Product of invertible operators of quadratic type

AbstractSuppose all invertible quadratic operators T are assumed to satisfy T2 + bT + I = 0. We show that a complex matrix T is the product of two invertible quadratic matrices if and only if T is similar to a matrix of the form (D ⊕ D−1) ⊕ (I + N) ⊕ (−I + ∑mi=1 ⊕ Ji) ⊕ β2I ⊕ β−2I ⊕ [(β2I + XY) ⊕ (β2I + YX)−1], where β = (−b + √b2 − 4)2, 0, ±1 and gb±2 are not eigenvalues of D; N, XY, and YX are nilpotent; and each Ji is a nilpotent Jordan block of even size. Also, we show that an n × n matrix T is the product of finitely many invertible quadratic matrices if and only if det T = βm, where m is an integer for n odd and is an even integer for n even. On the other hand, for operators on an infinite-dimensional Hilbert space, we characterize the products of two and four invertible quadratic operators among normal operators and show that every invertible operator is the product of six invertible quadratic operators