Some inequalities of Hölder type for quadratic weighted geometric mean of bounded linear operators in Hilbert spaces

Abstract

In this paper, we consider the quadratic weighted geometric mean

for bounded linear operators T,  V in the Hilbert space H with T invertible and . Using the celebrated McCarthy’s inequality we show that

for any where p,  with We also provide some norm inequalities such as

for any n-tuples of invertible operators and any n-tuple of positive weights .

Keywords:

• Weighted geometric mean
• weighted harmonic mean
• McCarthy’s inequality
• hölder’s inequality
• young’s inequality
• operator modulus
• arithmetic mean-geometric mean-harmonic mean inequality

AMS Subject Classifications:

• 47A63
• 47A30
• 15A60
• 26D15
• 26D10

Notes

No potential conflict of interest was reported by the author.