Linear operators T-invariant with respect to a congruence group (Q1101667)

Let \(\Omega\) be an open subset of \({\mathbb{R}}^ n\), \(W_ m(\Omega)\) the linear space of m-vector valued functions defined on \(\Omega\), \(G\equiv \{\gamma \}^ a \)group of orthogonal matrices mapping \(\Omega\) onto itself and \(T\equiv \{T(\gamma)\}^ a \)linear representation of order m of G. A suitable group \({\mathfrak C}(G,T)\) of linear operators of \(W_ m(\Omega)\), which leads to a general definition of T-invariant linear operator with respect to G, is here introduced. Characterization theorems concerning the linear differential and integral T-invariant operators are also given. When G is a finite group projection operators are explicitly obtained; they define a ``maximal'' decomposition of \(W_ m(\Omega)\) into a direct sum of subspaces each of them invariant with respect to any T-invariant linear operator of \(W_ m(\Omega)\). Some examples are given.