  In this paper, we define a new and broad family of vector-valued random
fields called tempered operator fractional operator-stable random fields (TRF,
for short). TRF is typically non-Gaussian and generalizes tempered fractional
stable stochastic processes. TRF comprises moving average and harmonizable-type
subclasses that are constructed by tempering (matrix-) homogeneous,
matrix-valued kernels in time- and Fourier-domain stochastic integrals with
respect to vector-valued, strictly operator-stable random measures. We
establish the existence and fundamental properties of TRF. Assuming both
Gaussianity and isotropy, we show the equivalence between certain moving
average and harmonizable subclasses of TRF. In addition, we establish sample
path properties in the scalar-valued case for several Gaussian instances.
