Trace of Multiplication Operator Restricted to Invariant Subspaces of some Weighted Bergman Space

Faruk Yılmaz

doi:10.17776/csj.1863606

Trace of Multiplication Operator Restricted to Invariant Subspaces of some Weighted Bergman Space

Abstract

Let ω be a logarithmically subharmonic weight that is radial and reproducing for the origin, and L_a^2 (D,ωdA) be the weighted Bergman space. Let f be a bounded holomorphic function on the open unit disc, I be a z-invariant subspace of L_a^2 (D,ωdA), and f(M_I) denotes the restriction to I of the multiplication operator M_f. This paper investigates the trace of the self-commutator of the operator f(M_I). More precisely, we compute the trace of the commutator [f(M_I )^*,f(M_I)] and show that it equals dim⁡(I⊝zI)∫_D |f^' (z)|^2 dA(z).

Keywords

References

[1] Berger, C. A., & Shaw, B. I. (1973). Self-commutators of multicyclic hyponormal operators are always trace class. Bulletin of the American Mathematical Society, 79(6), 1193–1199. https://doi.org/10.1090/S0002-9904-1973-13375-0
[2] Berger, C. A., & Shaw, B. I. (1973). Intertwining, analytic structure, and the trace norm estimate. In Proceedings of a Conference on Operator Theory (pp. 1–6). Springer. https://doi.org/10.1007/BFb0058914
[3] Fisher, S. D., Arazy, J., Janson, S., & Peetre, J. (1990). An identity for reproducing kernels in a planar domain and Hilbert-Schmidt Hankel operators. Journal für die reine und angewandte Mathematik (Crelles Journal), 1990(406), 179–199. https://doi.org/10.1515/crll.1990.406.179
[4] Aleman, A. (1996). Subnormal operators with compact self-commutator. Manuscripta Mathematica, 91(1), 353–367. https://doi.org/10.1007/BF02567960
[5] Feldman, N. S. (1997). The Berger-Shaw theorem for cyclic subnormal operators. Indiana University Mathematics Journal, 46(3), 741–751. https://doi.org/10.1512/IUMJ.1997.46.1408
[6] Elias, N. (1988). Toeplitz operators on weighted Bergman spaces. Integral Equations and Operator Theory, 11(3), 310–331. https://doi.org/10.1007/BF01202076
[7] Conway, J. B. (1991). The theory of subnormal operators (Mathematical Surveys and Monographs, Vol. 36). American Mathematical Society. https://doi.org/10.1090/surv/036
[8] Cowen, M. J., & Douglas, R. G. (1981). On moduli for invariant subspaces. In Invariant Subspaces and Other Topics: 6th International Conference on Operator Theory, Timişoara and Herculane (Romania), June 1–11 (pp. 65–73). Birkhäuser. https://doi.org/10.1007/978-3-0348-5445-0_5

[9] Zhu, K. (1997). Restriction of the Bergman shift to an invariant subspace. The Quarterly Journal of Mathematics, 48(4), 519–532. https://doi.org/10.1093/qmath/48.4.519
[10] Zhu, K. (2001). A trace formula for multiplication operators on invariant subspaces of the Bergman space. Integral Equations and Operator Theory, 40(2), 244–255. https://doi.org/10.1007/BF01301468
[11] Ni, J. (2020). A trace formula on invariant subspaces of Hardy space induced by rotation-invariant Borel measure. Complex Analysis and Operator Theory, 14(1). https://doi.org/10.1007/s11785-019-00958-3
[12] Hedenmalm, H., Korenblum, B., & Zhu, K. (2012). Theory of Bergman spaces (Graduate Texts in Mathematics, Vol. 199). Springer New York. https://doi.org/10.1007/978-1-4612-0497-8
[13] Koosis, P. (1998). The logarithmic integral I (Cambridge Studies in Advanced Mathematics, Vol. 12). Cambridge University Press. https://doi.org/10.1017/CBO9780511566196
[14] Hedenmalm, H., Jakobsson, S., & Shimorin, S. (2002). A biharmonic maximum principle for hyperbolic surfaces. Journal für die reine und angewandte Mathematik (Crelles Journal), 2002(550), 25–75. https://doi.org/10.1515/crll.2002.074
[15] Shimorin, S. M. (2001). Wold-type decompositions and wandering subspaces for operators close to isometries. Journal für die reine und angewandte Mathematik (Crelles Journal), 2001(531), 147–189. https://doi.org/10.1515/crll.2001.013
[16] Shimorin, S. M. (1996). Single-point extremal functions in weighted Bergman spaces. Journal of Mathematical Sciences, 80(4), 2349–2356. https://doi.org/10.1007/BF02362392
[17] Aleman, A., Hedenmalm, H., Richter, S., & Sundberg, C. (2001). Curious properties of canonical divisors in weighted Bergman spaces. In Israel Mathematical Conference Proceedings (Vol. 15, pp. 1–10). American Mathematical Society. https://doi.org/10.48550/arXiv.math/0406369
[18] Zhu, K. (2000). A sharp estimate for extremal functions. Proceedings of the American Mathematical Society, 128(9), 2577–2583. https://doi.org/10.1090/S0002-9939-00-05319-3
[19] Richter, S. (1987). Invariant subspaces in Banach spaces of analytic functions. Transactions of the American Mathematical Society, 304(2), 585–616. https://doi.org/10.1090/S0002-9947-1987-0911086-8
[20]Aleman, A., Richter, S., & Sundberg, C. (1996). Beurling’s theorem for the Bergman space. Acta Mathematica, 177(2), 275–310. https://doi.org/10.1007/BF02392623
[21] Shimorin, S. M. (1998). The Green functions for weighted biharmonic operators of the form $\Delta w^{-1} \Delta$ in the unit disk. Journal of Mathematical Sciences, 92(6), 4404–4411. https://doi.org/10.1007/BF02433445
[22] Martin, M., & Putinar, M. (1989). Lectures on hyponormal operators (Operator Theory: Advances and Applications, Vol. 39). Birkhäuser. https://doi.org/10.1007/978-3-0348-7466-3

Details

Primary Language

English

Subjects

Operator Algebras and Functional Analysis

Journal Section

Research Article

Authors

Faruk Yılmaz ^*
0000-0003-2742-7963
Türkiye

Publication Date

April 29, 2026

Submission Date

January 14, 2026

Acceptance Date

April 14, 2026

Published in Issue

Year 2026 Volume: 47 Number: 2

DOI

https://doi.org/10.17776/csj.1863606

IZ

https://izlik.org/JA29PM85EF

Cite

RIS / Bibtex

APA

Yılmaz, F. (2026). Trace of Multiplication Operator Restricted to Invariant Subspaces of some Weighted Bergman Space. Cumhuriyet Science Journal, 47(2), 361-365. https://doi.org/10.17776/csj.1863606