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No, the proof is wrong. Conceptual reason why it can't be correct: The argument in the paper, were it correct, would actually show much more, namely that for every bounded linear operator $T$ on a sep …

The operator $L = L_1 + L_f$ is closed. Proof. The operator $L_1$ generates a $C_0$-semigroup on $L^2(\mathbb{R})$ (namely, the left shift semigroup) and $L_f$ is a relatively compact perturbation of …

Since the OP has recently asked a related question for a different PDE and since this can all be answered in a completely general fashion, here is a general approach. General setting. Let us consider …

The answer is no in general (but it's not difficult to check that the answer is yes for injective operators). Counterexample. Let $H$ be a Hilbert space and $T_0: H \to H$ a bounded linear operator wi …

Yes, $C_0$-semigroups of Koopman operators and, more generally, lattice homomorphisms on $L^p$-spaces are studied in the paper "Measure-preserving semiflows and one-parameter Koopman semigroups" (2019 …

I quasi-nilpotent operator $T \in B(X)$ is nilpotent if and only if $0$ is a pole of its resolvent $(\cdot - T)^{-1}$, which means that there exists a number $M \ge 0$ and an integer $n \ge 1$ such th …

(In the following I assume that the word "invertible" in the question means "bijective".) Your assumptions do not imply that $F$ is bijective (however, they imply that $F$ is injective and has closed …

Here's a way to construct such an example: For each integer $n \ge 1$ consider the matrices $A_n, M_n \in \mathbb{C}^{2 \times 2}$ given by $$ M_n = e_1 e_1^T = \begin{pmatrix} 1 & 0 \\ …

Pazy's 1983 book on operator semigroups (link to zbMATH) Pazy, A., Semigroups of linear operators and applications to partial differential equations, Applied Mathematical Sciences, 44. New York etc.: …

As other users have indicated in the comments, for sufficiently smooth domains one can get it by combining, for instance, elliptic regularity with Hopf's boundary point lemma (and then go from the ell …

Integrals over strongly continuous functions with values in the compact operators are compact. That's a result by Jürgen Voigt.

The answer by Lars van der Laan gives a positive answer for the Hilbert space case (which was considered in the question), and it also works on reflexive Banach spaces. It might be worthwhile to add t …

Here's an algorithm for testing an ad-hoc conjecture $C$ about Hilbert space operators. :-) Set up the runtime environment correctly by loading the information "Most conjectures are false" into short …

If $V$ is merely an isomorphism from $D(A)$ to $D(B)$, the operator $V^{-1}BV$ is not well-defined (since $BV$ maps $D(A)$ to $Y$ rather than to $D(B)$). The "right" notion is a follows: Let's call th …

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 P …