spaces, revealing properties of a related maximization problem involving compact symmetric operators and providing conditions for the existence and uniqueness of solutions.

Contribution

It introduces a new structure theorem characterizing solutions to non-homogeneous linear equations in Hilbert spaces with compact symmetric operators, including properties of associated maximization functions.

Findings

The function γ(r) is C^1, increasing, and strictly concave.

Maximization of J over spheres S_r is well-posed with unique solutions.

The solutions satisfy a specific operator equation involving γ'(r).

Abstract

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 $T(x)-\|T\|x=z$ 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 $S_r=\{x\in X:\|x\|^2=r\}$. Then, the function $\gamma$ is $C^1$, increasing and strictly concave in $]0,+\infty[$, with $\gamma'(]0,+\infty[)=]\|T\|,+\infty[$; moreover, for each $r>0$, the problem of maximizing $J$ over $S_r$ is well-posed, and one has $$T(\hat x_r)-\gamma'(r)\hat x_r=z$$ where $\hat x_r$ is the only global maximum of $J_{|S_r}$.\par