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We determine the extrapolation law of rearrangement operators acting on the Haar system in vector‐valued  H   p   spaces: if 0< q ⩽ p <2, then,∥   T   τ  ,  q    ⊗   I  d    X    ∥   q    q   /   (  2  −  q  )    ⩽  A  (  p  ,     q  )  ∥   T   τ  ,  p    ⊗I  d    X    ∥   p    p   /   (  2  −  p  )For a fixed Banach space  X , the extrapolation range 0< q ⩽ p <2 is optimal. If, however, there exists 1< p  0 <∞, so that∥   T   τ  ,   p   0⊗    I  d    E∥    L   Ep   0<  ∞       f  o  r     e  a  c  h     U  M  D     s  p  a  c  e       E  ,then, for any 1< p <∞,∥   T   τ  ,  p    ⊗    I  d    E∥    L   E    p<  ∞for any UMD space  E . (The value  p  0 =2 is  not  excluded.) We characterize Hilbert spaces in terms of vector‐valued rearrangement operators. If∥   T   τ  ,  2    ⊗    I  d    X∥    L   Y    2<  ∞      a  n  d      ∥   T    τ   )    p∥   L  p    =  ∞  (   f  o  r      1  <  p  ≠  2  <  ∞  )  ,then  X  is isomorphic to a Hilbert space.

Extrapolation of vector‐valued rearrangement operators II