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Theory of $T$-invariant linear operators which was considered for a group of congruences [2], [3] is now extended to a group of homeomorphisms. An analysis is carried out in order to establish to what extent the main results of the previous theory still hold under the actual very general assumptions.

Let $A$ be an open subset of ${ℝ}^n$, $W_m(A)$ 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 $𝒯(G,T)$ 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...

The eigenvalue problem for the linear elasticity operator in a cube under the Dirichlet condition is considered. Lower and upper bounds for the eigenvalues, obtained in a previous paper, are compared here with the values derived from the classical asymptotic formula due to H. Weyl. The conclusion is the following: the asymptotic formula is not satisfactory for the first 170 eigenvalues.

Theory of $T$-invariant linear operators which was considered for a group of congruences [2], [3] is now extended to a group of homeomorphisms. An analysis is carried out in order to establish to what extent the main results of the previous theory still hold under the actual very general assumptions.

Let $A$ be an open subset of ${ℝ}^n$, $W_m(A)$ 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 $𝒯(G,T)$ 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...