$D$-symmetric operators: Comments and Some open problems

Abstract

In this paper, we show that the class of $D$-symmetric operators is norm dense in ${\mathcal L(H)}$. It is known that the direct sum of two $D$- symmetric operators are not $D$-symmetric in general. Here we will show that the direct sum of two $D$-symmetric operators is $D$-symmetric if their spectrums do not meet each other. As a consequence, we show that the set $T+K: T$ is $D-$symmetric and $K$ is compact is norm dense. Some open problems are also presented