LAUSR

Abstract W. Specht (1940) proved that two n × n complex matrices A and B are unitarily similar if and only if trace w ( A , A ⁎ ) = trace w ( 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.