On the Taylor's joint spectrum of 2n-tuple \((L_ A,R_ B)\) (Q1262513)

Let \((A_ 1,...,A_ n)\) and \((B_ 1,...,B_ n)\) be doubly commuting n-tuples of operators on a Hilbert space, and \(L_{A_ i}\), \(R_{B_ j}\) denote left and right multiplications by \(A_ i\) and \(B_ j\) respectively. It is proved that the Taylor joint spectrum \(Sp(L_ A,R_ B)\) is Sp(A)\(\times Sp(B)\), and for the essential joint spectrum, \[ Sp_ e(L_ A,R_ B)=Sp_ e(A)\times Sp(B)\cup Sp(A)\times Sp_ e(B). \] Beside this the paper contains some other results of the same kind.