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We construct three different families of commuting pairs of subnormaloperators, jointly hyponormal but not admitting commuting normal extensions.Each such family can be used to answer in the negative a 1988 conjecture of R.Curto, P. Muhly and J. Xia. We also obtain a sufficient condition under whichjoint hyponormality does imply joint subnormality.

Jointly hyponormal pairs of commuting subnormal operators need not be jointly subnormal