Quasi-Diagonal Operators - Lohaj, Muhib; Lohaj, Shqipe (Sarajevo Journal of Mathematics, 2024)
Let $H$ be a separable complex Hilbert space and let $B(H)$ denote the algebra of all bounded linear operators on $H.$ If $T$ is a quasi-normal Fredholm operator we prove that $TT^*\in (QD)(P_n)$ if and only if $T^*T\in (QD)(P_n).$ We also show that if $T$ is quasi-normal and $T(T^*T)$ is quasi-diagonal with respect to any sequence $(P_n)$ in $PF(H),$ such that $P_n→ I$ strongly, then $T=N+K,$ where $N$ is a normal operator and $K$ is a compact operator. 2000 Mathematics Subject Classification. 47Bxx, 47B20