# IJPAM: Volume 64, No. 1 (2010)

**ON NORM OF THE SUM OF OPERATORS**

EQUAL TO THE SUM OF THEIR NORMS

EQUAL TO THE SUM OF THEIR NORMS

Department of Mathematics

Bishop's University

2600 College St., Sherbrooke, QC, J1M 1Z7, CANADA

e-mail: plin@ubishops.ca

**Abstract.**In this paper we first characterize the norm of the sum of vectors in a normed linear space being equal to the sum of their norms. These results make it possible to characterize the norm of the sum of bounded linear operators in a normed linear space being equal to the sum of their norms. Many interesting results about approximate eigenvalues follows. In applications, these results are characterized in terms of numerical ranges for operators in Hilbert spaces.

an esteemed colleague and friend.

**Received: **August 16, 2010

**AMS Subject Classification: **47A10, 47A30, 47A50

**Key Words and Phrases: **uniformly convex space, strictly convex space, Daugavet equation, isometric isomorphism, numerical range of operator and its closure, complete vector for operator, norm attaining vector for operator, eigenvalue and approximate eigenvalue of operator

**Source:** International Journal of Pure and Applied Mathematics

**ISSN:** 1311-8080

**Year:** 2010

**Volume:** 64

**Issue:** 1