The Tuple On The Reproducing Kernel Space With U-invariant Kernels On The Polydisc

This paper will introduce the reproducing kernel space with u-invariant kernels on the polydisc, givesome properties of the m-tuple Mz = (M1, M2 Mm) defind on this space, then describe the Taylor spec-trum and essential Taylor spectrum. They have been introduced in references [4]and[6]. For convenience, inthis paper, we will give the definition and proposition of them in the second part.