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A mean counting function for Dirichlet series and compact composition operators. (English) Zbl 07358512

The authors introduce a mean counting function for Dirichlet series which plays the same role in the theory of Hardy spaces of Dirichlet series as the Nevanlinna counting function does in the classical theory over the unit disk. In the paper \(\mathcal H^2\) stand for the Hardy space of Dirichlet series and the composition operator \(C_\phi(f)=f\circ \phi\) defines an analytic function in \(\mathbb C_{1/2}\) for any \(f\in \mathcal H^2\) and any analytic function \(\phi: \mathbb C_{1/2}\to \mathbb C_{1/2}\) where \(\mathbb C_{\theta}=\{s\in \mathbb C: \operatorname{Re} s>\theta\}\),. It is known (see [J. Gordon and H. Hedenmalm, Mich. Math. J. 46, No. 2, 313–329 (1999; Zbl 0963.47021)]) that \(C_\phi\) defines a bounded operator if and only if \(\phi(s)=c_0s+ \phi_0(s)= c_0 s+\sum_{n=1}^\infty c_n n^{-s}\) where \(\phi_0\) converges uniformly in \(\mathbb C_\varepsilon\) for any \(\varepsilon>0\) and satisfies the conditions: (a) if \(c_0=0\) then \(\phi_0(\mathbb C_0)\subseteq \mathbb C_{1/2}\) and (b) if \(c_0\ge 1\) then either \(\phi_0(\mathbb C_0)\subseteq \mathbb C_{0}\) or \(\phi_0=0\). Classifying the symbols to obtain compact composition operators remained as an open problem for long time. Let us denote \(\mathcal G_0\) the symbols in the case (a) where \(c_0=0\).
If \(f\) is a Dirichlet series which converges uniformly in \(\mathbb C_\varepsilon\) for each \(w\ne f(+\infty)\) they introduce the mean counting function \[\mathcal M_f(w)=\lim_{\sigma\to 0^+} \lim_{T\to \infty} \frac{\pi}{T} \sum_{s\in f^{-1}(w), |\operatorname{Im} s|<T, \sigma<\operatorname{Re} s<\infty} \operatorname{Re} s .\]
One of their main result establishes that for symbols in \(\mathcal G_0\) the mean counting function \(\mathcal M_\phi\) exists for every \(w\in \mathbb C_{1/2}\setminus {\phi(+\infty)}\) and satisfies \(\mathcal M_\phi(w)\le \log\Big|\frac{\bar w+\phi(+\infty)-1}{w-\phi(+\infty)}\Big|\). They actually study when the upper bound is achieved. Another important result in the paper is the analogue to the Stanton formula showing that if \(\phi \in \mathcal G_0\) then \[\|C_\phi(f)\|_{\mathcal H^2}^2= |f(\phi(+\infty))|^2+\frac{2}{\pi}\int_{\mathbb C_{1/2}}|f'(w)|^2\mathcal M_\phi(w) dw.\] As a consequence they manage to obtain that that \(C_\phi\) is compact on \(\mathcal H^2\) if and only if \(\lim_{\operatorname{Re} w \to (1/2)^+} \frac{\mathcal M_\phi(w)}{\operatorname{Re} w -\frac{1}{2}}=0,\) solving an old standing question.

MSC:

47B33 Linear composition operators
30H10 Hardy spaces
42A75 Classical almost periodic functions, mean periodic functions

Citations:

Zbl 0963.47021
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References:

[1] Adams, D. R.; Hedberg, L. I., Function Spaces and Potential Theory, Grundlehren der Mathematischen Wissenschaften, vol. 314 (1996), Springer-Verlag: Springer-Verlag Berlin
[2] Aleman, A.; Olsen, J.-F.; Saksman, E., Fatou and brothers Riesz theorems in the infinite-dimensional polydisc, J. Anal. Math., 137, 1, 429-447 (2019) · Zbl 1431.46021
[3] Bailleul, M., Composition operators on weighted Bergman spaces of Dirichlet series, J. Math. Anal. Appl., 426, 1, 340-363 (2015) · Zbl 1380.47022
[4] Bayart, F., Hardy spaces of Dirichlet series and their composition operators, Monatshefte Math., 136, 3, 203-236 (2002) · Zbl 1076.46017
[5] Bayart, F., Compact composition operators on a Hilbert space of Dirichlet series, Ill. J. Math., 47, 3, 725-743 (2003) · Zbl 1059.47023
[6] Bayart, F.; Brevig, O. F., Compact composition operators with nonlinear symbols on the \(H^2\) space of Dirichlet series, Pac. J. Math., 291, 1, 81-120 (2017) · Zbl 1481.47037
[7] Bohnenblust, H. F.; Hille, E., On the absolute convergence of Dirichlet series, Ann. Math. (2), 32, 3, 600-622 (1931) · JFM 57.0266.05
[8] Bohr, H., Über die gleichmäßige Konvergenz Dirichletscher Reihen, J. Reine Angew. Math., 143, 203-211 (1913) · JFM 44.0307.01
[9] Bondarenko, A.; Brevig, O. F.; Saksman, E.; Seip, K., Linear space properties of \(H^p\) spaces of Dirichlet series, Trans. Am. Math. Soc., 372, 9, 6677-6702 (2019) · Zbl 1432.30036
[10] Brevig, O. F., Sharp norm estimates for composition operators and Hilbert-type inequalities, Bull. Lond. Math. Soc., 49, 6, 965-978 (2017) · Zbl 06826674
[11] Brevig, O. F.; Perfekt, K.-M., Norms of composition operators on the \(H^2\) space of Dirichlet series, J. Funct. Anal., 278, 2, Article 108320 pp. (2020) · Zbl 1483.47045
[12] Carlson, F., Contributions à la théorie des séries de Dirichlet. IV, Ark. Mat., 2, 293-298 (1952) · Zbl 0048.05303
[13] Cole, B. J.; Gamelin, T. W., Representing measures and Hardy spaces for the infinite polydisk algebra, Proc. Lond. Math. Soc. (3), 53, 1, 112-142 (1986) · Zbl 0624.46032
[14] Duren, P. L., Theory of \(H^p\) Spaces, Pure and Applied Mathematics, vol. 38 (1970), Academic Press: Academic Press New York-London · Zbl 0215.20203
[15] Favorov, S. Yu., Lagrange’s mean motion problem, Algebra Anal., 20, 2, 218-225 (2008) · Zbl 1206.33004
[16] Finet, C.; Queffélec, H.; Volberg, A., Compactness of composition operators on a Hilbert space of Dirichlet series, J. Funct. Anal., 211, 2, 271-287 (2004) · Zbl 1070.47013
[17] Gordon, J.; Hedenmalm, H., The composition operators on the space of Dirichlet series with square summable coefficients, Mich. Math. J., 46, 2, 313-329 (1999) · Zbl 0963.47021
[18] Hardy, G. H., The mean value of the modulus of an analytic function, Proc. Lond. Math. Soc. (2), 14, 1, 269-277 (1915) · JFM 45.1331.03
[19] Hedenmalm, H., Dirichlet series and functional analysis, (The Legacy of Niels Henrik Abel (2004), Springer: Springer Berlin), 673-684 · Zbl 1068.30005
[20] Hedenmalm, H.; Lindqvist, P.; Seip, K., A Hilbert space of Dirichlet series and systems of dilated functions in \(L^2(0, 1)\), Duke Math. J., 86, 1, 1-37 (1997) · Zbl 0887.46008
[21] Jessen, B., Über die Nullstellen einer analytischen fastperiodischen Funktion. Eine Verallgemeinerung der Jensenschen Formel, Math. Ann., 108, 1, 485-516 (1933) · JFM 59.1027.04
[22] Jessen, B.; Tornehave, H., Mean motions and zeros of almost periodic functions, Acta Math., 77, 137-279 (1945) · Zbl 0061.16504
[23] Littlewood, J. E., On the zeros of the Riemann zeta-function, Proc. Lond. Math. Soc. (2), 22, 3, 295-318 (1924) · JFM 50.0230.02
[24] Littlewood, J. E., On inequalities in the theory of functions, Proc. Lond. Math. Soc. (2), 23, 7, 481-519 (1925) · JFM 51.0247.03
[25] Luecking, D. H.; Zhu, K. H., Composition operators belonging to the Schatten ideals, Am. J. Math., 114, 5, 1127-1145 (1992) · Zbl 0792.47032
[26] Olsen, J.-F.; Saksman, E., On the boundary behaviour of the Hardy spaces of Dirichlet series and a frame bound estimate, J. Reine Angew. Math., 663, 33-66 (2012) · Zbl 1239.46020
[27] Queffélec, H., Espaces de séries de Dirichlet et leurs opérateurs de composition, Ann. Math. Blaise Pascal, 22, S2, 267-344 (2015) · Zbl 1409.47004
[28] Queffélec, H.; Queffélec, M., Diophantine Approximation and Dirichlet Series, Harish-Chandra Research Institute Lecture Notes, vol. 2 (2013), Hindustan Book Agency: Hindustan Book Agency New Delhi · Zbl 1317.11001
[29] Queffélec, H.; Seip, K., Approximation numbers of composition operators on the \(H^2\) space of Dirichlet series, J. Funct. Anal., 268, 6, 1612-1648 (2015) · Zbl 1308.47032
[30] Rudin, W., A generalization of a theorem of frostman, Math. Scand., 21, 136-143 (1967), 1968 · Zbl 0185.33301
[31] Rudin, W., Function Theory in Polydiscs (1969), W. A. Benjamin, Inc.: W. A. Benjamin, Inc. New York-Amsterdam · Zbl 0177.34101
[32] Saksman, E.; Seip, K., Integral means and boundary limits of Dirichlet series, Bull. Lond. Math. Soc., 41, 3, 411-422 (2009) · Zbl 1180.30002
[33] Saksman, E.; Seip, K., Some Open Questions in Analysis for Dirichlet Series, Recent Progress on Operator Theory and Approximation in Spaces of Analytic Functions, Contemp. Math., vol. 679, 179-191 (2016), Amer. Math. Soc.: Amer. Math. Soc. Providence, RI · Zbl 1377.30048
[34] Shapiro, J. H., The essential norm of a composition operator, Ann. Math. (2), 125, 2, 375-404 (1987) · Zbl 0642.47027
[35] Shapiro, J. H., Composition Operators and Classical Function Theory, Universitext: Tracts in Mathematics (1993), Springer-Verlag: Springer-Verlag New York · Zbl 0791.30033
[36] Titchmarsh, E. C., The Theory of the Riemann Zeta-Function (1986), The Clarendon Press, Oxford University Press: The Clarendon Press, Oxford University Press New York, Edited and with a preface by D.R. Heath-Brown · Zbl 0601.10026
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