The Commutants of Relatively Prime Powers in Banach Algebras

Al-Moajil, Abdullah H.

doi:10.2307/2041197

Cited by 3 publications

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“…In [8], extending a result of Al-Moajil [1], the author has proved the following results. By a slight modification of the proof of Theorem C given in [10], Duggal [5,Theorem 5] has extended this result to relatively prime powers other than 2 and 3.…”

Section: Intertwining Relations Modulo Arbitrary Idealsmentioning

confidence: 65%

“…However, these conditions cannot be replaced by the symmetric conditions that ran A ⊆ ran A * and ran B ⊆ ran B * , or even more strongly, that A and B are hyponormal operators. This may be concluded from Remark 3.2 (a) in [1] or by considering the following example. EXAMPLE 1.…”

Section: Here We Have Respectively Used the Facts That T = 0 If Andmentioning

confidence: 76%

“…The following lemma, which is in the spirit of Lemma 1.1 in [1], indicates that in order to generalize Theorems A, B and C cited above, it is sufficient to consider two consecutive powers rather than two relatively prime powers.…”

Section: Theorem a Let A B And X ∈ B(h)mentioning

confidence: 99%

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Some Intertwining Relations Modulo Operator Ideals

Kittaneh¹

2006

Glasgow Math. J.

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Abstract. Let B(H)denote the algebra of all bounded linear operators on a separable, infinite-dimensional, complex Hilbert space H. Let I be a two-sided ideal in B(H). For operators A, B and X ∈ B(H), we say that Xintertwines A and B modulo I if AX − XB ∈ I. It is easy to see that if X intertwines A and B modulo I, then it intertwines A n and B n modulo I for every integer n >1. However, the converse is not true. In this paper, sufficient conditions on the operators A and B are given so that any operator X which intertwines certain powers of A and B modulo I also intertwines A and B modulo J for some two-sided ideal J ⊇ I.2000 Mathematics Subject Classification. 47A53, 47B10, 47B15, 47B20, 47B47.

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Section: Intertwining Relations Modulo Arbitrary Idealsmentioning

confidence: 65%

Section: Here We Have Respectively Used the Facts That T = 0 If Andmentioning

confidence: 76%

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Some Intertwining Relations Modulo Operator Ideals

Kittaneh¹

2006

Glasgow Math. J.

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“…Hence C 2 is the Hilbert-Schmidt class and C x is the class of compact operators. Relations between the commutant of an operator A and the commutants of powers of A have been investigated by many authors, for example, see [1] and [2]. In [1], Al-Moajil proved the following result.…”

mentioning

confidence: 99%

“…Relations between the commutant of an operator A and the commutants of powers of A have been investigated by many authors, for example, see [1] and [2]. In [1], Al-Moajil proved the following result. The purpose of this paper is to extend Theorem 1 to intertwining relations between a subnormal and a co-subnormal operator and to present the commutator modulo C p versions of these results.…”

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confidence: 99%

On the commutants modulo C_p of A² and A³

Kittaneh

1986

J Aust Math Soc A

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The Commutant Modulo Cp of Co-prime Powers of Operators on a Hilbert Space

Duggal

2001

Journal of Mathematical Analysis and Applications

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Let H be a separable infinite-dimensional complex Hilbert space and let A B ∈ B H , where B H is the algebra of operators on H into itself. Let δ A B B H → B H denote the generalized derivation δ AB X = AX − XB. This note considers the relationship between the commutant of an operator and the commutant of coprime powers of the operator. Let m n be some co-prime natural numbers and let p denote the Schatten p-class, 1 ≤ p < ∞. We prove (i) If δ A m B m X = 0 for some X ∈ B H and if either of A and B * is injective, then a necessary and sufficient condition for δ AB X = 0 is that A r XB n−r − A n−r XB r = 0 for (any) two consecutive values of r 1 ≤ r < n. (ii) If δ A m B m X and δ A n B n X ∈ p for some X ∈ B H , and if m = 2 or 3, then either δ n AB X or δ n+3 AB X ∈ p ; for general m and n, if A and B * are normal or subnormal, then there exists a natural number t such that δ AB X ∈ 2 tn p . (iii) If δ A m B m X and δ A n B n X ∈ p for some X ∈ B H , and if either A is semi-Fredholm with ind A ≤ 0 or 1 − A * A ∈ p , then δ AB X ∈ p .

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The Commutants of Relatively Prime Powers in Banach Algebras

Cited by 3 publications

References 0 publications

Some Intertwining Relations Modulo Operator Ideals

Some Intertwining Relations Modulo Operator Ideals

On the commutants modulo C_p of A² and A³

The Commutant Modulo Cp of Co-prime Powers of Operators on a Hilbert Space

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The Commutants of Relatively Prime Powers in Banach Algebras

Cited by 3 publications

References 0 publications

Some Intertwining Relations Modulo Operator Ideals

Some Intertwining Relations Modulo Operator Ideals

On the commutants modulo Cp of A2 and A3

The Commutant Modulo Cp of Co-prime Powers of Operators on a Hilbert Space

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On the commutants modulo C_p of A² and A³