Weighted Shift Matrices: Unitary Equivalence, Reducibility and Numerical Ranges

An $n$-by-$n$ ($n\ge 3$) weighted shift matrix $A$ is one of the form [{array}{cccc}0 & a_1 & & & 0 & \ddots & & & \ddots & a_{n-1} a_n & & & 0{array}], where the $a_j$’s, called the weights of $A$, are complex numbers. Assume that all $a_j$’s are nonzero and $B$ is an $n$-by-$n$ weighted shift matrix with weights $b_1,\mathrm{.}\mathrm{.}\mathrm{.},b_n$. We show that $B$ is unitarily equivalent to $A$ if and only if $b_1\mathrm{.}\mathrm{.}\mathrm{.}{b}_n=a_1\mathrm{.}\mathrm{.}\mathrm{.}{a}_n$ and, for some fixed $k$, $1\le k \le n$,$|b_j| = |a_{k+j}|$($a_{n+j}\equiv a_j$)forall$j$\mathrm{.}Next,weshowthat$A$isreducibleifandonlyif$A$ has periodic weights, that is, for some fixed $k$, $1\le k\le \lfloor n\mathrm{/}2\rfloor$, $n$ is divisible by $k$, and $\mathrm{\mid }{a}_j\mathrm{\mid }=\mathrm{\mid }{a}_{k+j}\mathrm{\mid }$ for all $1\le j\le n-k$. Finally, we prove that $A$ and $B$ have the same numerical range if and only if $a_1\mathrm{.}\mathrm{.}\mathrm{.}{a}_n=b_1\mathrm{.}\mathrm{.}\mathrm{.}{b}_n$ and $S_r(\mathrm{\mid }{a}_1{\mathrm{\mid }}^2,\mathrm{.}\mathrm{.}\mathrm{.},\mathrm{\mid }{a}_n{\mathrm{\mid }}^2)=S_r(\mathrm{\mid }{b}_1{\mathrm{\mid }}^2,\mathrm{.}\mathrm{.}\mathrm{.},\mathrm{\mid }{b}_n{\mathrm{\mid }}^2)$ for all $1\le r\le \lfloor n/2\rfloor$,where$S_r$'s are the circularly symmetric functions.