The Cowen-Douglas Theory for Operator Tuples and Similarity

We are concerned with the similarity problem for Cowen-Douglas operator tuples. The unitary equivalence of single Cowen-Douglas operators was already investigated in the 1970’s and geometric concepts including vector bundles and curvature appeared in the description. Soon after, a relationship between the corresponding reproducing kernels was formulated to determine the unitary equivalence of Cowen-Douglas operator tuples. As the Cowen-Douglas conjecture indicates, the similarity problem study has not been so successful until now. The latest results reveal the close correlation between complex geometry, the corona problem, and the similarity problem for single Cowen-Douglas operators. Without making use of the corona theorems that no longer hold in the multi-variable setting, we prove that the single operator results for similarity remain true for Cowen-Douglas operator tuples as well.