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February 2022 On some functional equation in standard operator algebras
Irena Kosi-Ulbl, Joso Vukman
Rocky Mountain J. Math. 52(1): 171-181 (February 2022). DOI: 10.1216/rmj.2022.52.171

Abstract

In this paper we prove the following result. Let n3 be some fixed integer, let X be a real or complex Banach space, let (X) be the algebra of all bounded linear operators on X, and let 𝒜(X)(X) be a standard operator algebra. Suppose there exists a linear mapping D:𝒜(X)(X) satisfying the relation $${2^{n - 2}}D({A^n}) = \sum\nolimits_{i = 0}^{n - 2} {\left( {\matrix{ {n - 2} i } } \right)} {A^i}D({A^2}){A^{n - 2 - i}} + ({2^{n - 2}} - 1)(D(A){A^{n - 1}} + {A^{n - 1}}D(A)) + \sum\nolimits_{i = 1}^{n - 2} {\left( {\sum\nolimits_{k = 2}^i {({2^{k - 1}} - 1)\left( {\matrix{ {n - k - 2} {i - k} } } \right)} + \sum\nolimits_{k = 2}^{n - 1 - i} {({2^{k - 1}} - 1)\left( {\matrix{ {n - k - 2} {n - i - k - 1} } } \right)} } \right)} {A^i}D(A){A^{n - 1 - i}}$$ for all A𝒜(X). In this case D is of the form D(A)=[B,A] for all A𝒜(X), and some fixed B(X). In particular, D is continuous. This result is related to a classical result of Chernoff.

Citation

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Irena Kosi-Ulbl. Joso Vukman. "On some functional equation in standard operator algebras." Rocky Mountain J. Math. 52 (1) 171 - 181, February 2022. https://doi.org/10.1216/rmj.2022.52.171

Information

Received: 15 February 2021; Accepted: 15 June 2021; Published: February 2022
First available in Project Euclid: 19 April 2022

Digital Object Identifier: 10.1216/rmj.2022.52.171

Subjects:
Primary: 16W25
Secondary: 39B52‎ , 47B01

Keywords: Banach space , derivation‎ , functional equation , generalized inner derivation , inner derivation , Jordan derivation , Jordan triple derivation , Prime ring , semiprime ring , standard operator algebra

Rights: Copyright © 2022 Rocky Mountain Mathematics Consortium

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Vol.52 • No. 1 • February 2022
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