Factorization of matrices into partial isometries
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- by Kung Hwang Kuo and Pei Yuan Wu
- Proc. Amer. Math. Soc. 105 (1989), 263-272
- DOI: https://doi.org/10.1090/S0002-9939-1989-0977922-1
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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 $(\left \| T \right \| \leq 1)$ 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 $(\left \| S \right \| < 1)$. As contrasted to the previous case, the number of factors can be arbitrarily large.References
- C. S. Ballantine, Products of idempotent matrices, Linear Algebra Appl. 19 (1978), no. 1, 81–86. MR 472862, DOI 10.1016/0024-3795(78)90006-x
- Lawrence G. Brown, Almost every proper isometry is a shift, Indiana Univ. Math. J. 23 (1973/74), 429–431. MR 328634, DOI 10.1512/iumj.1973.23.23035
- Ivan Erdélyi, Partial isometries closed under multiplication on Hilbert spaces, J. Math. Anal. Appl. 22 (1968), 546–551. MR 228998, DOI 10.1016/0022-247X(68)90193-5
- Paul Richard Halmos, A Hilbert space problem book, 2nd ed., Encyclopedia of Mathematics and its Applications, vol. 17, Springer-Verlag, New York-Berlin, 1982. MR 675952
- Heydar Radjavi and James P. Williams, Products of self-adjoint operators, Michigan Math. J. 16 (1969), 177–185. MR 244801
- Béla Sz.-Nagy and Ciprian Foiaş, Harmonic analysis of operators on Hilbert space, North-Holland Publishing Co., Amsterdam-London; American Elsevier Publishing Co., Inc., New York; Akadémiai Kiadó, Budapest, 1970. Translated from the French and revised. MR 0275190
Bibliographic Information
- © Copyright 1989 American Mathematical Society
- 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