On certain functional equation in semiprime rings and standard operator algebras

• Nejc Širovnik ^[1]
  1. [1] University of Maribor Faculty of Natural Sciences and Mathematics Department of Mathematics and Computer Science
• Localización: Cubo: A Mathematical Journal, ISSN 0716-7776, ISSN-e 0719-0646, Vol. 16, Nº. 1, 2014, págs. 73-80
• Idioma: inglés
• DOI: 10.4067/S0719-06462014000100007
• Enlaces
  • Texto completo
• Resumen
  • español

    El propósito principal de este artículo es probar el siguiente resultado, el cual se relaciona a un resultado clásico de Chernoff. Sea X un espacio de Banach real o complejo, sea L (X) el álgebra de todos los operadores lineales acotados en X y sea A (X) C L (X) una álgebra de operadores estándar. Supongamos que existe una aplicación lineal D : A (X) ͢ L (X) satisfaciendo la relación 2D (An) = D (An-1) A + An-1D (A) + D (A) An-1 + AD (An-1) para todo A e A (X), donde n > 2 es algún entero fijo. En este caso D es de la forma D (A) = [A, B] para todo A e A (X) y algún B e L (X) fijo, lo que significa que D es una derivación lineal. En particular, D es continua.

  • English

    The main purpose of this paper is to prove the following result, which is related to a classical result of Chernoff. Let X be a real or complex Banach space, let L (X) be the algebra of all bounded linear operators on X and let A(X) C L (X) be a standard operator algebra. Suppose there exists a linear mapping D : A (X) ͢ L (X) satisfying the relation 2D (An) = D (An-1) A+An-1 D (A) + D (A) An-1+ AD (An-1) for all A e A (X), where n > 2 is some fixed integer. In this case D is of the form D (A) = [A, B] for all A e A (X) and some fixed B e L (X), which means that D is a linear derivation. In particular, D is continuous.

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