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Products of positive operators. (English) Zbl 1515.47025

Denote by \(\mathcal{L}^{+2}\) the set of all bounded operators on separable infinite-dimensional Hilbert spaces that can be written as the product of two bounded positive operators. The paper deals with the following natural questions of general interest: Classify the elements of \(\mathcal{L}^{+2}\), and determine the properties \(\mathcal{L}^{+2}\).
In the case of matrices, the answer is straight, which is due to P. Wu [Linear Algebra Appl. 111, 53–61 (1988; Zbl 0658.15020)]: A square matrix \(T\) is in \(\mathcal{L}^{+2}\) if and only if it is similar to a positive square matrix. See also [C. S. Ballantine, Linear Algebra Appl. 3, 79–114 (1970; Zbl 0192.37002)] for a classification of square matrices that can be expressed as the product of two, three, or four positive square matrices.
As the results of the paper suggest, the problem is far more complicated for infinite-dimensional Hilbert spaces. For instance, part of one of the main results of the paper states: Let \(T\) be a bounded linear operator on a Hilbert space \(\mathcal{H}\). Then \(T\) is quasi-similar to a positive operator if and only if \(T = AB\) for some closed surjective positive operator \(A\) and a positive semidefinite operator \(B\); and \(T^* = B' A'\) for some closed surjective positive operator \(B'\) and a positive semidefinite operator \(A'\).
The authors point out, however, that a bounded linear operator which is quasi-similar to a positive operator is not necessarily the product of two bounded positive bounded operators. The paper gives considerable space for the properties (including spectral properties) of operators in the class \(\mathcal{L}^{+2}\).
A stronger version of Sebestyén’s factorization theorem [Z. Sebestyén, Acta Sci. Math. 46, 299–301 (1983; Zbl 0551.47005)] is an additional result of the paper that also plays an important part in its own analysis. In summary, it states the following: Given bounded linear operators \(A\) and \(T\) on a Hilbert space \(\mathcal{H}\), there exists a positive operator \(X\) on \(\mathcal{H}\) such that \[ AX = T \] if and only \[ T T^* \leq \lambda A T^* \] for some \(\lambda > 0\).
In addition to the preceding result, the paper is a treasure trove of applications of numerous classical operator theory results developed by many giants.

MSC:

47B02 Operators on Hilbert spaces (general)
47A65 Structure theory of linear operators
47A11 Local spectral properties of linear operators
47A05 General (adjoints, conjugates, products, inverses, domains, ranges, etc.)
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