Some weighted operator geometric mean inequalities

Authors
Alemeh Sheikhhosseini 1
Asma Ilkhanizadeh Manesh 2
Maryam Khosravi 1
1 Shahid Bahonar university of Kerman
2 Vali-e-asr university of Rafsanjan
10.29252/mmr.3.2.113
Abstract
In this paper, using the extended Holder- -McCarthy inequality, several inequalities involving the α-weighted geometric mean (0<α<1) of two positive operators are established. In particular, it is proved that if A,B,X,Y∈B(H) such that A and B are two positive invertible operators, then for all r ≥1, ‖X^* (A⋕_α B)Y‖^r≤‖〖(X〗^* AX)^r ‖^((1-α)/2) ‖〖(Y〗^* AY)^r ‖^((1-α)/2) ‖〖(X〗^* BX)^r ‖^(α/2) ‖〖(Y〗^* BY)^r ‖^(α/2), and ‖X^* (A⋕_α B)X‖^r≤‖α(X^* BX)^r+(1-α)(X^* AX)^r ‖^ -Ω(X) where Ω(X)=inf┬(‖x‖=1)⁡〖〖(√(<(X^* BX)^r x,x>^ )-√(<(X^* AX)^r x,x>^ ))〗^2 〗.min⁡{α^ ,(1-α)^ }../files/site1/files/0Abstract3.pdf
Keywords
Holder-McCarthy inequality
numerical range
operator norm
positive invertible operator
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Mathematical Research
Volume 3, Issue 3 - Serial Number 7
Autumn 2017
Pages 113-118