On Bounded Linear Operations in b-Hilbert Spaces and their Numerical Ranges


  • Mahnaz Khanehgir Department of Mathematics, Mashhad Branch, Islamic Azad University, Mashhad, Iran.
  • Firoozeh Hasanvand


bounded linear operator, b-Hilbert space, 2-inner product space, 2-normed space, numerical range, numerical radius.


In this paper, we introduce the notions of b-bounded linear operator, b-numericalrange and b-numerical radius in a b-Hilbert space and describe some of their properties. Thenwe will show that this new numerical range (radius) can be considered as a usual numericalrange (radius) in a Hilbert space, so it shares many useful properties with numerical range(radius).


A. Arejamaal and Gh. Sadeghi, Frames in 2-inner product spaces, Iranian J. Math. Sci. and informatics,

(2013), no. 2, 123{130.

R. Bahatia, Matrix analysis, Grad. Texts in Math. 169, Springer-Verlag, New York, 1997.

Y. J. Cho, P. C. S. Lin, S. S. Kim and A. Misiak, Theory of 2-inner product spaces, Nova Science

Publishers, Inc. New York, 2001.

S. S. Dragomir, Some numerical radius inequalities for power series of operators in Hilbert spaces, J.

Inequal. Appl. 298(2013) 1{12.

S. Gahler, Linear 2-normierte Raume, Math. Nachr. 28(1964) 1{43.

I. Golet, On probabilistic 2-normed spaces, Novi. Sad J. Math. 35(2005) no. 1, 95{102.

K. E. Gustafson and D. K. M. Rao, Numerical range, Springer-Verlag, New York, 1997.

P. R. Halmos, A Hilbert space problem book(2nd ed.), Springer-Verlag, New York, 1982.

P. K. Harikrishnan, P. Riyas and K. T. Ravindran, Riesz theorems in 2-inner product spaces, Novi. Sad

J. Math. 41(2011) no. 2, 57{61.

O. Hirzallah, F. Kittaneh and K. Shebrawi, Numerical radius inequalities for commutators of Hilbert

space operators, Numer. Funct. Anal. Optim. 32(7), (2011) 739{749.

R. V. Kadison and J. R. Ringrose, Fundamentals of the theory of operator algebras, Acadmic Press. Inc.

New York, 1983.

F. Kittaneh, M. S. Moslehian and T. Yamazaki, Cartesian decomposition and numerical radius inequal-

ities, Linear Algebra Appl. 471(2015) 46{53.

Z. Lewandowska, Bounded 2-linear operators on 2-normed sets, Glas. Math. Ser. III, 39(59), (2004)


H. Mazaheri and R. Kazemi, Some results on 2-inner product spaces, Novi. Sad J. Math. 37(2007) no.

, 35{40.

Sh. Rezapour and I. Kupka, 1-Type Lipschitz selections in generalized 2-Normed Spaces, Anal. Theory

Appl. 24(2008) no. 3, 205{210.

N. Srivastava, S. Bhattacharya and S. N. Lal, 2-Normed algebras-I, Publ. de L'institut math. Nouvelle

srie, tome, 88(102) (2010), 111{121.




How to Cite

Khanehgir, M., & Hasanvand, F. (2015). On Bounded Linear Operations in b-Hilbert Spaces and their Numerical Ranges. Asian Journal of Fuzzy and Applied Mathematics, 3(4). Retrieved from https://ajouronline.com/index.php/AJFAM/article/view/2875