Linear operator has one-dimensional kernel

Question

Let $S_{\lambda}$ be a family of linear bounded operator on $L^2(\mathbb{R}^n)$ depending on some parameter $\lambda$, I have recently encountered several problems that dealt with the question whether $S_{\lambda}x=x$ has at most a one-dimensional space of solutions or equivalently if $Q_{\lambda}=S_{\lambda}-\operatorname{id}$ has at most a one-dimensional nullspace.

These were applications in PDEs and $S_{\lambda}$ was an integral operator. In general, I am very aware of fixed point methods which provide existence and uniqueness, however existence is not expected for all $\lambda$.

Thus, fixed point theorems are too restrictive for my purposes.

I ask: Is anybody aware of theorems/methods that imply that such an equation $$S_{\lambda}x=x$$ has at most a one-dimensional space of solutions which are not so strong that they provide existence?

Answer 1

As I don't know your precise setting, the following might or might not be useful to you.

There are results that show that if $\lambda \mapsto Q_\lambda$ is analytic, where, say $\lambda\in(a,b)$ belongs to an interval, then the set of $\lambda$ s.t. the dimension of the kernel of $Q_\lambda$ is not minimal, is discrete. (Provided that $Q_\lambda$ fulfills certain properties.)

More precisely, if one defines $\mu:=\min\{\mathrm{dim}ker(Q_\lambda)\text{ }|\text{ } \lambda \in (a,b)\}$, then the set $\{\lambda\in(a,b)\text{ }|\text{ }\mathrm{dim}ker(Q_\lambda)>\mu\}$ is discrete.

You can find two versions of this (with proofs) in the following papers. If this sounds useful to you, you might be able to adapt their proofs.

• Generic vanishing for harmonic spinors of twisted Dirac Operators (Nicolae Anghel), Theorem 1.1.
• Generic Metrics and Connections on Spin- and $\mathrm{Spin}^c$-Manifolds (Stephan Maier), Proposition 11.4.

This means in particular, that, if your situation fits in the above setting, and you can find at least one $\lambda_0$ s.t. $Q_{\lambda_0}$ has zero-dimensional kernel, then the kernel of $Q_\lambda$ is zero-dimensional for almost all $\lambda$.