Joint q-Numerical Ranges of Continuous Operator-Valued Functions
Abstract
This paper develops a unified theory of joint q-numerical ranges Fq(F) for continuous functions F:Om → B(cH), where Om is a compact Hausdorff space and B(cH) denotes the C*-algebra of bounded linear operators on a complex Hilbert space cH. We prove that Fq(F) is totally bounded in cC(Om), the space of continuous complex-valued functions on Om, and establish the upper semicontinuity of the set-valued map F↦ Fq(F), and show that Fq(F) is compact when cH is finite-dimensional. The main result of the paper is a realization theorem. We prove that, for every totally bounded subset K of cC(Om) and every q∈ C with |q|=1, there exist a Hilbert space cH and a continuous function F:Om→B(cH) such that qK⊂ Fq(F)⊂ qcok. Consequently, if K is closed and convex, then Fq(F)=qK. As applications, we derive analogous realization results for bounded subsets in Cn and for totally bounded subsets in arbitrary Banach spaces.
Related
- Bohr-type inequalities for certain integral transforms of operator-valued analytic functions on shifted disc
- Some Generalized Numerical Ranges of Real-linear Operators on Hilbert Spaces
- Dilation Theory for Scaled q-Numerical Ranges
- Block numerical ranges for linear relations
- Spectral constants for algebraic numerical ranges
BibTeX
@misc{azadi2026joint,
title = {{Joint q-Numerical Ranges of Continuous Operator-Valued Functions}},
author = {Amin Azadi},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.32504},
eprint = {2610.32504},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.32504},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Joint q-Numerical Ranges of Continuous Operator-Valued Functions
AU - Amin Azadi
PY - 2026
DA - 2026/10/23
PB - arXiv
DO - 10.48550/arxiv.2610.32504
UR - https://arxiv.org/abs/2610.32504
AB - This paper develops a unified theory of joint q-numerical ranges Fq(F) for continuous functions F:Om → B(cH), where Om is a compact Hausdorff space and B(cH) denotes the C^*-algebra of bounded linear operators on a complex Hilbert space cH. We prove that Fq(F) is totally bounded in cC(Om), the space of continuous complex-valued functions on Om, and establish the upper semicontinuity of the set-valued map F↦ Fq(F), and show that Fq(F) is compact when cH is finite-dimensional. The main result of the paper is a realization theorem. We prove that, for every totally bounded subset K of cC(Om) and every q∈ C with |q|=1, there exist a Hilbert space cH and a continuous function F:Om→B(cH) such that qK⊂ Fq(F)⊂ qcok. Consequently, if K is closed and convex, then Fq(F)=qK. As applications, we derive analogous realization results for bounded subsets in C^n and for totally bounded subsets in arbitrary Banach spaces.
ER -