On the closedness, the self-adjointness and the normality of the product of two unbounded operators

Abstract

Abstract. Let A and B be two unbounded densely defined operators on a Hilbert space H. The purpose of this work is to give simple conditions that make the product AB closed, self-adjoint and normal provided the two operators are so. IntroductionFirst, we assume the reader is very familiar with notions, definitions and results on unbounded operators. Some general references are [2,8,10].Second, we recall the following theorems which will be needed to prove some of our results. Theorem 1.[11] Let B be a closed operator. If A is B-bounded with relative bound "a" smaller than one, then A + B is closed.The analog of the previous theorem for self-adjoint operators is the KatoRellich theorem (see e.g. [9]).Theorem 2. [Kato-Rellich] Let B be an unbounded self-adjoint operator with domain D(B). If A is B-bounded with relative bound "a" smaller than one, such that A is symmetric, then A + B is self-adjoint on D(B).If A and B are self-adjoint commuting bounded operators, then it is straightforward that AB = BA is self-adjoint.If A and B are two commuting bounded normal operators, then a simple use of the Fuglede theorem yields the normality of BA = AB. In the unbounded case (at least one operator is unbounded, A say), things get a little tricky in the sense that the same hypotheses do not necessarily yield The following example will be used on several occasions throughout this paper and it shows that the product of two commuting self-adjoint (and hence it also includes the class of normal operators) operators need not be self-adjoint and it may not be even normal.Example 1. Let A be an unbounded self-adjoint operator having a trivial kernel, for example takeNow set B = A −1 (observe that both A and B are positive on their respective domains). Then BA, defined on D(BA) = D(A), is not closed as BA ⊂ I. Thus it can neither be self-adjoint nor it can be normal and yetSo, the main purpose of this paper is to give conditions that force the product of two unbounded (one of them at most is bounded) closed, selfadjoint or normal operators to be closed, self-adjoint or normal.We draw the attention of the reader to the paper [3] where some conditions implying the closedness of the product of two closed operators are established.Finally, it is worth mentioning that the author has similar papers about the sum of unbounded operators (see [6,7]). Since rB − I < 1, we can say that (rB − I)A is A-bounded with relative bound smaller than one. Since A is closed, Theorem 1 yields the closedness of rBA or BA and hence the proposition is proved. Then BA is closed.Remark. Although Condition 1 seems to be a bit strong but the power of the previous theorem lies in the fact that B is not assumed to be closed.and so BA is densely defined. Then we can write for all ϕ ∈ D(BA) rBAϕ = (rB − I)Aϕ + Aϕ.Since (rB − I)Aϕ ≤ a Aϕ + b ϕ and a < 1, this shows that (rB − I)A is A-bounded with relative bound smaller than one and hence Theorem 1 allows us to establish the closedness of rBA = (rB −I)A+A or that of BA. The self-adjointness and the normality question...

Commutativity up to a factor for bounded and unbounded operators

“…It also generalizes known results for two normal operators (where at least one of them is bounded, see e.g. [7] and [13]). Proof.…”

“…The proof uses the celebrated Fuglede theorem (we note that this question has been generalized to the case of unbounded operators in e.g. [7], [13], [14] and [15]). In a very similar manner, we also notice that -this time via the Fuglede-Putnam theorem-the product of two anti-commuting normal operators remains normal.…”

Unbounded products of operators and connections to Dirac-type operators

“…This question has recently been studied, assuming some boundedness, in a number of papers. See for example [22,26,30] and literature citations therein. We state some of those results in the following theorem.…”

Conditions Implying Normality and Hypo-Normality of Operators in Hilbert Space

“…For more details, the interested reader is referred to [2,[6][7][8][9]. For other works related to products of normal (bounded and unbounded) operators, the reader may consult [10][11][12][13][14][15] We start by this result on the normality.…”