# Specht's criterion for systems of linear mappings

## TL;DR

This paper generalizes Specht's criterion for unitary similarity of matrices to systems of linear mappings between inner product spaces, using graph-based invariants involving traces of closed walks.

## Contribution

It extends Specht's criterion to arbitrary systems of linear mappings represented by directed graphs, incorporating adjoint mappings and trace invariants for classification.

## Key findings

- Systems are unitarily equivalent if traces of all closed walks match.
- The criterion applies to complex and real inner product spaces.
- Graph-based invariants fully characterize system equivalence under isometries.

## Abstract

W.Specht (1940) proved that two $n\times n$ complex matrices $A$ and $B$ are unitarily similar if and only if $\operatorname{trace} w(A,A^{\ast}) = \operatorname{trace} w(B,B^{\ast})$ 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 $\mathcal A$ consisting of complex or real inner product spaces and linear mappings among them. We represent such a system by the directed graph $Q(\mathcal A)$, whose vertices are inner product spaces and arrows are linear mappings. Denote by $\widetilde Q(\mathcal A)$ the directed graph obtained by enlarging to $Q(\mathcal A)$ the adjoint linear mappings. We prove that a system $\mathcal A$ is transformed by isometries of its spaces to a system $\mathcal B$ if and only if the traces of all closed directed walks in $\widetilde Q(\mathcal A)$ and $\widetilde Q(\mathcal B)$ coincide.