Abstract

The operator <TEX>$A \; {\in} \; L(H_{i})$</TEX>, the Banach algebra of bounded linear operators on the complex infinite dimensional Hilbert space <TEX>$\cal H_{i}$</TEX>, is said to be p-hyponormal if <TEX>$(A^\ast A)^P \geq (AA^\ast)^p$</TEX> for <TEX>$p\; \in \; (0,1]$</TEX>. Let (equation omitted) denote the completion of (equation omitted) with respect to some crossnorm. Let <TEX>$I_{i}$</TEX> be the identity operator on <TEX>$H_{i}$</TEX>. Letting (equation omitted), where each <TEX>$A_{i}$</TEX> is p-hyponormal, it is proved that the commuting n-tuple T = (<TEX>$T_1$</TEX>,..., <TEX>$T_{n}$</TEX>) satisfies Bishop's condition (<TEX>$\beta$</TEX>) and that if T is Weyl then there exists a non-singular commuting n-tuple S such that T = S + F for some n-tuple F of compact operators.