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

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?”

### 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

Zbl 0641.46037
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