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Characterizations and models for the $C_{1,r}$ class and quantum annulus

Part of: General theory of linear operators

Published online by Cambridge University Press:  07 February 2025

Sourav Pal Open the ORCID record for Sourav Pal [Opens in a new window]  and
Nitin Tomar
Sourav Pal*
Affiliation:
Mathematics Department, Indian Institute of Technology Bombay, Powai, Mumbai-400076, India e-mail: tnitin@math.iitb.ac.in
Nitin Tomar
Affiliation:
Mathematics Department, Indian Institute of Technology Bombay, Powai, Mumbai-400076, India e-mail: tnitin@math.iitb.ac.in
*
e-mail: sourav@math.iitb.ac.in
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Abstract

For fixed $0<r<1$, let $A_r=\{z \in \mathbb {C} : r<|z|<1\}$ be the annulus with boundary $\partial \overline {A}_r=\mathbb {T} \cup r\mathbb {T}$, where $\mathbb T$ is the unit circle in the complex plane $\mathbb C$. An operator having $\overline {A}_r$ as a spectral set is called an $A_r$-contraction. Also, a normal operator with its spectrum lying in the boundary $\partial \overline {A}_r$ is called an $A_r$-unitary. The $C_{1,r}$ class was introduced by Bello and Yakubovich in the following way:

$$\begin{align*}C_{1, r}=\{T: T \ \text{is invertible and} \ \|T\|, \|rT^{-1}\| \leq 1\}. \end{align*}$$

McCullough and Pascoe defined the quantum annulus $\mathbb Q \mathbb A_r$ by

$$\begin{align*}\mathbb Q\mathbb A_r = \{T \,:\, T \text{ is invertible and } \, \|rT\|, \|rT^{-1}\| \leq 1 \}. \end{align*}$$

If $\mathcal A_r$ denotes the set of all $A_r$-contractions, then $\mathcal A_r \subsetneq C_{1,r} \subsetneq \mathbb Q \mathbb A_r$. We first find a model for an operator in $C_{1,r}$ and also characterize the operators in $C_{1,r}$ in several different ways. We prove that the classes $C_{1,r}$ and $\mathbb Q\mathbb A_r$ are equivalent. Then, via this equivalence, we obtain analogous model and characterizations for an operator in $\mathbb Q \mathbb A_r$.

Keywords

C1, r classquantum annulusA r-contractionA r-unitary

MSC classification

Primary: 47A20: Dilations, extensions, compressions 47A25: Spectral sets 47A63: Operator inequalities

Information

Type
Article
Information
Canadian Mathematical Bulletin , Volume 68 , Issue 3 , September 2025 , pp. 818 - 833
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Copyright
© The Author(s), 2025. Published by Cambridge University Press on behalf of Canadian Mathematical Society

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Footnotes

The first named author is supported by the Seed Grant of IIT Bombay, the CDPA and the “Core Research Grant” with Award No. CRG/2023/005223 of Science and Engineering Research Board (SERB), India. The second named author is supported by the Prime Minister’s Research Fellowship (PMRF ID 1300140), Government of India.

References

Agler, J., Rational dilation on an annulus . Ann. Math. 121(1985), 537–563.CrossRefGoogle Scholar
Agler, J., Harland, J., and Raphael, B. J., Classical function theory, operator dilation theory, and machine computation on multiply-connected domains . Mem. Amer. Math. Soc. 191(2008), 289–312.Google Scholar
Ando, T., On a pair of commutative contractions . Acta Sci. Math. 24(1963), 88–90.Google Scholar
Bello, G. and Yakubovich, D., An operator model in the annulus . J. Oper. Theory 90(2023), 25–40.CrossRefGoogle Scholar
Conway, J. B., The theory of subnormal operators, American Mathematical Society, Providence, RI, 1991.CrossRefGoogle Scholar
Dritschel, M. A., Jury, M. T., and McCullough, S., Dilations and constrained algebras . Oper. Matrices 10(2016), 829–861.CrossRefGoogle Scholar
Dritschel, M. and Undrakh, B., Rational dilation problems associated with constrained algebras . J. Math. Anal. Appl. 467(2018), 95–131.CrossRefGoogle Scholar
Mair, B. A. and McCullough, S., Invariance of extreme harmonic functions on an annulus; applications to theta functions . Houston J. Math. 20(1994), 453–473.Google Scholar
McCullough, S. and Shen, L., On the Szegő kernel of an annulus . Proc. Amer. Math. Soc. 121(1994), 1111–1121.Google Scholar
McCullough, S., Matrix functions of positive real part on an annulus . Houston J. Math. 21(1995), 489–506.Google Scholar
McCullough, S. and Sultanic, S., Agler-commutant lifting on an annulus . Integral Equations Operator Theory 72(2012), 449–482.CrossRefGoogle Scholar
McCullough, S. and Pascoe, J. E., Geometric dilations and operator annuli . J. Funct. Anal. 285(2023), Paper No. 110035, 20 pp.CrossRefGoogle Scholar
Misra, G., Curvature inequalities and extremal properties of bundle shifts . J. Oper. Theory 11(1984), 305–317.Google Scholar
Mittal, M., Function theory on the quantum annulus and other domains. Ph.D. thesis, University of Houston, 2010, 141 pp.Google Scholar
Taylor, J. L., The analytic-functional calculus for several commuting operators . Acta Math. 125(1970), 1–38.CrossRefGoogle Scholar
Tsikalas, G., A von Neumann type inequality for an annulus . J. Math. Anal. Appl. 506(2022), Paper No. 125714, 12 pp.CrossRefGoogle Scholar
Tsikalas, G., A note on a spectral constant associated with an annulus . Oper. Matrices 16(2022), 95–99.CrossRefGoogle Scholar
Williams, J. P., Minimal spectral sets of compact operators . Acta Sci. Math. 28(1967), 93–106.Google Scholar