Spectral properties of elementary operators. II

Abstract:

Let $A = ({A_1}, \ldots ,{A_n})$ and $B = ({B_1}, \ldots ,{B_n})$ denote commutative $n$-tuples of operators on a Hilbert space $\mathcal {H}$. Let ${R_{AB}}$ denote the elementary operator on $\mathcal {L}(\mathcal {H})$ defined by ${R_{AB}}(X) = {A_1}X{B_1} + \cdots + {A_n}X{B_n}$. We obtain new expressions for the essential spectra of ${R_{AB}}$ and ${R_{AB}}|\mathcal {J}$ (the restriction of ${R_{AB}}$ to a norm ideal $\mathcal {J}$ of $\mathcal {L}(\mathcal {H})$). We also study isolated points of joint spectra defined in the sense of ${\text {R}}$. Harte.