Backward Shift Operators on Bergman-Besov Spaces as Bergman Projections

Abstract

We express backward shift operators on all Bergman-Besov spaces in terms of Bergman projections in one and several variables including the Banach function spaces and the special Hilbert spaces such as Drury-Arveson and Dirichlet spaces. These operators are adjoints of the shift operators and their definitions for the case 𝑝 = 1 and proper Besov spaces require the use of nontrivial imbeddings of the spaces into Lebesgue classes. Our results indicate that the backward shifts are compositions of imbeddings into Lebesgue classes followed by multiplication operators by the conjugates of the coordinate variables followed by Bergman projections on appropriate spaces. We apply our results to the wandering subspace property of invariant subspaces of the shift operators on certain of our Hilbert spaces.

Keywords

• Backward shift operator
• Bergman projection
• Wandering subspace property
• Bergman-Besov space
• Hardy space
• Dirichlet space
• Drury-Arveson space

References

1. Aleman, A., Richter, S. and Sundberg, C., 1996, Beurling’s Theorem for the Bergman Space, Acta Math. 177, 275-310. google scholar
2. Alpay, D. and Kaptanoğlu, H. T., 2001, Integral Formulas for a Sub-Hardy Hilbert Space on the Ball with Complete Nevanlinna-Pick Reproducing Kernel, C. R. Acad. Sci. Paris Ser. I Math. 333, 285-290. google scholar
3. Gallardo-Gutierrez, E. A., Partington, J. R. and Seco, D., 2020, On the Wandering Property in Dirichlet Spaces, Integral Equations Operator Theory 92, 11 pp. google scholar
4. Gu, C. and Luo, S., 2024, Analytic Representations Backward Shifts on Weighted Bergman and Dirichlet Spaces, J. Math. Anal. Appl. 539, #128502. google scholar
5. Kaptanoğlu, H. T., 2005, Bergman Projections on Besov Spaces on Balls, Illinois J. Math. 49, 385-403. google scholar
6. Kaptanoğlu, H. T., 2014, Aspects of Multivariable Operator Theory on Weighted Symmetric Fock Spaces, Commun. Contemp. Math. 16, #1350034. google scholar
7. Kaptanoğlu, H. T. and Tülü, S., 2011, Weighted Bloch, Lipschitz, Zygmund, Bers, and Growth Spaces of the Ball: Bergman Projections and Characterizations, Taiwanese J. Math. 15, 101-127. google scholar
8. Kaptanoğlu, H. T. and Üreyen, A. E., 2008, Analytic Properties of Besov Spaces via Bergman Projections, Contemp. Math. 455, 169-182. google scholar

9. Richter, S., 1988, Invariant Subspaces of the Dirichlet Shift, J. Reine Angew. Math. 386, 205-220. google scholar
10. Shimorin, S., 2001, Wold-Type Decompositions and Wandering Subspaces for Operators Close to Isometries, J. Reine Angew. Math. 531, 147-189. google scholar