Biquasitriangularity and derivations
Abstract
A Banach space operator is biquasitriangular if its
essential spectrum has no holes or pseudo holes. Biquasitriangular
Banach space operators $A, B$ have a biquasitriangular tensor
product, a biquasitriangular left-right multiplication operator
$L_AR_B$, and a biquasitriangular generalised derivation $L_A-R_B.$
Moreover, the Weyl spectral identity: $\sigma_w(A\otimes
B)=\sigma(A)\cdot\sigma_w(B)\cup\sigma_w(A)\cdot\sigma(B)$, the
a-Weyl spectral identity: $\sigma_{aw}(A\otimes
B)=\sigma(A)\cdot\sigma_{aw}(B)\cup\sigma_{aw}(A)\cdot\sigma(B)$,
the $\delta$-Weyl spectral identity:
$\sigma_w(L_A-R_B)=(\sigma(A)-\sigma_w(B))\cup(\sigma_w(A)-\sigma(B))$,
and the a-$\delta$-Weyl spectral identity:
$\sigma_{aw}(L_A-R_B)=(\sigma(A)-\sigma_{aw}(B^*))\cup(\sigma_{aw}(A)-\sigma(B))$
hold.
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