# Correction to: Unbounded Hankel Operators and Moment Problems

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## 1 Correction to: Integr. Equ. Oper. Theory 85 (2016), 289–300 https://doi.org/10.1007/s00020-016-2289-y

We use freely the notation of [5]. It was observed in [1] that some a priori condition on moments \(q_{n}\) was omitted in Theorem 1.2 of [5]. Our goal is to give a corrected version of this theorem.

Let us now give a corrected version of Theorem 1.2 of [5].

### Theorem 1

- (i)
- (ii)
The matrix elements \( q_n\rightarrow 0\) as \(n\rightarrow \infty \).

- (iii)The measure \(dM (\mu )\) defined by equations (2) satisfies the condition(to put it differently, \({{\,\mathrm{supp}\,}}M\subset [-\,1,1]\) and \(M(\{-1\}) = M(\{1\})=0)\).$$\begin{aligned} M({\mathbb R}{\setminus } (-\,1,1) )=0 \end{aligned}$$(8)

### Remark 2

- (i)
In the previous version of this paper [5], condition (7) was omitted. It was pointed out in [1] that, without some kind of an a priori assumption, the closability of

*q*[*g*,*g*] does not imply (ii) or (iii). - (ii)
A priori conditions (5), (7) permit very rapid growth of the moments \(q_{n}\) as \(n\rightarrow \infty \), for example, as \((n \ln n)^{n}\). However for closable forms

*q*[*g*,*g*], we prove that \( q_n\rightarrow 0\) as \(n\rightarrow \infty \). - (iii)Let the Carleman conditionbe satisfied. In accordance with (7) set$$\begin{aligned} \sum _{n\ge 1} q_{2n}^{-1/(2n)} =\infty \end{aligned}$$(9)It follows from the Stirling formula that$$\begin{aligned} \varkappa _{n}= (n!)^{-1/n}q_{2n}^{1/(2n)}. \end{aligned}$$and hence condition (9) implies (6).$$\begin{aligned} \lim _{n\rightarrow \infty }\frac{q_{2n} ^{1/(2n)}}{n\varkappa _{n}}=e^{-1}, \end{aligned}$$

### Lemma 3

The form *q*[*g*, *g*] defined on \(\mathcal D\) is closable in the space \(\ell ^2 ({\mathbb Z}_{+})\) if and only if the set \({\mathcal D}_{*}\) is dense in \( L^2 (M)\).

Thus, for the proof of (10), we only have to check that if the set \({\mathcal D}_{*}\) is dense in \( L^2 (M)\) and condition (7) is satisfied, then relation (8) holds.

*f*(

*x*) belong to the class \(C^{\infty } (\Delta ; \{ \varkappa _{n}\})\) for any bounded interval \(\Delta \subset {\mathbb R}\), they coincide on \(\Delta \) and hence for all \(x\in {\mathbb R}\). Using the Phragmén–Lindelöf principle, it is easy to deduce from estimates (13) for \(\widetilde{f}(x)\) and (15) that

### Remark 4

*f*(

*z*) given by (11) is analytic and bounded in the strip \(|{\mathrm{Im}}\,z| <\epsilon \). Therefore the functions \(\widetilde{f}(z)\) (defined by (14)) and

*f*(

*z*) coincide as analytic functions so that the theory of quasi-analytic functions is not required.

Finally, we note that, in Propositions 4.1 and 4.3, the phrase “(or, equivalently, the form (1) is closable)” should be replaced by “(or, equivalently, the form (1) is closable and condition (7) is satisfied)”.

I thank the authors of [1] who observed the omission in [5].

## Notes

## References

- 1.Berg, C., Szwarc, R.: Closable Hankel operators and moment problems. arXiv:1905.03010
- 2.Carleman, T.: Les Fonctions Quasi-Analytiques. Gauthier-Villars, Paris (1926)zbMATHGoogle Scholar
- 3.Cohen, P .J.: A simple proof of Denjoy–Carleman theorem. Am. Math. Mon.
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**85**, 289–300 (2016)MathSciNetCrossRefGoogle Scholar