Relation between the solutions $v_t=Lv$ and $v_t=Av$ if $A$ is a relatively compact perturbation of the linear operator $L$

Question

In a nutshell, here is my question. I read and know about the relation between the spectra of $L$ and $A$ if $A$ is a relatively compact perturbation of $L$. However, for my purpose, I am interested in the relation between the solutions of the equations $v_t=Lv$ and $v_t=Av$. For example, if the first has an unbounded solution, can we say the same about the second? Or, maybe, if all the solutions of the first are bounded, can we say the same for the second? Given the relation between the spectra of $A$ and $L$, one might think it is possible to make a connection, but I did not find one. Note that in my case, the operators $A$ and $L$ have the same spectrum in the sense that their resolvent sets coincide. Really, I am looking for references that would treat this kind of questions, not a proof.

The precise version of my question is below.

Let $L$ be a compactly defined differential operator on $L^2(\mathbb{R})$ such that it is possible to prove that the solutions to the IVP $$ v_t=Lv,\;\;v(0)=v_0\in {\rm Dom}(L)\;\;\;\;\;\;(1) $$ admits for any $v_0\in {\rm Dom}(L)$ the unique solution $v \in C(\mathbb{R},{\rm Dom}(L))$ such that the $L^2$ norm $v(t)$ is bounded by the $L^2$ norm of $v_0$. Then I would like the same to be true for $$ v_t=Av,\;\;v(0)=v_0\in {\rm Dom}(L)\;\;\;\;\;\;\;(2) $$ where $A$ is a relatively compact perturbation for $L$. More specifically, I am thinking of $A-L$ as a Hilbert-Schmidt integral operator.

Also, I would like the result that if there is a $v_0$ such that the solution to (1) has a norm that is not bounded, then there is also a solution to (2) that is not bounded.

If that helps, in my case, I can solve (1) by the method of characteristic and thus establish the boundedness or unboundedness of the solutions and I would like to extend that property to (2). Intuitively, this might be possible for a relatively compact perturbation. One important fact though is that the operators $A$ and $L$ have the same spectrum in the sense that their resolvent sets coincide.

Again, I am not interested in a proof but rather a reference where this type of problems could possibly be discussed.

Answer 1

General references.

The references that you're probably looking for are books on the theory of $C_0$-semigroups. Some classics are:

[1] Amnon Pazy: Semigroups of Linear Operators and Applications to Partial Differential Equations, 1983

[2] Klaus-Jochen Engel and Rainer Nagel: One-Parameter Semigroups of Linear Evolution Equations, 2000 (my personal favourite)

[3] Klaus-Jochen Engel and Rainer Nagel: A Short Course on Operator Semigroups, 2006 (the short version, naturally, of the previous reference)

Well-posedness.

Bounded perturbations of semigroup generators are again semigroup generators (see for instance [2, Section III.1].

Boundedness of the solutions.

That's essentially the reason why I wrote this answer: all solutions to $v_t = Lv$ are bounded if and only if the semigroup generated by $L$ is bounded. However, this property is not preserved by compact perturbations, even if they don't change the spectrum:

Counterexample. Let $L$ be the zero operator on $L^2(0,1)$ (which is everyhwere defined), and let $V: L^2(0,1) \to L^2(0,1)$ be the Volterra operator given by $$ (Vf)(x) = \int_0^x f(y) \, dy. $$ Then $V$ is compact and both $L=0$ and $L+V=V$ have spectrum $\{0\}$. But the semigroup generated by $L$ is bounded, while the semigroup generated by $L+V$ is not.