  We consider a Hilbert space that is a product of a finite number of Hilbert
spaces and operators that are represented by "componental operators" acting on
the Hilbert spaces that form the product space. We attribute operatorial
properties to the componental operators rather than to the full operators. The
operatorial properties that we discuss include nonexpansivity, firm
non-expansivity, relaxed firm nonexpansivity, averagedness, being a cutter,
quasi-nonexpansivity, strong quasi-nonexpansivity, strict quasi-nonexpansivity
and contraction. Some relationships between operators whose componental
operators have such properties and operators that have these properties on the
product space are studied. This enables also to define componental fixed point
sets and to study their properties. For componental contractions we offer a
variant of the Banach fixed point theorem. Our motivation comes from the desire
to extend a fully-simultaneous method that takes into account sparsity of the
linear system in order to accelerate convergence [Censor et al., On diagonally
relaxed orthogonal projection methods, SIAM J. Sci. Comput. 30 (2008),
473-504]. This was originally applicable to the linear case only and gives rise
to an iterative process that uses dfferent componental operators during
iterations.
