Let $A$ be an open subset of $\mathbb{R}^{n}$, $W_{m}(A)$ the linear space of $m$-vector valued functions defined on $A$, $G \equiv \{\gamma\}$ a group of orthogonal matrices mapping $A$ onto itself and $T \equiv \{T_{\gamma}\}$ a linear representation of order $m$ of $G$. A suitable group $\mathcal{T}(G,T)$ of linear operators of $W_{m}(A)$ is introduced which leads to a general definition of $T$-invariant linear operator with respect to $G$. When $G$ is a finite group, projection operators are explicitly obtained which define a "maximal" decomposition of the function space into a direct sum of subspaces each of them invariant with respect to any $T$-invariant linear operator. The theory includes as special cases previous results obtained for $m=1$, $m=n$.