Right (Or Left) Invertibility of Bounded and Unbounded Operators and Applications to the Spectrum of Products

Abstract

Abstract. This paper is mainly concerned with proving σ(AB) = σ(BA) for two linear and non necessarily bounded operators A and B. The main tool is left and right invertibility of bounded and unbounded operators.

Unbounded generalizations of the Fuglede-Putnam theorem and applications to the commutativity of self-adjoint operators

“…Generalizations to weaker classes than normality vary. Notice in passing that in [7], the self-adjointness of BA was established for a positive B ∈ B(H) and an unbounded self-adjoint A such that BA is hyponormal and σ(BA) = C. The next result is of the same kind. (1) Since B n is positive for all n and AB n is normal, it follows by Theorem 4.3 that AB n is self-adjoint and B n A ⊂ AB n .…”

“…In particular, B 2 |A * | 2 is closed. So, Proposition 3.7 in [7] implies that We now have all the necessary tools to establish the self-adjointness of A. Indeed,…”

“…In [1], [7], [13], [14], [16], [18], [19], [20], [25], and [29], the question of the self-adjointness of the normal product of two self-adjoint operators was tackled in different settings (cf. [2]).…”

“…(Cf. Proposition 3.7 in[7]) Let B ∈ B(H) and let A be an arbitrary operator such that B n A is closed for some integer n ≥ 2. Suppose further that BA is closable.…”

On The Absolute Value of Unbounded Operators

“…Thus, the invertibility of |ReA| + |ImA| implies the invertibility of |A|. Since the latter means that the normal operator A is left invertible, by [5] we get that A is invertible.…”