International Journal of Nonlinear Analysis and Applications

Boundary curvature of the numerical range of self-inverse operators

Articles in Press

Document Type : Research Paper

Authors

1 Department of Mathematics, College of Sciences, Yasouj University, Yasouj, 75918 Iran

2 Department of Mathematics, College of Sciences, Yasouj Univer- sity, Yasouj, 75918, Iran

10.22075/ijnaa.2024.33147.4935

Abstract

In this article, firstly we investigate some properties of the boundary curvature of the numerical range. In the next, we define $M(T)$ as the smallest constant such that $dist(\lambda,\sigma(T))\leq M(T) R_\lambda (T),$ for all $\lambda \in \partial W(T)$, where $R_\lambda (T)$, the radius curvature at the point $\lambda$, is defined. Also, we investigate for non-convexoid $T$, $M(T)=\sup\frac{dist(\lambda,\sigma(T))}{R_{\lambda}(T)}$, where the supremum on the right-hand side is taken along all points $\lambda\in \partial W(T)$ with finite non-zero curvature. Finally, the value of $M(T)$ will be calculated for the self-inverse operators.

Keywords

[1] A. Abdollahi, On the self-inverse operators, Rendi. Circ. Mate. Palermo Ser. 2 58 (2009), 257–264.
[2] J. Agler, Geometric and topological properties of the numerical range, Indiana Univ. Math. J. 31 (1982), 767–777.
[3] A. Baklouti and M. Mabrouk, Essential numerical ranges of operators in semi-Hilbertian spaces, Ann. Funct. Anal. 13 (2022), 16.
[4] F.L. Bauer, On the field of values subordinate to a norm, Numer. Math. 4 (1962), 103—113.
[5] S.K. Berberian and G.H. Orland, On the closure of the numerical range of an operator, Proc. Amer. Math. Soc. 18 (1967) 499–503.
[6] P. Bhunia, S.S. Dragomir, M.S. Moslehian, and P. Kallol, Lectures on Numerical Radius Inequalities, Springer Nature, 2022.
[7] F.F. Bonsall and J. Duncan, Numerical Ranges of Operators on Normed Spaces and of Elements of normed Algebras, London Math. Soc. Lecture Notes Series 2, Cambridge, 1971.
[8] M. Campen, R. Capouellez, H. Shen, L. Zhu, D. Panozzo, and D. Zorin, Efficient and robust discrete conformal equivalence with boundary, ACM Trans. Graphics 40 (2021), no. 6, 1–16.
[9] L. Caston and M. Savova, I. Spitkovsky, and N. Zobin, On eigenvalues and boundary curvature of the numerical range, Linear Algebra Appl. 322 (2001), 129–140.
[10] F. Facchinei and J.S. Pang, Finite-Dimensional Variational Inequalities and Complementarity Problems, Springer Science and Business Media. 2007.
[11] K. Fan, A generalization of Tychonoff’s fixed point theorem, Math. Ann. 142 (1961), 305–310.
[12] K.E. Gustafon and K.M. Rao, The Numerical Range, the Field of Values of Linear Operators and Matrices, Springer, New York, 1997.
[13] L. Hauswirth, H. Rosenberg, and J. Spruck, Infinite boundary value problems for constant mean curvature graphs in H2 × R and S2 × R, Amer. J. Math. 131 (2009), no. 1, 195–226.
[14] L. He and Z. Tan, Inverse boundary value problem for the magnetohydrodynamics equations, J. Funct. Spaces 2021 (2021), Article ID 9966687, 10 pages.
[15] J.H. Lightbourne and R.H. Martin, Projection seminorms and the field of values of linear operators, Numer. Math. 24 (1975), 151–161.
[16] G. Lumer, Semi-inner-product spaces, Trans. Amer. Math. Soc. 100 (1961), 29–43.
[17] F.J. Narcowich, Analytic properties of the boundary of the numerical range, Indiana Univ. Math. J. 29 (1980), 67–77.
[18] S. Niewczas, M. Sitko, and L. Madej, Influence of the grain boundary curvature model on cellular automata static recrystallization simulations, Mater. Sci. Engin. 926 (2022), 1977–1985.
[19] M. Pengzi and T. Luen-Fai, On the volume functional of compact manifolds with boundary with constant scalar curvature, Cal. Variat. Partial Differ. Equ. 36 (2009), 141–171.
[20] D. Pokorny and J. Rataj, Normal cycles and curvature measures of sets with d.c. boundary, Adv. Math. 248 (2013), 963–985.
[21] F.M. Pollack, Numerical range and convex sets, Canad. Math. Bull. 17 (1974), 295–296.
[22] M. Radjabalipour and H. Radjavi, On the geometry of numerical ranges, Pacific J. Math. 61 (1975), no. 2, 507–511.
[23] M.H.M. Rashid, On the isolated points of the operators satisfying absolute condition inequality, Rend. Circ. Mat. Palermo Ser 2 73 (2024), no. 3, 1217–1230.
[24] K. B. Sabitov and S. N. Sidorov, Initial–boundary value problem for a three-dimensional equation of the parabolic-hyperbolic type, Differ. Equ. 57 (2017), 1042–1052.
[25] K.B. Sabitov and S.N. Sidorov, Initial-boundary problem for a three-dimensional inhomogeneous equation of parabolic-hyperbolic type, Lobachevskii J. Math. 41 (2020), 2257–2268.
[26] S.N. Sidorov, Three-dimensional initial-boundary value problem for a parabolic-hyperbolic equation with a degenerate parabolic part, Azerbaijan J. Math. 12 (2022), no. 1, 49–67.
[27] M. Sprlak and P. Novak, Spherical gravitational curvature boundary-value problem, J. Geod. 90 (2016), 727–739.
[28] J.G. Stampfli and J.P. Williams, Growth conditions and the numerical range in a Banach algebra, Tohoku Math. J. 20 (1968), 417–424.
[29] O. Toeplitz, Das algebraische Analogon zu einem Satze von Fejer (German), Math. Z. 2 (1918), no. 1-2, 187–197.
[30] Y. Zhang, Toeplitz Operator and Carleson Measure on Weighted Bloch Spaces, J. Funct. Spaces 2019 (2019), Article ID 4358959, 5 pages.
[31] Y. Zhang, G. Cao, and L. He, Toeplitz operators with IMOs symbols between generalized Fock spaces, J. Funct. Spaces 2021 (2021), Article ID 8044854, 9 pages.
[32] X. Zhong, D. J. Rowenhorst, H. Beladi, and G.S. Rohrer, The five-parameter grain boundary curvature distribution in an austenitic and ferritic steel, Acta Mater. 123 (2017), 136–145.
International Journal of Nonlinear Analysis and Applications

Articles in Press, Corrected Proof
Available Online from 02 June 2024
  • Receive Date: 31 January 2024
  • Revise Date: 09 April 2024
  • Accept Date: 29 April 2024