  We consider a Cauchy problem for a fractional anisotropic parabolic equation
in anisotropic H\"{o}lder spaces. The equation generalizes the heat equation to
the case of fractional power of the Laplace operator and the power of this
operator can be different with respect to different groups of space variables.
The time derivative can be either fractional Caputo - Jrbashyan derivative or
usual derivative. Under some necessary conditions on the order of the time
derivative we show that the operator of the whole problem is an isomorphism of
appropriate anisotropic H\"{o}lder spaces. Under some another conditions we
prove unique solvability of the Cauchy problem in the same spaces.
  The final version of this paper is published by AIMS Evolution Equations and
Control Theory at https://www.aimsciences.org/article/doi/10.3934/eect.2022029,
doi: 10.3934/eect.2022029.
