Abstract
In this paper we prove the following result. Let be some fixed integer, let be a real or complex Banach space, let be the algebra of all bounded linear operators on , and let be a standard operator algebra. Suppose there exists a linear mapping 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 . In this case is of the form for all , and some fixed . In particular, is continuous. This result is related to a classical result of Chernoff.
Citation
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
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