We give a systematic study on the Hardy spaces of functions with values in the non-commutative L-spaces associated with a semifinite von Neumann algebra M. This is motivated by the works on matrix valued Harmonic Analysis (operator weighted norm inequalities, operator Hilbert transform), and on the other hand, by the recent development on the non-commutative martingale inequalities. Our non-com...

In this paper we extend the classical Bohr’s inequality to setting of non-commutative Hardy space $H^1$ associated with a semifinite von Neumann algebra. As consequence, obtain for operators in Neumann-Schatten class $\mathcal C_1$ and square matrices any finite order. Interestingly, establish that optimal bound $r$ above mentioned concerning is 1/3 whereas it 1/2 case $2\times 2$ reduces $\s...