A generalized Hilbert matrix acting on Hardy spaces. (English) Zbl 1308.42021
Summary: If \(\mu\) is a positive Borel measure on the interval \([0,1)\), the Hankel matrix \(\mathcal H_\mu=(\mu_{n,k})_{n,k\geqslant 0}\) with entries \(\mu_{n,k}=\int_{[0,1)}t^{n+k}d\mu(t)\) induces formally the operator
\[
\mathcal H_\mu(f)(z)=\sum\limits_{n=0}^\infty\left(\sum\limits_{k=0}^\infty\mu_{n,k}a_k\right)z^n
\]
on the space of all analytic functions \(f(z)=\sum_{k=0}^\infty a_kz^k\), in the unit disc \(\mathbb D\). In this paper we describe those measures \(\mu\) for which \(\mathcal H_\mu\) is a bounded (compact) operator from \(H^p\) into \(H^q\), \(0<p,q<\infty\). We also characterize the measures \(\mu\) for which \(\mathcal H_\mu\) lies in the Schatten class \(S_p(H^2)\), \(1<p<\infty\).
MSC:
| 42B30 | \(H^p\)-spaces |
Keywords:
Hilbert matrices; Hardy spaces; BMOA; Carleson measures; integration operators; Hankel operators; Besov spaces; Schatten classesReferences:
| [1] | Aleman, A.; Montes-Rodríguez, A.; Sarafoleanu, A., The eigenfunctions of the Hilbert matrix, Constr. Approx., 36, 3, 353-374 (2012) · Zbl 1268.47040 |
| [2] | Anderson, J. M.; Clunie, J.; Pommerenke, Ch., On Bloch functions and normal functions, J. Reine Angew. Math., 270, 12-37 (1974) · Zbl 0292.30030 |
| [3] | Baernstein, A., Analytic functions of bounded mean oscillation, (Brannan, D. A.; Clunie, J. G., Aspects of Contemporary Complex Analysis (1980), Academic Press: Academic Press London, New York), 3-36 · Zbl 0492.30026 |
| [4] | Blasco, O.; Jarchow, H., A note on Carleson measures for Hardy spaces, Acta Sci. Math. (Szeged), 71, 1-2, 371-389 (2005) · Zbl 1101.47022 |
| [5] | Carleson, L., Interpolation by bounded analytic functions and the corona problem, Ann. of Math., 76, 547-559 (1962) · Zbl 0112.29702 |
| [6] | Diamantopoulos, E.; Siskakis, A. G., Composition operators and the Hilbert matrix, Studia Math., 140, 191-198 (2000) · Zbl 0980.47029 |
| [7] | Dostanic, M.; Jevtic, M.; Vukotic, D., Norm of the Hilbert matrix on Bergman and Hardy spaces and a theorem of Nehari type, J. Funct. Anal., 254, 2800-2815 (2008) · Zbl 1149.47017 |
| [8] | Duren, P. L., Extension of a theorem of Carleson, Bull. Amer. Math. Soc., 75, 143-146 (1969) · Zbl 0184.30503 |
| [9] | Duren, P. L., Theory of \(H^p\) Spaces (2000), Academic Press: Academic Press New York, London: Dover: Academic Press: Academic Press New York, London: Dover Mineola, New York, Reprint: · Zbl 0215.20203 |
| [10] | Duren, P. L.; Romberg, B. W.; Shields, A. L., Linear functionals on \(H^p\) spaces with \(0 < p < 1\), J. Reine Angew. Math., 238, 32-60 (1969) · Zbl 0176.43102 |
| [11] | Fefferman, C., Characterizations of bounded mean oscillation, Bull. Amer. Math. Soc., 77, 587-588 (1971) · Zbl 0229.46051 |
| [12] | Fefferman, C.; Stein, E. M., \(H^p\) spaces of several variables, Acta Math., 129, 3-4, 137-193 (1972) · Zbl 0257.46078 |
| [13] | Galanopoulos, P.; Peláez, J. A., A Hankel matrix acting on Hardy and Bergman spaces, Studia Math., 200, 3, 201-220 (2010) · Zbl 1206.47024 |
| [14] | Galanopoulos, P.; Girela, D.; Peláez, J. A.; Siskakis, A., Generalized Hilbert operators, Ann. Acad. Sci. Fenn. Math., 39 (2014) · Zbl 1297.47030 |
| [15] | Garnett, J. B., Bounded Analytic Functions (1981), Academic Press: Academic Press New York, London · Zbl 0469.30024 |
| [16] | Girela, D., Analytic functions of bounded mean oscillation, (Aulaskari, R., Complex Function Spaces. Complex Function Spaces, Mekrijärvi 1999. Complex Function Spaces. Complex Function Spaces, Mekrijärvi 1999, Univ. Joensuu Dept. Math. Rep. Ser., vol. 4 (2001), Univ. Joensuu: Univ. Joensuu Joensuu), 61-170 · Zbl 0981.30026 |
| [17] | Lanucha, B.; Nowak, M.; Pavlovic, M., Hilbert matrix operator on spaces of analytic functions, Ann. Acad. Sci. Fenn. Math., 37, 161-174 (2012) · Zbl 1258.47047 |
| [18] | Luecking, D. H., Trace ideal criteria for Toeplitz operators, J. Funct. Anal., 73, 2, 345-368 (1987) · Zbl 0618.47018 |
| [19] | Luecking, D. H., Embedding derivatives of Hardy spaces into Lebesgue spaces, Proc. London Math. Soc., 63, 3, 595-619 (1991) · Zbl 0774.42011 |
| [20] | Mateljevic, M.; Pavlovic, M., \(L^p\) behaviour of the integral means of analytic functions, Studia Math., 77, 219-237 (1984) · Zbl 1188.30004 |
| [21] | Pavlović, M., Introduction to Function Spaces on the Disk, Posebna Izdanja [Special Editions], vol. 20 (2004), Matematički Institut SANU: Matematički Institut SANU Beograd · Zbl 1107.30001 |
| [22] | Pavlović, M., Analytic functions with decreasing coefficients and Hardy and Bloch spaces, Proc. Edinb. Math. Soc. (2), 56, 2, 623-635 (2013) · Zbl 1327.30062 |
| [23] | Peláez, J. A.; Rättyä, J., Generalized Hilbert operators on weighted Bergman spaces, Adv. Math., 240, 227-267 (2013) · Zbl 1288.30060 |
| [24] | Peller, V., Hankel Operators and Their Applications, Springer Monogr. Math. (2003), Springer-Verlag: Springer-Verlag New York · Zbl 1030.47002 |
| [25] | Power, S. C., Hankel operators on Hilbert space, Bull. Lond. Math. Soc., 12, 422-442 (1980) · Zbl 0446.47015 |
| [26] | Videnskii, I. V., An analogue of Carleson measures, Dokl. Akad. Nauk SSSR. Dokl. Akad. Nauk SSSR, Soviet Math. Dokl., 37, 1, 186-190 (1988), (in Russian); translation in · Zbl 0687.30026 |
| [27] | Zhao, R., On logarithmic Carleson measures, Acta Sci. Math. (Szeged), 69, 3-4, 605-618 (2003) · Zbl 1050.30024 |
| [28] | Zhu, K., Operator Theory in Function Spaces, Math. Surveys Monogr., vol. 138 (2007) · Zbl 1123.47001 |
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