The natural BMO (bounded mean oscillation) conditions suggested by scalar-valued results are known to be insufficient for the boundedness of operator-valued paraproducts. Accordingly, the boundedness of operator-valued singular integrals has only been available under versions of the classical “$T(1)\in \mathrm{BMO}$” assumptions that are not easily checkable. Recently, Hong, Liu and Mei (J. Funct. Anal. 2020) observed that the situation improves remarkably for singular integrals with a symmetry assumption, so that a classical $T(1)$ criterion still guarantees their $L^2$-boundedness on Hilbert space -valued functions. Here, these results are extended to general UMD (unconditional martingale differences) spaces with the same natural BMO condition for symmetrised paraproducts, and requiring in addition only the usual replacement of uniform bounds by R-bounds in the case of general singular integrals. In particular, under these assumptions, we obtain boundedness results on non-commutative $L^p$ spaces for all $1<p<\infty$, without the need to replace the domain or the target by a related non-commutative Hardy space as in the results of Hong et al. for $p\ne 2$.

已知标量值结果所建议的自然BMO（有界平均振荡）条件对于算子值副产品的有界性是不足的。因此，仅在经典“$Ť（\mathrm{1个}）\in \mathrm{BMO}$难以验证的假设。最近，Hong，Liu和Mei（J. Funct。Anal。2020）观察到，具有对称假设的奇异积分的情况显着改善。$Ť（\mathrm{1个}）$ 标准仍然保证他们 ${\mathrm{大号}}^2$Hilbert空间值函数的有界性。在这里，这些结果扩展到对对称副产品具有相同自然BMO条件的一般UMD（无条件mar差）空间，并且在一般奇异积分的情况下，仅需要通常用R边界替换均匀界。特别是，在这些假设下，我们获得了非交换性的有界性结果${\mathrm{大号}}^p$ 所有空间 $\mathrm{1个}<p<\infty$，而不需要像Hong等人的结果那样用相关的非交换Hardy空间替换域或目标。对于$p\ne 2$。