Discussion on Lecture 14

Markus Borkowski, 2020/02/14 13:24, 2020/02/14 13:25

Dear virtual lecturers,

here are some comments from our meeting in Darmstadt.

The first sentence of the proof of Lemma 14.1.2: We cannot show the statement by ourselves and haven’t seen it in the Lecture before. Do you have suggestions, how the statement can be shown?
in the second line of Lemma 14.1.2 we guess there is a „for“ missing in front of „all $n \in N$ “
in the second line from below on page 176 in Example 14.2.4 there is a „p“ missing in „assumptions“.

Best regards Markus B

Marcus Waurick, 2020/02/14 13:39, 2020/02/14 13:39

Dear Markus, dear Darmstadt team,

thank you for your remarks. I start off with 2 and 3: Thanks for this! Concerning Lemma 14.1.2: Unfortunately (sorry for that!), we omitted mentioning a result from the lecture notes here. In fact, one can use Proposition 6.3.1 and apply this to $B = T_{n} + A,$ with $dom (B) = dom (A)$ . Then, by the boundedness of $T_{n}$ , we obtain $B^{*} = (T_{n} + A)^{*} = T_{n}^{*} - A$ with $dom (B) = dom (A)$ . Furthermore, we have for all $ϕ \in dom (A)$ $ℜ ⟨ ϕ, B ϕ ⟩ = ℜ ⟨ ϕ, (T_{n} + A) ϕ ⟩ = ℜ ⟨ ϕ, T_{n} ϕ ⟩ \geq c ⟨ ϕ, ϕ ⟩,$ where we used that $ℜ ⟨ ϕ, A ϕ ⟩ = 0$ since $A$ is skew-selfadjoint. Thus, Proposition 6.3.1 implies the statements in the first line of the proof of Lemma 14.1.2.

Does this help?

Best regards,

MM

Markus Borkowski, 2020/02/27 10:27, 2020/02/27 10:27

Yes, Of Course. Thanks

ISem23

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Discussion on Lecture 14

Discussion on Lecture 14