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Journal of Operator Theory

Volume 78, Issue 2, Fall 2017  pp. 417-433.

Projective spectrum and kernel bundle. II

Authors:  Wei He (1), Xiaofeng Wang (2), and Rongwei Yang (3)
Author institution:(1) School of Mathematics, Southeast University, Nanjing, Jiangsu 211189, China
(2) School of Mathematics and Information Science $\&$ Key Laboratory of Mathematics\, \textit{and}\, Interdisciplinary Sciences of the Guang-dong Higher Education Institute, Guangzhou University, Guangzhou, Gu-angdong 510006, China
(3) Department of Mathematics and Statistics, SUNY at Albany, Albany, NY 12222, U.S.A.

Summary: In this paper, we study the projective joint spectrum $P(A)$ and $P(A_{*})$ of the operator tuple $A=(A_1, A_2, \dots, A_n)$. We first compute the joint spectrum for the Cuntz tuple. Then we study tuples of compact operators on an infinite dimensional Banach space. We show that if $P(A_{*})$ is smooth, then $\bigvee\limits_{z\in P(A_{*})}\ker A_{*}(z)$ forms a holomorphic line bundle over $P(A_{*})$. For linearly independent vectors $e_1, e_2, e_3$ and $A_i=e_i\otimes e_i,\ i=1, 2, 3$, the smoothness of $P(A_*)$ depends rather subtly on the relative position of the vectors. As an example, we compute the Chern character of the line bundle in the two vector case and show that it is nontrivial.

Keywords: projective spectrum, Cuntz tuple, compact operator tuple, Hermitian bundle

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