On the Normality of the Sum of Two Normal Operators

Mortad, Mohammed Hichem

doi:10.1007/s11785-010-0072-7

Cited by 13 publications

(8 citation statements)

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“…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.…”

Section: Introductionmentioning

confidence: 98%

“…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.…”

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confidence: 98%

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

Mortad

2012

Demonstratio Mathematica

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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...

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Section: Introductionmentioning

confidence: 98%

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confidence: 98%

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

Mortad

2012

Demonstratio Mathematica

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“…In [18,Theorem 3] The study of operators satisfying Kaplansky theorem is of significant interest and is currently being done by a number of mathematicians around the world.Some developments toward this subject have been done in [6,10,12,15,16,17] and the references therein.…”

Section: A Densely Defined Operatormentioning

confidence: 99%

Generalizations of Kaplansky Theorem for some (p,k)-Quasihyponormal Operators

Benali,

Ahmed

2016

Preprint

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“…It is well-known that the sum of two bounded commuting normal operators is normal. This was generalized to the case where one operator is unbounded (see [10]). It is also known that the sum of two strongly anti-commuting unbounded self-adjoint operators is self-adjoint (see [21]).…”

Section: Proposition 2 Let a And B Be Two Unbounded Normal Operators mentioning

confidence: 99%

“…Then we prove a result on the normality of the sum of two unbounded normal operators (cf [10]). A certain form of commutativity (not strong commutativity though) is required.…”

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A criterion for the normality of unbounded operators and applications to self-adjointness

Mortad

2014

Rend. Circ. Mat. Palermo

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In this paper, we give and prove a criterion for the normality of unbounded closed operators, which is a sort of a maximality result which will be called "double maximality". As applications, we show, under some assumptions, that the sum of two symmetric operators is essentially self-adjoint; and that the sum of two unbounded normal operators is essentially normal. Some other important results are also established.

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On the Normality of the Sum of Two Normal Operators

Abstract: The aim of this paper is to give sufficient conditions on two normal operators (bounded or not), defined on a Hilbert space, which make their algebraic sum normal. The results are accompanied by some interesting examples and counter examples.

Cited by 13 publications

References 10 publications

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

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

Generalizations of Kaplansky Theorem for some (p,k)-Quasihyponormal Operators

A criterion for the normality of unbounded operators and applications to self-adjointness

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