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Products of Normal Operators

Published online by Cambridge University Press:  20 November 2018

Pei Yuan Wu
Pei Yuan Wu*
Affiliation:
National Chiao Tung University, Hsinchu, Taiwan
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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]).

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 40 , Issue 6 , 01 December 1988 , pp. 1322 - 1330
Copyright
Copyright © Canadian Mathematical Society 1988

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