In this paper, we give necessary and sufficient conditions for weighted
L
2
L^2
estimates with matrix-valued measures of well localized operators. Namely, we seek estimates of the form
\[
‖
T
(
W
f
)
‖
L
2
(
V
)
≤
C
‖
f
‖
L
2
(
W
)
,
\| T(\mathbf {W} f)\|_{L^2(\mathbf {V})} \le C\|f\|_{L^2(\mathbf {W})},
\]
where
T
T
is formally an integral operator with additional structure,
W
,
V
\mathbf {W}, \mathbf {V}
are matrix measures, and the underlying measure space possesses a filtration. The characterization we obtain is of Sawyer type; in particular, we show that certain natural testing conditions obtained by studying the operator and its adjoint on indicator functions suffice to determine boundedness. Working in both the matrix-weighted setting and the setting of measure spaces with arbitrary filtrations requires novel modifications of a T1 proof strategy; a particular benefit of this level of generality is that we obtain polynomial estimates on the complexity of certain Haar shift operators.