Factorisation of orthogonal projectors

  • N. S. Sushchyk Ivan Franko National University of Lviv, Lviv, Ukraine
  • V. M. Degnerys Peeklogic, Lviv, Ukraine
Keywords: special factorisation, orthogonal projector

Abstract

We study the problem of a special factorisation of an orthogonal projector~$P$ acting in the Hilbert space $L_2(\mathbb R)$ with $\dim\ker P<\infty$. In particular, we prove that the orthogonal projector~$P$ admits a special factorisation in the form
$P=VV^*$, where $V$ is an isometric upper-triangular operator in the Banach algebra of all linear continuous operators in $L_2(\mathbb R)$. Moreover, we
give an explicit formula for the operator $V$.

References

I. Gohberg, M. Krein, Theory of Volterra operators in Hilbert space and its applications, Nauka Publ., Moscow, 1967 (in Russian); Engl. transl.: Amer. Math. Soc. Transl. Math. Monographs, V.24, Amer. Math. Soc., Providence, RI, 1970.

S. Albeverio, R. Hryniv, Ya. Mykytyuk, Factorisation of non-negative Fredholm operators and inverse spectral problems for Bessel operators, Integr. equ. oper. theory, 64 (2009), 301–323.

D.R. Larson, Nest algebras and similarity transformations, Ann of Math. (2), 121 (1985), №2, 409–427.

Published
2021-06-22
How to Cite
1.
Sushchyk NS, Degnerys VM. Factorisation of orthogonal projectors. Mat. Stud. [Internet]. 2021Jun.22 [cited 2021Sep.17];55(2):181-7. Available from: http://matstud.org.ua/ojs/index.php/matstud/article/view/210
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Articles