Discussion on Lecture 10

Johann Beurich, 2020/01/15 13:37

Dear all,

here are two remarks from our meeting in Dresden.

1. In the proof of Proposition 10.1.6 the inequality $sup_{| z | \geq r} ‖ (z M_{0} + M_{1})^{- 1} ‖ \leq K_{1} sup_{| z | \geq r} ‖ (z N + 1)^{- 1} ‖$ should rather be formulated pointwise like $‖ (z M_{0} + M_{1})^{- 1} ‖ \leq K_{1} ‖ (z N + 1)^{- 1} ‖ (| z | \geq r) .$

2. In Proposition 10.2.7 we only assume $ν > 0$ , but then existence seems to be not guaranteed. Should we only assume $ν > 0$ , for which a solution exists?

Best regards,

Johann

Sascha Trostorff, 2020/01/15 14:29

Dear Johann,

thanks for your remarks.

1. Your are right, it is better to formulate the estimate point-wise, since we need it in that way later on.

2. The formulation is a bit misleading, sorry for that. If $U_{0} \in I V (M_{0}, M_{1})$ , there exists a solution $U \in L_{2, ν} (R_{\geq 0}; H) \cap C (R_{\geq 0}; H)$ for some $ν > 0$ by definition and this $ν$ is meant in the proposition. So the statement is not an existence statement, but only states the uniqueness of the solution if it exists.

Best regards

Sascha

Sahiba Arora, 2020/01/14 13:06

Dear ISEM Team,

Since $(L_{ρ} 1_{R \geq 0}) (t) = \frac{1}{\sqrt{2 π} (i t + ρ)},$ so isn't the equation in Proposition $10.2.7$ missing a $\frac{1}{\sqrt{2 π}}$ ?

Regards,

Sahiba

Sascha Trostorff, 2020/01/14 13:58

Dear Sahiba,

yes, your are completely right. Thanks for pointing out!

Best regards

Sascha

Sahiba Arora, 2020/01/11 20:23, 2020/01/11 20:23

Dear all,

I believe there is a missing $M_{1}$ in the statement of Lemma $10.2.3 (c) .$ The equality according to me should be: $(z M_{0} + M_{1})^{- 1} x = \frac{1}{z} x + \sum_{l = 1}^{k} \frac{1}{z^{l + 1}} x_{l} + \frac{1}{z^{k + 1}} (z M_{0} + M_{1})^{- 1} M_{1} x_{k + 1} .$

Regards,

Sahiba

Marcus Waurick, 2020/01/13 08:30

Dear Sahiba,

thank you for careful reading. The statement in Lemma 10.2.3© is not missing an $M_{1}$ . The reason for this is that we stated a less precise statement than you did. We only asserted the existence of some $x_{1}, \dots, x_{k + 1} \in H$ . You provided a more precise statement telling that some of these elements might be chosen to lie in the range of $M_{1}$ . We did not need this observation. Hence, we sticked to the formulation we have.

Does this clarify the matter?

Best regards,

MM

Sahiba Arora, 2020/01/13 10:52

Dear Marcus,

Yes, that makes sense. Sorry for the silly error.

Regards,

Sahiba

Fabian Bleitner, 2020/01/10 12:39, 2020/01/10 12:39

Hey all,

there is a minor mistake in the notes. In the proof of Proposition 10.2.1 the scalar products under the integral have to be taken in $H$ instead of $L_{2, ν}$ .

Best regards

Fabian

Marcus Waurick, 2020/01/10 12:42

Dear Fabian,

You are absolutely right, thank you for pointing this out!

Best regards,

MM

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Discussion on Lecture 10

Discussion on Lecture 10