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Approximation numbers of composition operators on the \(H^2\) space of Dirichlet series. (English) Zbl 1308.47032

Summary: By a theorem of J. Gordon and H. Hedenmalm [Mich. Math. J. 46, No. 2, 313–329 (1999; Zbl 0963.47021)], \(\phi\) generates a bounded composition operator on the Hilbert space \(\mathcal{H}^2\) of Dirichlet series \(\sum_n b_n n^{- s}\) with square-summable coefficients \(b_n\) if and only if \(\varphi(s) = c_0 s + \psi(s)\), where \(c_0\) is a nonnegative integer and {\(\psi\)} a Dirichlet series with the following mapping properties: {\(\psi\)} maps the right half-plane into the half-plane \(\operatorname{Re} s > 1 / 2\) if \(c_0 = 0\) and is either identically zero or maps the right half-plane into itself if \(c_0\) is positive. It is shown that the nth approximation numbers of bounded composition operators on \(\mathcal{H}^2\) are bounded below by a constant times \(r^n\) for some \(0 < r < 1\) when \(c_0 = 0\) and bounded below by a constant times \(n^{- A}\) for some \(A > 0\) when \(c_0\) is positive. Both results are best possible. The case when \(c_0 = 0\), {\(\psi\)} is bounded and smooth up to the boundary of the right half-plane, and \(\sup \operatorname{Re} \psi = 1 / 2\), is discussed in depth; it includes examples of non-compact operators as well as operators belonging to all Schatten classes \(S_p\). For \(\varphi(s) = c_1 + \sum_{j = 1}^d c_{q_j} q_j^{- s}\) with \(q_j\) independent integers, it is shown that the nth approximation number behaves as \(n^{-(d - 1) / 2}\), possibly up to a factor \((\log n)^{(d - 1) / 2}\). Estimates rely mainly on a general Hilbert space method involving finite linear combinations of reproducing kernels. A key role is played by a recently developed interpolation method for \(\mathcal{H}^2\) using estimates of solutions of the \(\overline{\partial}\) equation. Finally, by a transference principle from \(H^2\) of the unit disc, explicit examples of compact composition operators with approximation numbers decaying at essentially any sub-exponential rate can be displayed.

MSC:

47B33 Linear composition operators
30B50 Dirichlet series, exponential series and other series in one complex variable
30H10 Hardy spaces

Citations:

Zbl 0963.47021
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References:

[1] Bayart, F., Hardy spaces of Dirichlet series and their composition operators, Monatsh. Math., 136, 203-236 (2002) · Zbl 1076.46017
[2] Bayart, F., Compact composition operators on a Hilbert of Dirichlet series, Illinois J. Math., 47, 725-743 (2003) · Zbl 1059.47023
[3] Boas, R. P., A general moment problem, Amer. J. Math., 63, 361-370 (1941) · Zbl 0025.25404
[4] Bohr, H., Über die gleichmässige Konvergenz Dirichletscher Reihen, J. Reine Angew. Math., 143, 203-211 (1912) · JFM 44.0307.01
[5] Carl, B.; Stephani, I., Entropy, Compactness and the Approximation of Operators, Cambridge Tracts in Math., vol. 98 (1990), Cambridge University Press: Cambridge University Press Cambridge · Zbl 0705.47017
[6] Carleson, L., An interpolation problem for bounded analytic functions, Amer. J. Math., 80, 921-930 (1958) · Zbl 0085.06504
[7] Carleson, L., Interpolations by bounded analytic functions and the corona problem, Ann. of Math. (2), 76, 547-559 (1962) · Zbl 0112.29702
[8] Carlson, F., Contributions à la théorie des séries de Dirichlet, Ark. Mat. (Note I), 16, 1-19 (1922) · JFM 48.0338.02
[9] Carroll, T.; Cowen, C. C., Compact composition operators not in the Schatten classes, J. Operator Theory, 26, 109-120 (1991) · Zbl 0792.47031
[10] Finet, C.; Queffélec, H.; Volberg, A., Compactness of composition operators on a Hilbert space of Dirichlet series, J. Funct. Anal., 211, 271-287 (2004) · Zbl 1070.47013
[11] Garnett, J. B., Bounded Analytic Functions, Grad. Texts in Math., vol. 236 (2007), Springer: Springer New York
[12] Gordon, J.; Hedenmalm, H., The composition operators on the space of Dirichlet series with square-summable coefficients, Michigan Math. J., 46, 313-329 (1999) · Zbl 0963.47021
[13] 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, 1-37 (1997) · Zbl 0887.46008
[14] Li, D.; Queffélec, H.; Rodriguez-Piazza, L., On approximation numbers of composition operators, J. Approx. Theory, 164, 431-459 (2012) · Zbl 1246.47007
[15] Li, D.; Queffélec, H.; Rodriguez-Piazza, L., Estimates for approximation numbers of some classes of composition operators on the Hardy space, Ann. Acad. Sci. Fenn. Math., 38, 1-18 (2013)
[16] MacCluer, B. D., Compact composition operators on \(H^p(B_N)\), Michigan Math. J., 32, 237-248 (1985) · Zbl 0585.47022
[17] Montgomery, H. L.; Vaughan, R. C., Hilbert’s inequality, J. Lond. Math. Soc., 8, 73-82 (1974) · Zbl 0281.10021
[18] Nikolskii, N., Treatise on the Shift Operator (1986), Springer-Verlag: Springer-Verlag Berlin, Heidelberg, New York
[19] Olsen, J.-F.; Seip, K., Local interpolation in Hilbert spaces of Dirichlet series, Proc. Amer. Math. Soc., 136, 203-212 (2008) · Zbl 1146.30003
[20] Pietsch, A., \(s\)-numbers of operators in Banach spaces, Studia Math., 51, 201-223 (1974) · Zbl 0294.47018
[21] Queffélec, H.; Seip, K., Decay rates for approximation numbers of composition operators, J. Anal. Math. (2014), in press; available at
[22] Rudin, W., Function Theory in Polydiscs (1969), W. A. Benjamin, Inc.: W. A. Benjamin, Inc. New York-Amsterdam · Zbl 0177.34101
[23] Schuster, A. P.; Seip, K., A Carleson-type condition for interpolation in Bergman spaces, J. Reine Angew. Math., 497, 223-233 (1998) · Zbl 0916.30037
[24] Seip, K., Zeros of functions in Hilbert spaces of Dirichlet series, Math. Z., 274, 1327-1339 (2013) · Zbl 1281.30005
[25] Shapiro, J. H., Composition Operators and Classical Function Theory, Universitext: Tracts in Mathematics (1993), Springer-Verlag: Springer-Verlag New York · Zbl 0791.30033
[26] Shapiro, H. S.; Shields, A. L., On some interpolation problems for analytic functions, Amer. J. Math., 83, 513-532 (1961) · Zbl 0112.29701
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