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In this paper, the authors study the matrix-valued harmonic functions andcharacterize them by the Poisson integral of functions in non-commutative BMO(bounded mean oscillation) spaces. This provides a very satisfactorynon-commutative analogue of the beautiful result due to Fabes, Johnson and Neri[Indiana Univ. Math. J. {\bf25} (1976) 159-170; MR0394172].

A Representation of Matrix-Valued Harmonic Functions by the Poisson Integral of Non-commutative BMO Functions