Schatten-class generalized Volterra companion integral operators

Tesfa Mengestie

doi:10.1215/17358787-3492743

Banach Journal of Mathematical Analysis

Banach J. Math. Anal.
Volume 10, Number 2 (2016), 267-280.

Schatten-class generalized Volterra companion integral operators

Tesfa Mengestie

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Abstract

We study the Schatten-class membership of generalized Volterra companion integral operators on the standard Fock spaces $\mathcal{F}_{\alpha}^{2}$ . The Schatten $\mathcal{S}_{p}(\mathcal{F}_{\alpha}^{2})$ membership of the operators are characterized in terms of $L^{p/2}$ -integrability of certain generalized Berezin-type integral transforms on the complex plane. We also give a more simplified and easy-to-apply description in terms of $L^{p}$ -integrability of the symbols inducing the operators against super-exponentially decreasing weights. Asymptotic estimates for the $\mathcal{S}_{p}(\mathcal{F}_{\alpha}^{2})$ norms of the operators have also been provided.

Article information

Source
Banach J. Math. Anal., Volume 10, Number 2 (2016), 267-280.

Dates
Received: 11 April 2015
Accepted: 7 June 2015
First available in Project Euclid: 15 March 2016

Permanent link to this document
https://projecteuclid.org/euclid.bjma/1458053862

Digital Object Identifier
doi:10.1215/17358787-3492743

Mathematical Reviews number (MathSciNet)
MR3474839

Zentralblatt MATH identifier
1341.47026

Subjects
Primary: 47B32: Operators in reproducing-kernel Hilbert spaces (including de Branges, de Branges-Rovnyak, and other structured spaces) [See also 46E22]
Secondary: 46E22: Hilbert spaces with reproducing kernels (= [proper] functional Hilbert spaces, including de Branges-Rovnyak and other structured spaces) [See also 47B32] 46E20: Hilbert spaces of continuous, differentiable or analytic functions 47B33: Composition operators

Keywords
Fock space Schatten class Berezin transform Volterra operator generalized Volterra companion operator

Citation

Mengestie, Tesfa. Schatten-class generalized Volterra companion integral operators. Banach J. Math. Anal. 10 (2016), no. 2, 267--280. doi:10.1215/17358787-3492743. https://projecteuclid.org/euclid.bjma/1458053862

References

[1] A. Aleman and J. Cima, An integral operator on $H^{p}$ and Hardy’s inequality, J. Anal. Math. 85 (2001), no. 1, 157–176.
Mathematical Reviews (MathSciNet): MR1869606
Digital Object Identifier: doi:10.1007/BF02788078
[2] A. Aleman and A. Siskakis, An integral operator on $H^{p}$, Complex Variables Theory Appl. 28 (1995), no. 2, 149–158.
Mathematical Reviews (MathSciNet): MR1700079
Digital Object Identifier: doi:10.1080/17476939508814844
[3] A. Aleman and A. Siskakis, Integration operators on Bergman spaces, Indiana Univ. Math. J. 46 (1997), no. 2, 337–356.
Mathematical Reviews (MathSciNet): MR1481594
Digital Object Identifier: doi:10.1512/iumj.1997.46.1373
[4] O. Constantin, A Volterra-type integration operators on Fock spaces, Proc. Amer. Math. Soc. 140 (2012), no. 12, 4247–4257.
Mathematical Reviews (MathSciNet): MR2957216
Digital Object Identifier: doi:10.1090/S0002-9939-2012-11541-2
[5] R. Fleming and J. Jamison, Isometries on Banach Spaces: Function Spaces, Monogr. Surveys Pure Appl. Math. 129, Chapman and Hall/CRC, Boca Raton, FL, 2003.
Mathematical Reviews (MathSciNet): MR1957004
[6] J. Isralowitz and K. Zhu, Toeplitz operators on the Fock space, Integral Equations Operator Theory 66 (2010), no. 4, 593–611.
Mathematical Reviews (MathSciNet): MR2609242
Digital Object Identifier: doi:10.1007/s00020-010-1768-9
[7] S. Janson, J. Peetre, and R. Rochberg, Hankel forms and the Fock space, Rev. Mat. Iberoam. 3 (1987), no. 1, 61–138.
Mathematical Reviews (MathSciNet): MR1008445
Digital Object Identifier: doi:10.4171/RMI/46
[8] S. Li, Volterra composition operators between weighted Bergman spaces and Bloch type spaces, J. Korean Math. Soc. 45 (2008), no. 1, 229–248.
Mathematical Reviews (MathSciNet): MR2375133
Digital Object Identifier: doi:10.4134/JKMS.2008.45.1.229
[9] S. Li and S. Stević, Generalized composition operators on Zygmund spaces and Bloch type spaces, J. Math. Anal. Appl. 338 (2008), no. 2, 1282–1295.
Mathematical Reviews (MathSciNet): MR2386496
Digital Object Identifier: doi:10.1016/j.jmaa.2007.06.013
[10] S. Li and S. Stević, Products of Volterra type operator and composition operator from $H^{\infty}$ and Bloch spacesto the Zygmund space, J. Math. Anal. Appl. 345 (2008), no. 1, 40–52.
[11] T. Mengestie, Volterra type and weighted composition operators on weighted Fock spaces, Integral Equations Operator Theory 76 (2013), no 1, 81–94.
Mathematical Reviews (MathSciNet): MR3041722
Digital Object Identifier: doi:10.1007/s00020-013-2050-8
[12] T. Mengestie, Product of Volterra type integral and composition operators on weighted Fock spaces, J. Geom. Anal. 24 (2014), no. 2, 740–755.
Mathematical Reviews (MathSciNet): MR3192295
Digital Object Identifier: doi:10.1007/s12220-012-9353-x
[13] T. Mengestie, Generalized Volterra companion operators on Fock spaces, Potential Anal., published electronically 20 November 2015.
Mathematical Reviews (MathSciNet): MR3489856
Digital Object Identifier: doi:10.1007/s11118-015-9520-3
[14] J. Pau and J. Peláez, Embedding theorems and integration operators on Bergman spaces with rapidly decreasing weights, J. Funct. Anal. 259 (2010), no. 10, 2727–2756.
Mathematical Reviews (MathSciNet): MR2679024
Digital Object Identifier: doi:10.1016/j.jfa.2010.06.019
[15] C. Pommerenke, Schlichte Funktionen und analytische Funktionen von beschránkter mittlerer Oszillation, Comment. Math. Helv. 52 (1977), no. 4, 591–602.
Mathematical Reviews (MathSciNet): MR454017
Digital Object Identifier: doi:10.1007/BF02567392
[16] A. Siskakis, “Volterra operators on spaces of analytic functions—A survey” in Proceedings of the First Advanced Course in Operator Theory and Complex Analysis, Univ. Sevilla Secr., Seville, 2006, 51–68.
Mathematical Reviews (MathSciNet): MR2290748
[17] M. Smith, Testing Schatten class Hankel operators and Carleson embeddings via reproducing kernels, J. Lond. Math. Soc. 71 (2005), no. 1, 172–186.
Mathematical Reviews (MathSciNet): MR2108255
Digital Object Identifier: doi:10.1112/S0024610704005988
[18] K. Zhu, Operator Theory on Function Spaces, 2nd ed., Math. Surveys Monogr. 138, Amer. Math. Soc., Providence, 2007.
Mathematical Reviews (MathSciNet): MR2311536
[19] K. Zhu, Analysis on Fock Spaces, Springer, New York, 2012.
Mathematical Reviews (MathSciNet): MR2934601

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Banach Journal of Mathematical Analysis

Schatten-class generalized Volterra companion integral operators

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