Journal of Operator Theory

Volume 56, Issue 1, Summer 2006 pp. 3-15.

Norm estimations for finite sums of positive operators

Authors: Dan Popovici (1) and Zoltan Sebestyen
Author institution: (1) Department of Mathematics and Computer Science, University of the West Timisoara, Bd. Vasile Parvan 4, RO-300223 Timisoara, Romania
(2) Department of Applied Analysis, Lorand Eotvos University, Pazmany Peter setany 1/C, H-11 17 Budapest, Hungary

Summary: We propose some norm estimations for sums of positive operators on Hilbert spaces, extending the ones given by Davidson-Power and Kittaneh for two operators. Such inequalities are useful in the theory of best approximations in $C^*$-algebras, complex interpolation, the theory of generalized inverses and operator approximation. We prove that the equality case in generalized triangle inequalities is obtained when equality holds in the corresponding Cauchy-Schwarz type inequalities, extending a recent result of Kittaneh. Certain applications concerning orthogonal projections or operators having orthogonal ranges are given.

Keywords: norm estimation, operator matrix, triangle inequality, positive operator

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