Non-Hermitian interactions and the evidence for violation of T invariance

It is shown that the usual argument based on the Bell-Steinberger sum rule to the effect that the data on neutral-$K$-meson decay provide evidence that $T$ invariance must be violated contains a logical weakness because $T$-invariant, $CPT$-violating interactions that are non-Hermitian would abrogate the sum rule, which is a unitarity condition. The required modification of the sum rule for such an interaction is given explicitly but it introduces several quantities that cannot be measured directly; therefore the modified sum rule does not provide useful general information concerning the validity of $T$ invariance.