Geodesic
Products of Normal Operators
Canadian journal of mathematics, Tome 40 (1988) no. 6, pp. 1322-1330

Voir la notice de l'article provenant de la source Cambridge University Press

Which bounded linear operator on a complex, separable Hilbert space can be expressed as the product of finitely many normal operators? What is the answer if “normal” is replaced by “Hermitian”, “nonnegative” or “positive”? Recall that an operator T is nonnegative (resp. positive) if (Tx, x) ≧ 0 (resp. (Tx, x) ≥ 0) for any x ≠ 0 in the underlying space. The purpose of this paper is to provide complete answers to these questions.If the space is finite-dimensional, then necessary and sufficient conditions for operators expressible as such are already known. For normal operators, this is easy. By the polar decomposition, every operator is the product of two normal operators. An operator is the product of Hermitian operators if and only if its determinant is real; moreover, in this case, 4 Hermitian operators suffice and 4 is the smallest such number (cf. [10]). 
Wu, Pei Yuan. Products of Normal Operators. Canadian journal of mathematics, Tome 40 (1988) no. 6, pp. 1322-1330. doi: 10.4153/CJM-1988-059-5
@article{10_4153_CJM_1988_059_5,
     author = {Wu, Pei Yuan},
     title = {Products of {Normal} {Operators}},
     journal = {Canadian journal of mathematics},
     pages = {1322--1330},
     year = {1988},
     volume = {40},
     number = {6},
     doi = {10.4153/CJM-1988-059-5},
     url = {http://geodesic.mathdoc.fr/articles/10.4153/CJM-1988-059-5/}
}
TY  - JOUR
AU  - Wu, Pei Yuan
TI  - Products of Normal Operators
JO  - Canadian journal of mathematics
PY  - 1988
SP  - 1322
EP  - 1330
VL  - 40
IS  - 6
UR  - http://geodesic.mathdoc.fr/articles/10.4153/CJM-1988-059-5/
DO  - 10.4153/CJM-1988-059-5
ID  - 10_4153_CJM_1988_059_5
ER  - 
%0 Journal Article
%A Wu, Pei Yuan
%T Products of Normal Operators
%J Canadian journal of mathematics
%D 1988
%P 1322-1330
%V 40
%N 6
%U http://geodesic.mathdoc.fr/articles/10.4153/CJM-1988-059-5/
%R 10.4153/CJM-1988-059-5
%F 10_4153_CJM_1988_059_5

[1] 1. Ballantine, C. S., Products of positive definite matrices, IV, Linear Alg. Appl. 3 (1970), 79–114. Google Scholar

[2] 2. Bonsall, F. F. and Duncan, J., Complete normed algebras (Springer-Verlag, New York, 1973). Google Scholar | DOI

[3] 3. Bouldin, R., The essential minimum modulus, Indiana Univ. Math. J. 30 (1981), 513–517. Google Scholar

[4] 4. Conway, J. B., A course in functional analysis (Springer-Verlag, New York, 1985). Google Scholar | DOI

[5] 5. Feldman, J. and Kadison, R. V., The closure of the regular operators in a ring of operators, Proc. Amer. Math. Soc. 5 (1954), 909–916. Google Scholar

[6] 6. Fillmore, P. A. and Williams, J. P., On operator ranges, Adv. in Math. 7 (1971), 254–281. Google Scholar

[7] 7. Gray, L. J., Products of Hermitian operators, Proc. Amer. Math. Soc. 59 (1976), 123–126. Google Scholar

[8] 8. Halmos, P. R., A Hilbert space problem book, 2nd ed. (Springer-Verlag, New York, 1982). Google Scholar | DOI

[9] 9. Izumino, S. and Kato, Y., The closure of invertible operators on a Hilbert space, Acta Sci. Math. (Szeged) 49 (1985), 321–327. Google Scholar

[10] 10. Radjavi, H., Products of Hermitian matrices and symmetries, Proc. Amer. Math. Soc. 21 (1969), 369–372;26 (1970), 701. Google Scholar

[11] 11. Radjavi, H., On self adjoint factorization of operators, Can. J. Math. 21 (1969), 1421–1426. Google Scholar

[12] 12. Radjavi, H. and Williams, J. P., Products of self adjoint operators, Michigan Math. J. 16 (1969), 177–185. Google Scholar

[13] 13. Wu, P. Y., Products of positive semi-definite matrices, Linear Alg. Appl., to appear. Google Scholar

Cité par Sources :