Putnam-Fuglede theorem and the range-kernel orthogonality of derivations

B. P. Duggal

International Journal of Mathematics and Mathematical Sciences, 2001, vol. 27, 1-10

Abstract:

Let ℬ ( H ) denote the algebra of operators on a Hilbert space H into itself. Let d = δ or Δ , where δ A B : ℬ ( H ) → ℬ ( H ) is the generalized derivation δ A B ( S ) = A S − S B and Δ A B : ℬ ( H ) → ℬ ( H ) is the elementary operator Δ A B ( S ) = A S B − S . Given A , B , S ∈ ℬ ( H ) , we say that the pair ( A , B ) has the property PF ( d ( S ) ) if d A B ( S ) = 0 implies d A ∗ B ∗ ( S ) = 0 . This paper characterizes operators A , B , and S for which the pair ( A , B ) has property PF ( d ( S ) ) , and establishes a relationship between the PF ( d ( S ) ) -property of the pair ( A , B ) and the range-kernel orthogonality of the operator d A B .

Date: 2001
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Persistent link: https://EconPapers.repec.org/RePEc:hin:jijmms:193506

DOI: 10.1155/S0161171201006159

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