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In this paper we prove the following result. 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 ) ⊂ L ( X ) be a standard operator algebra. Suppose we have a linear mapping D : A ( X ) → L ( X ) satisfying the relation D ( A 3 ) = D ( A ) A 2 + AD ( A ) A + A 2 D ( A ), for all AA ( X ). In this case D is of the form D ( A ) = ABBA , for all AA ( X ) and some BL ( X ). We apply this result, which generalizes a classical result of Chernoff, to semisimple H *-algebras.

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Abstract  

The main purpose of this paper is to prove the following result. Let H be a complex Hilbert space, let
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(H) be the algebra of all bounded linear operators on H, and let
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(H) ⊂
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(H) be a standard operator algebra which is closed under the adjoint operation. Suppose that T:
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(H) →
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(H) is a linear mapping satisfying T(AA* A) = T(A)A* AAT(A*)A + AA*T(A) for all A
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(H). Then T is of the form T(A) = AB + BA for all A
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(H), where B is a fixed operator from
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(H). A result concerning functional equations related to bicircular projections is proved
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