On the Numerical Range of Operators on some Special Banach Spaces

Kalidas Mandal; Aniket Bhanja; Santanu Bag; Kallol Paul

doi:10.68381/jca29023

Abstract

The numerical range of a bounded linear operator on a complex Banach space need not be convex unlike that on a Hilbert space. The aim of this paper is to study operators

T

on

\ell^2_p

for which the numerical range is convex. We also obtain a nice relation between

V(T)

and

V(T^t)

considering

T\in\mathbb{L}(\ell_p^2)

and

T^t\in\mathbb{L}(\ell_q^2)

, where

T^t

denotes the transpose of

T

and

p

and

q

are conjugate real numbers, i.e.,

1 <p,q< \infty

and

\frac{1}{p}+\frac{1}{q}=1

.

Author information

Contact details are reproduced from the original publication and may be historical.

Kalidas Mandal

Dept. of Mathematics, Jadavpur University, Kolkata, West Bengal, India

kalidas.mandal14@gmail.com

Aniket Bhanja

Dept. of Mathematics, Vivekananda College Thakurpukur, Kolkata, West Bengal, India

aniketbhanja219@gmail.com

Santanu Bag

Dept. of Mathematics, Vivekananda College for Women, Barisha, Kolkata, West Bengal, India

santanumath84@gmail.com

Kallol Paul

Dept. of Mathematics, Jadavpur University, Kolkata, West Bengal, India

kalloldada@gmail.com

Keywords

Semi-inner-product
numerical range
convex set

Mathematics Subject Classification

47A12; 46A55

References

F. L. Bauer: On the field of values subordinate to a norm, Numer. Math. 4 (1962) 103–113.https://doi.org/10.1007/BF01386300 Google Scholar
F. F. Bonsall, J. Duncan: Numerical Ranges of Operators on Normed Spaces and of Elements of Normed Algebras, London Math. Soc. Lecture Note Series 2, Cambridge University Press, London (1971).https://doi.org/10.1017/CBO9781107359895 Google Scholar
F. F. Bonsall, J. Duncan: Numerical Ranges II, London Math. Soc. Lecture Note Series 10, Cambridge University Press, London (1973).https://doi.org/10.1017/CBO9780511662515 Google Scholar
S. S. Dragomir: Some functionals associated to semi-inner products on complex Banach spaces, Bull. Allahabad Math. Soc. 34/1 (2019) 85–103.Google Scholar
K. E. Gustafson, D. K. M. Rao: Numerical Range. The Field of Values of Linear Operators and Matrices, Springer-Verlag, New York, 1997.https://doi.org/10.1007/978-1-4613-8498-4_1 Google Scholar
J. R. Giles: Classes of semi-inner-product spaces, Trans. Amer. Math. Soc. 129 (1967) 436–446.https://doi.org/10.1090/S0002-9947-1967-0217574-1 Google Scholar
P. R. Halmos: A Hilbert Space Problem Book, D. Van Nostrand, Princeton (1967).Google Scholar
G. Lumer: Semi-inner-product spaces, Trans. Amer. Math. Soc. 100 (1961) 29–43.https://doi.org/10.1090/S0002-9947-1961-0133024-2 Google Scholar
O. Toeplitz: Das algebraische Analogon zu einem Satze von Fejér, Math. Zeitschrift 2/1-2 (1918) 187–197.https://doi.org/10.1007/BF01212904 Google Scholar

Suggested citation

Published by Heldermann Verlag, 2022. Rights now held by Banach Press LLC.

[1] F. L. Bauer: On the field of values subordinate to a norm, Numer. Math. 4 (1962) 103–113.https://doi.org/10.1007/BF01386300 Google Scholar

[2] F. F. Bonsall, J. Duncan: Numerical Ranges of Operators on Normed Spaces and of Elements of Normed Algebras, London Math. Soc. Lecture Note Series 2, Cambridge University Press, London (1971).https://doi.org/10.1017/CBO9781107359895 Google Scholar

[3] F. F. Bonsall, J. Duncan: Numerical Ranges II, London Math. Soc. Lecture Note Series 10, Cambridge University Press, London (1973).https://doi.org/10.1017/CBO9780511662515 Google Scholar

[4] S. S. Dragomir: Some functionals associated to semi-inner products on complex Banach spaces, Bull. Allahabad Math. Soc. 34/1 (2019) 85–103.Google Scholar

[5] K. E. Gustafson, D. K. M. Rao: Numerical Range. The Field of Values of Linear Operators and Matrices, Springer-Verlag, New York, 1997.https://doi.org/10.1007/978-1-4613-8498-4_1 Google Scholar

[6] J. R. Giles: Classes of semi-inner-product spaces, Trans. Amer. Math. Soc. 129 (1967) 436–446.https://doi.org/10.1090/S0002-9947-1967-0217574-1 Google Scholar

[7] P. R. Halmos: A Hilbert Space Problem Book, D. Van Nostrand, Princeton (1967).Google Scholar

[8] G. Lumer: Semi-inner-product spaces, Trans. Amer. Math. Soc. 100 (1961) 29–43.https://doi.org/10.1090/S0002-9947-1961-0133024-2 Google Scholar

[9] O. Toeplitz: Das algebraische Analogon zu einem Satze von Fejér, Math. Zeitschrift 2/1-2 (1918) 187–197.https://doi.org/10.1007/BF01212904 Google Scholar