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Invertibility and Fredholmness of linear combinations of quadratic, k-potent and nilpotent operators

Hwa-Long Gau Chih-Jen Wang Ngai-Ching Wong

Spectral Theory and Operator Algebra mathscidoc:1910.43644

Operators and Matrices, 2, (2), 193-199, 2008.1
Recently, the invertibility of linear combinations of two idempotents has been studied by several authors. Let P and Q be idempotents in a Banach algebra. It was shown that the invertibility of P+ Q is equivalent to that of aP+ bQ for nonzero a, b with a+ b= 0. In this note, we obtain a similar result for square zero operators and those operators satisfying x2= dx for some scalar d. More generally, we show that if P, Q satisfy a quadratic polynomial (x c)(x d) then the linear combination aP+ bQ c (a+ b) being invertible or Fredholm (and the index) is independent of the choice of the nonzero scalars a, b. However, this is not the case when P and Q are involutions, unitaries, partial isometries, k-potents (k 3) and other nilpotents, as counterexamples are provided.
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@inproceedings{hwa-long2008invertibility,
  title={Invertibility and Fredholmness of linear combinations of quadratic, k-potent and nilpotent operators},
  author={Hwa-Long Gau, Chih-Jen Wang, and Ngai-Ching Wong},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20191020205323191805173},
  booktitle={Operators and Matrices},
  volume={2},
  number={2},
  pages={193-199},
  year={2008},
}
Hwa-Long Gau, Chih-Jen Wang, and Ngai-Ching Wong. Invertibility and Fredholmness of linear combinations of quadratic, k-potent and nilpotent operators. 2008. Vol. 2. In Operators and Matrices. pp.193-199. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20191020205323191805173.
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