  Linear equivalences of norms of vector-valued singular integral operators and
vector-valued martingale transforms are studied. In particular, it is shown
that the UMD(p)-constant of a Banach space X equals the norm of the real (or
the imaginary) part of the Beurling-Ahlfors singular integral operator, acting
on the X-valued L^p-space on the plane. Moreover, replacing equality by a
linear equivalence, this is found to be the typical property of even
multipliers. A corresponding result for odd multipliers and the Hilbert
transform is given.
