Numerical range and orthogonality in normed spaces. (English) Zbl 1199.46038

Bachir, A.; Segres, A.

Filomat 23, No. 1, 21-41 (2009).

The authors describe the numerical range of two operators and investigate the orthogonality in the sense of Birkhoff-James. They also give a positive answer to the following question of M. Mathieu [Can. Math. Bull. 32, No. 4, 490–497 (1989; Zbl 0641.46037)]: “Does the inequality \(\| M_{a,b}+M_{b,a}\| \geq\| a\| \,\| b\| \) hold for any elements \(a,b\) in a prime \(C^*\)-algebra?”

Reviewer: Mohammad Sal Moslehian (Mashhad, Iran)

MSC:

46B20	Geometry and structure of normed linear spaces
47A12	Numerical range, numerical radius
47B47	Commutators, derivations, elementary operators, etc.
46L35	Classifications of \(C^*\)-algebras

Keywords:

normalized duality mapping; Birkhoff-James orthogonality; maximal numerical range; norm of an elementary operator

Citations:

Zbl 0641.46037