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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

Factorization of matrices into partial isometries


Authors: Kung Hwang Kuo and Pei Yuan Wu
Journal: Proc. Amer. Math. Soc. 105 (1989), 263-272
MSC: Primary 15A23; Secondary 47A68
DOI: https://doi.org/10.1090/S0002-9939-1989-0977922-1
MathSciNet review: 977922
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Abstract: In this paper, we characterize complex square matrices which are expressible as products of partial isometries and orthogonal projections. More precisely, we show that a matrix $ T$ is the product of $ k$ partial isometries $ (k \geq 1)$ if and only if $ T$ is a contraction $ (\left\Vert T \right\Vert \leq 1)$ and rank $ (1 - {T^*}T) \leq k \cdot $ nullity $ T$. It follows, as a corollary, that any $ n \times n$ singular contraction is the product of $ n$ partial isometries and $ n$ is the smallest such number. On the other hand, $ T$ is the product of finitely many orthogonal projections if and only if $ T$ is unitarily equivalent to $ 1 \oplus S$, where $ S$ is a singular strict contraction $ (\left\Vert S \right\Vert < 1)$. As contrasted to the previous case, the number of factors can be arbitrarily large.


References [Enhancements On Off] (What's this?)

  • [1] C. S. Ballantine, Products of idempotent matrices, Linear Algebra Appl. 19 (1978), 81-86. MR 0472862 (57:12551)
  • [2] L. Brown, Almost every proper isometry is a shift, Indiana Univ. Math. J. 23 (1973), 429-431. MR 0328634 (48:6976)
  • [3] I. Erdelyi, Partial isometries closed under multiplication on Hilbert spaces, J. Math. Anal. Appl. 22 (1968), 546-551. MR 0228998 (37:4577)
  • [4] P. R. Halmos, A Hilbert space problem book, 2nd ed., Springer-Verlag, New York, 1982. MR 675952 (84e:47001)
  • [5] H. Radjavi and J. P. Williams, Products of self-adjoint operators, Michigan Math. J. 16 (1969), 177-185. MR 0244801 (39:6115)
  • [6] B. Sz.-Nagy and C. Foiaş, Harmonic analysis of operators on Hilbert space, North-Holland, Amsterdam, 1970. MR 0275190 (43:947)

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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1989-0977922-1
Keywords: Partial isometry, orthogonal projection, contraction, idempotent matrix
Article copyright: © Copyright 1989 American Mathematical Society

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