Factorization of regular Hermitian valued trigonometric polynomials (on the unit circle) and Hermitian valued polynomials (on the real line) have been studied well. In this paper we drop the condition of regularity and study factorization of singular Hermitian valued (trigonometric) polynomials. We subsequently apply the results to obtain a characterization of matrices with a circular higher rank numerical range and derive a new version of Anderson's theorem. As a special case, we obtain a new characterization of matrices with a circular numerical range.
- Hermitian valued trigonometric polynomials
- J-spectral factorization
- circular numerical range
- higher rank numerical range
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