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Unitary weighted composition operators on Bergman-Besov and Hardy Hilbert spaces on the ball

Year 2023, Volume: 1 Issue: 1, 12 - 19, 21.06.2023
https://doi.org/10.26650/ijmath.2023.00004

Abstract

On weighted Bergman and Hardy Hilbert spaces on the unit ball of the complex 𝑁-space, we consider weighted compositon operators 𝑇𝜓 in which the composition is by an automorphism 𝜓 of the unit ball and the weight is a power of the Jacobian of 𝜓 in such a way that the operator is unitary. Assuming that the homogeneous expansion of an 𝑓 in one of these spaces contains only terms with total degree even (odd, respectively) and the homogeneous expansion of 𝑇𝜓 𝑓 contains only terms with total degree odd (even, respectively), we prove that 𝑓 is the zero function. We also find related operators on the remaining Bergman-Besov Hilbert spaces including the Drury-Arveson space and the Dirichlet space for which the same result holds. Our results generalize the results obtained in Montes-Rodríguez (2023) on three function spaces on the unit disc to a wider family of function spaces on the unit ball.

Thanks

The author thanks A. Ersin Üreyen of Eskişehir Technical University for useful discussions and an anonymous referee for careful attention to details.

References

  • Alpay, D., Kaptanoğlu, H.T., 2007, Toeplitz operators on Arveson and Dirichlet spaces, Integral Equations Operator Theory, 58, 1-33. google scholar
  • Beatrous, F., Burbea, J., 1989, Holomorphic Sobolev spaces on the ball, Dissertationes Math., 276, 57 pp. google scholar
  • Cowen, C.C., MacCluer, B.D., 1991, Composition Operators on Spaces of Analytic Functions, Stud. Adv. Math., CRC Press, Boca Raton. google scholar
  • Kaptanoğlu, H.T., 2005, Bergman projections on Besov spaces on balls, Illinois J. Math., 49, 385-403. google scholar
  • Kaptanoğlu , H.T., Üreyen, A.E., 2018, Precise inclusion relations among Bergman-Besov and Bloch-Lipschitz spaces and H'x on the unit ball of C*, Math. Nachr., 291, 2236-2251. google scholar
  • Montes-Rodrfguez, A., 2023, Zeros under unitary weighted composition operators in the Hardy and Bergman spaces, Mediterr. J. Math., 20, #82, 9 pp. google scholar
  • Rudin, W., 1980, Function Theory in the Unit Ball of C", Grundlehren Math. Wiss., vol. 241, Springer, New York. google scholar
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Year 2023, Volume: 1 Issue: 1, 12 - 19, 21.06.2023
https://doi.org/10.26650/ijmath.2023.00004

Abstract

References

  • Alpay, D., Kaptanoğlu, H.T., 2007, Toeplitz operators on Arveson and Dirichlet spaces, Integral Equations Operator Theory, 58, 1-33. google scholar
  • Beatrous, F., Burbea, J., 1989, Holomorphic Sobolev spaces on the ball, Dissertationes Math., 276, 57 pp. google scholar
  • Cowen, C.C., MacCluer, B.D., 1991, Composition Operators on Spaces of Analytic Functions, Stud. Adv. Math., CRC Press, Boca Raton. google scholar
  • Kaptanoğlu, H.T., 2005, Bergman projections on Besov spaces on balls, Illinois J. Math., 49, 385-403. google scholar
  • Kaptanoğlu , H.T., Üreyen, A.E., 2018, Precise inclusion relations among Bergman-Besov and Bloch-Lipschitz spaces and H'x on the unit ball of C*, Math. Nachr., 291, 2236-2251. google scholar
  • Montes-Rodrfguez, A., 2023, Zeros under unitary weighted composition operators in the Hardy and Bergman spaces, Mediterr. J. Math., 20, #82, 9 pp. google scholar
  • Rudin, W., 1980, Function Theory in the Unit Ball of C", Grundlehren Math. Wiss., vol. 241, Springer, New York. google scholar
  • Zhu, K., 2005, Spaces of Holomorphic Functions in the Unit Ball, Grad. Texts in Math., vol. 226, Springer, New York. google scholar
There are 8 citations in total.

Details

Primary Language English
Subjects Pure Mathematics (Other)
Journal Section Research Articles
Authors

H. Turgay Kaptanoğlu 0000-0002-8795-4426

Publication Date June 21, 2023
Published in Issue Year 2023 Volume: 1 Issue: 1

Cite

APA Kaptanoğlu, H. T. (2023). Unitary weighted composition operators on Bergman-Besov and Hardy Hilbert spaces on the ball. Istanbul Journal of Mathematics, 1(1), 12-19. https://doi.org/10.26650/ijmath.2023.00004
AMA Kaptanoğlu HT. Unitary weighted composition operators on Bergman-Besov and Hardy Hilbert spaces on the ball. Istanbul Journal of Mathematics. June 2023;1(1):12-19. doi:10.26650/ijmath.2023.00004
Chicago Kaptanoğlu, H. Turgay. “Unitary Weighted Composition Operators on Bergman-Besov and Hardy Hilbert Spaces on the Ball”. Istanbul Journal of Mathematics 1, no. 1 (June 2023): 12-19. https://doi.org/10.26650/ijmath.2023.00004.
EndNote Kaptanoğlu HT (June 1, 2023) Unitary weighted composition operators on Bergman-Besov and Hardy Hilbert spaces on the ball. Istanbul Journal of Mathematics 1 1 12–19.
IEEE H. T. Kaptanoğlu, “Unitary weighted composition operators on Bergman-Besov and Hardy Hilbert spaces on the ball”, Istanbul Journal of Mathematics, vol. 1, no. 1, pp. 12–19, 2023, doi: 10.26650/ijmath.2023.00004.
ISNAD Kaptanoğlu, H. Turgay. “Unitary Weighted Composition Operators on Bergman-Besov and Hardy Hilbert Spaces on the Ball”. Istanbul Journal of Mathematics 1/1 (June 2023), 12-19. https://doi.org/10.26650/ijmath.2023.00004.
JAMA Kaptanoğlu HT. Unitary weighted composition operators on Bergman-Besov and Hardy Hilbert spaces on the ball. Istanbul Journal of Mathematics. 2023;1:12–19.
MLA Kaptanoğlu, H. Turgay. “Unitary Weighted Composition Operators on Bergman-Besov and Hardy Hilbert Spaces on the Ball”. Istanbul Journal of Mathematics, vol. 1, no. 1, 2023, pp. 12-19, doi:10.26650/ijmath.2023.00004.
Vancouver Kaptanoğlu HT. Unitary weighted composition operators on Bergman-Besov and Hardy Hilbert spaces on the ball. Istanbul Journal of Mathematics. 2023;1(1):12-9.