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Abstract
We reprove Proposition 3.8 in our paper that was published in [Electron. J. Qual. Theory Differ. Equ. 2021, No. 7, 1–24], to fill a gap in the proof of Corollary 3.7 where the density of one of the embeddings does not follow by the original arguments. We further carry out some minor corrections in the proof of Corollary 3.7, in Remark 3.1 and in the formula (3.23) of the original paper.
Highlights
The proof of [4, Corollary 3.7] is incomplete as only the continuity of the embedding Eθp → B is verified in the proof but the density of the embedding is not
At the end of the note, we correct some minor inaccuracies that use the injectivity of the system operator Ap which is not true in general
In the step we prove that (Tp(t)|B)t≥0 is a strongly continuous semigroup
Summary
The proof of [4, Corollary 3.7] is incomplete as only the continuity of the embedding Eθp → B is verified in the proof but the density of the embedding is not (for the explanation of notation we refer to [4]). For p ∈ (1, ∞) the part of (Ap, D(Ap)) in B generates a positive strongly continuous semigroup of contractions on B. 2. In the step we prove that (Tp(t)|B)t≥0 is a strongly continuous semigroup. 2., and using that clearly B is continuously embedded in Ep, we can apply [2, Proposition in Section II.2.3] for (A2, D(A2)) and Y = B, and obtain that the part of (A2, D(A2)) in B generates a positive strongly continuous semigroup of contractions on B. We may present the complete proof of [4, Corollary 3.7] including the density of the embeddings in the statement using the above, independently proven, Proposition 2.1.
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