A Mixed-Adjoint Framework for 2 × 2 Operator Matrices and Their Numerical Radii
4 affiliations
- 1Department of Mathematics and Statistics, College of Science, Imam Mohammad Ibn Saud Islamic University (IMSIU), Riyadh 11564, Saudi Arabia
- 2Department of Mathematics, College of Science and Arts, Najran University, Najran 66462, Saudi Arabia
- 3Science and Engineering Research Center, Najran University, Najran, Saudi Arabia
- 4Department of Mathematical Sciences, College of Science, Princess Nourah bint Abdulrahman University, P.O. Box 84428, Riyadh 11671, Saudi Arabia
Abstract
We develop a semi-inner product framework for 2×2 block operator matrices acting on the direct sum K=H⊕H of a complex Hilbert space H, generated by the diagonal operator matrix A=diag(A,B), where A and B are distinct positive operators acting on H, neither assumed to be injective. This extends the setting of Bhunia, Paul, and Nayak [Electron. J. Linear Algebra 36 (2020), 143–157], who treated the case A=diag(A,A) with A injective. Using Douglas’ majorization theorem, we employ and develop the mixed (A,B)- and (B,A)-adjoint formalism in the block operator matrix setting; these adjoints relate the two distinct seminorms induced by A and B and take the place of the ordinary adjoint when these seminorms differ. Within this framework, we establish upper and lower bounds for the A-numerical radius of 2×2 operator matrices belonging to the appropriate A-bounded or A-adjointable classes, extending and refining the bounds of the above-mentioned paper to the case of two distinct, not necessarily injective, positive generators.
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BibTeX
@article{baazem2026mixedadjoint,
title = {{A Mixed-Adjoint Framework for 2 × 2 Operator Matrices and Their Numerical Radii}},
author = {Amani Baazem and Kais Feki and Hala H. Taha},
journal = {Mathematics},
year = {2026},
month = nov,
doi = {10.3390/math14214048},
url = {https://www.mdpi.com/2227-7390/14/21/4048},
note = {CC BY 4.0},
}
RIS
TY - JOUR
TI - A Mixed-Adjoint Framework for 2 × 2 Operator Matrices and Their Numerical Radii
AU - Amani Baazem
AU - Kais Feki
AU - Hala H. Taha
PY - 2026
DA - 2026/11/05
JO - Mathematics
DO - 10.3390/math14214048
UR - https://www.mdpi.com/2227-7390/14/21/4048
AB - We develop a semi-inner product framework for 2×2 block operator matrices acting on the direct sum K=H⊕H of a complex Hilbert space H, generated by the diagonal operator matrix A=diag(A,B), where A and B are distinct positive operators acting on H, neither assumed to be injective. This extends the setting of Bhunia, Paul, and Nayak [Electron. J. Linear Algebra 36 (2020), 143–157], who treated the case A=diag(A,A) with A injective. Using Douglas’ majorization theorem, we employ and develop the mixed (A,B)- and (B,A)-adjoint formalism in the block operator matrix setting; these adjoints relate the two distinct seminorms induced by A and B and take the place of the ordinary adjoint when these seminorms differ. Within this framework, we establish upper and lower bounds for the A-numerical radius of 2×2 operator matrices belonging to the appropriate A-bounded or A-adjointable classes, extending and refining the bounds of the above-mentioned paper to the case of two distinct, not necessarily injective, positive generators.
SN - 2227-7390
ER -