Spectral pictures of 2-variable weighted shifts

We study the spectral pictures of (jointly) hyponormal 2-variable weighted shifts with commuting subnormal components. By contrast with all known results in the theory of subnormal single and 2-variable weighted shifts, we show that the Taylor essential spectrum can be disconnected. We do this by obtaining a simple sufficient condition that guarantees disconnectedness, based on the norms of the horizontal slices of the shift. We also show that for every k ≥ 1 there exists a k-hyponormal 2-variable weighted shift whose horizontal and vertical slices have 1or 2-atomic Berger measures, and whose Taylor essential spectrum is disconnected. To cite this article: A. Name1, A. Name2, C. R. Acad. Sci. Paris, Ser. I 340 (2005).