Joint Defect Index of a Cyclic Tuple of Symmetric Operators

Abstract

Von Neumann showed that the defect indices of a symmetric operator are invariant in each the upper half and lower half complex planes, and if the operator commutes with a conjugation operator, the indices have the same value in $\mathbb{C}\setminus\mathbb{R}.$ This leads to self-adjoint extensions for the operator. We prove an anlogous invariance result in $\mathbb{C}^{d}\setminus\mathbb{R}^{d}$ for a class of operator tuples. We also apply this to give a result regarding reproducing kernels.

Mathematics Subject Classification (2000). Primary 47A13; Secondary 47B32.