Hyponormal operators with infinite essential spectrum

Abstract:

It is shown that if $T$ is an essentially hyponormal operator (i.e., the image of ${T^ \ast }T - T{T^ \ast }$ in the Calkin algebra is a positive element) in $\mathfrak {L}(\mathcal {H})$, and if the left essential spectrum of $T$ is infinite, then $R{({\delta _T})^ - } + \{ {T^ \ast }\} ’$ is not norm dense in $\mathfrak {L}(\mathcal {H})$, where $R{({\delta _T})^ - }$ denotes the norm closure of the range of derivation induced by $T$, and $\{ {T^ \ast }\} ’$ denotes the commutant of ${T^ \ast }$.