  Consider three normal operators $A,B,C$ on separable Hilbert space $\H$ as
well as scalar-valued spectral measures $\lambda_A$ on $\sigma(A)$, $\lambda_B$
on $\sigma(B)$ and $\lambda_C$ on $\sigma(C)$. For any $\phi\in
L^\infty(\lambda_A\times \lambda_B\times \lambda_C)$ and any $X,Y\in S^2(\H)$,
the space of Hilbert-Schmidt operators on $\H$, we provide a general definition
of a triple operator integral $\Gamma^{A,B,C}(\phi)(X,Y)$ belonging to
$S^2(\H)$ in such a way that $\Gamma^{A,B,C}(\phi)$ belongs to the space
$B_2(S^2(\H)\times S^2(\H), S^2(\H))$ of bounded bilinear operators on
$S^2(\H)$, and the resulting mapping $\Gamma^{A,B,C}\colon
L^\infty(\lambda_A\times \lambda_B\times \lambda_C) \to B_2(S^2(\H)\times
S^2(\H), S^2(\H))$ is a $w^*$-continuous isometry. Then we show that a function
$\phi\in L^\infty(\lambda_A\times \lambda_B\times \lambda_C)$ has the property
that $\Gamma^{A,B,C}(\phi)$ maps $S^2(\H)\times S^2(\H)$ into $S^1(\H)$, the
space of trace class operators on $\H$, if and only if it has the following
factorization property: there exist a Hilbert space $H$ and two functions $a\in
L^{\infty}(\lambda_A \times \lambda_B ; H)$ and $b\in
L^{\infty}(\lambda_B\times \lambda_C ; H)$ such that $\phi(t_1,t_2,t_3)=
\left\langle a(t_1,t_2),b(t_2,t_3) \right\rangle$ for a.e. $(t_1,t_2,t_3) \in
\sigma(A) \times \sigma(B) \times \sigma(C).$ This is a bilinear version of
Peller's Theorem characterizing double operator integral mappings $S^1(\H)\to
S^1(\H)$. In passing we show that for any separable Banach spaces $E,F$, any
$w^*$-measurable esssentially bounded function valued in the Banach space
$\Gamma_2(E,F^*)$ of operators from $E$ into $F^*$ factoring through Hilbert
space admits a $w^*$-measurable Hilbert space factorization.
