Products of diagonal operators (Q1210245)

The main result of the paper is as follows. Let \(T\) be a bounded linear operator on a separable complex Hilbert space. The following conditions are equivalent. 1. \(T\) is the product of finitely many normal operators. 2. \(T\) is the product of finitely many Hermitian operators. 3. \(T\) is the product of finitely many positive operators. 4. \(T\) is the product of finitely many diagonal operators. 5. \(T\) is the product of finitely many positive diagonal operators. 6. \(\dim(\text{ker } T)=\dim(\text{ker } T^*)\) or \(\text{ran }T\) is not closed. Further, if these conditions hold, then \(T\) can be expressed as the product of at most 3 normal operators, 6 Hermitian operators, 18 positive operators, 6 diagonal operators or 20 positive diagonal operators. For earlier work on this theme, see the first author, Products of normal operators, Can. J. Math. 40, No. 6, 1322-1330 (1988; Zbl 0671.47013).