Specht's criterion for systems of linear mappings

W. Specht (1940) proved that two n×n complex matrices A and B are unitarily similar if and only if tracew(A,A⁎)=tracew(B,B⁎) for every word w(x,y) in two noncommuting variables. We extend his criterion and its generalizations by N.A. Wiegmann (1961) and N. Jing (2015) to an arbitrary system A consisting of complex or real inner product spaces and linear mappings among them. We represent such a system by the directed graph Q(A), whose vertices are inner product spaces and arrows are linear mappings. Denote by Q˜(A) the directed graph obtained by enlarging to Q(A) the adjoint linear mappings. We prove that a system A is transformed by isometries of its spaces to a system B if and only if the traces of all closed directed walks in Q˜(A) and Q˜(B) coincide.