Abstract
Let K1, . . . , K n be positive kernel operators on a Banach function space. We prove that the Hadamard weighted geometric mean of K1, . . . , K n , the operator K, satisfies the following inequalities
where || · ||and r(·) denote the operator norm and the spectral radius, respectively.
In the case of completely atomic measure space we show some additional results. In particular, we prove an infinite-dimensional extension of the known characterization of those functions 
satisfying
for all non-negative matrices A1, . . . , A n of the same order.
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Drnovšek, R., Peperko, A. Inequalities for the Hadamard Weighted Geometric Mean of Positive Kernel Operators on Banach Function Spaces. Positivity 10, 613–626 (2006). https://doi.org/10.1007/s11117-006-0048-z
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DOI: https://doi.org/10.1007/s11117-006-0048-z