Rank two perturbations of matrices and operators and operator model for t-transformation of probability measures
For researchers in operator theory and free probability, this provides a new connection between rank two perturbations and t-transformations, though the results are largely theoretical and incremental.
The paper derives formulas for rank two perturbations of matrices and operators, linking them to the t-transformation of probability measures from free probability, and computes large parameter asymptotics of spectra and singular values.
Rank two parametric perturbations of operators and matrices are studied in various settings. In the finite dimensional case the formula for a characteristic polynomial is derived and the large parameter asymptotics of the spectrum is computed. The large parameter asymptotics of a rank one perturbation of singular values and condition number are discussed as well. In the operator case the formula for a rank two transformation of the spectral measure is derived and it appears to be the t-transformation of a probability measure, studied previously in the free probability context. New transformation of measures is studied and several examples are presented.