  The article is devoted to the application of multiple Fourier-Legendre series
to implementation of strong exponential Milstein and Wagner-Platen methods for
non-commutative semilinear stochastic partial differential equations with
multiplicative trace class noise. These methods have strong orders of
convergence $1.0-\varepsilon$ and $1.5-\varepsilon$ correspondingly (here
$\varepsilon$ is an arbitrary small positive real number) with respect to the
temporal discretization. The theorem on mean-square convergence of
approximations of iterated stochastic integrals of multiplicities 1 to 3 with
respect to the infinite-dimensional $Q$-Wiener process is formulated and
proved. The results of the article can be applied to implementation of
exponential Milstein and Wagner-Platen methods for non-commutative semilinear
stochastic partial differential equations with multiplicative trace class
noise.
