Continuous spectrum-shrinking maps and applications to preserver problems

Source abstract

Abstract For a positive integer n $n$ n let script upper X n $\mathcal{X}_n$ 𝒳 n be either the algebra Mn $M_n$ M n of n times n $n \times n$ n × n complex matrices, the set Nn $N_n$ N n of all n times n $n \times n$ n × n normal matrices, or any of the matrix Lie groups GL left parenthesis n right parenthesis $\mathrm{GL}(n)$ G L ( n ) , SL left parenthesis n right parenthesis $\mathrm{SL}(n)$ S L ( n ) and upper U left parenthesis n right parenthesis $\mathrm{U}(n)$ U ( n ) . We first give a short and elementary argument that for two positive integers m $m$ m and n $n$ n there exists a continuous spectrum-shrinking map phi colon script upper X n right arrow Mm $\phi : \mathcal{X}_n \to M_m$ ϕ : 𝒳 n → M m (i.e., sp left parenthesis phi left parenthesis upper X right parenthesis right parenthesis subset of or equal to sp left parenthesis upper X right parenthesis $\mathrm{sp}(\phi(X))\subseteq \mathrm{sp}(X)$ s p ( ϕ ( X ) ) ⊆ s p ( X ) for all upper X element of script upper X n $X \in \mathcal{X}_n$ X ∈ 𝒳 n ) if and only if n $n$ n divides m $m$ m . Moreover, in that case we have the equality of characteristic polynomials k phi left parenthesis upper X right parenthesis left parenthesis dot right parenthesis equals kX left parenthesis dot right parenthesis mn $k_{\phi(X)}(\cdot) = k_{X}(\cdot)^\frac{m}{n}$ k ϕ ( X ) ( · ) = k X ( · ) m n for all upper X element of script upper X n $X \in \mathcal{X}_n$ X ∈ 𝒳 n , which in particular shows that phi $\phi$ ϕ preserves spectra. Using this, we show that whenever n greater than or equals 3 $n \geq 3$ n ≥ 3 , any continuous commutativity-preserving and spectrum-shrinking map phi colon script upper X n right arrow Mn $\phi : \mathcal{X}_n \to M_n$ ϕ : 𝒳 n → M n is of the form phi left parenthesis dot right parenthesis equals upper T left parenthesis dot right parenthesis upper T minus 1 $\phi(\cdot)=T(\cdot)T^{-1}$ ϕ ( · ) = T ( · ) T − 1 or phi left parenthesis dot right parenthesis equals upper T left parenthesis dot right parenthesis tT minus 1 $\phi(\cdot)=T(\cdot)^tT^{-1}$ ϕ ( · ) = T ( · ) t T − 1 , for some upper T element of GL left parenthesis n right parenthesis $T\in \mathrm{GL}(n)$ T ∈ G L ( n ) . The analogous results fail for the special unitary group SU left parenthesis n right parenthesis $\mathrm{SU}(n)$ S U ( n ) but hold for the sets of semisimple matrices in either GL left parenthesis n right parenthesis $\mathrm{GL}(n)$ G L ( n ) or SL left parenthesis n right parenthesis $\mathrm{SL}(n)$ S L ( n ) . As a consequence, we also recover (a strengthened version of) Šemrl’s influential characterization of Jordan automorphisms of Mn $M_n$ M n via preserving properties.