On the real spectrum of differential operators with PT‐symmetric periodic matrix coefficients

We study the spectrum of the operator T$T$ generated by the differential expression of order n>2$n>2$ with the m×m$m\times m$ Parity‐Time (PT)‐symmetric periodic matrix coefficients. The case when m$m$ and n$n$ are the odd numbers was investigated in [18]. In this paper, we consider the all remained cases: (a) n$n$ is an odd number and m$m$ is an even number, (b) n$n$ is an even number and m$m$ is an arbitrary positive integer. We find conditions on the coefficients under which in the cases (a) and (b) the spectrum of T$T$ contains the sets (−∞,−H]$(-\infty,-H]$ ∪[H,∞)$\cup [H,\infty)$ and [H,∞)$[H,\infty)$ respectively for some H>0$H>0$.