Let G be a compact group, let X be a Banach space, and let P : L 1 (G) → X be an orthogonally additive, continuous n-homogeneous polynomial. Then we show that there exists a unique continuous linear map Φ : L 1 (G) → X such that P (f ) = Φ f *

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Let A and B be unital rings. An additive map T : A → B is called a weighted Jordan homomorphism if c = T (1) is an invertible central element and cT (x 2 ) = T (x) 2 for all x ∈ A. We provide assumptions, which are in particular fulfilled when A = B = Mn(R) with n ≥ 2 and R any unital ring with 1 2 , under which every surjective additive map T : A → B with the property that T (x)T (y) + T (y)T (x) = 0 whenever xy = yx = 0 is a weighted Jordan homomorphism. Further, we show that if A is a prime ring with char(A) = 2, 3, 5, then a bijective additive map T : A → A is a weighted Jordan homomorphism provided that there exists an additive map S : A → A such that S(x 2 ) = T (x) 2 for all x ∈ A.

Let X and Y be Banach spaces, let A(X) stands for the algebra of approximable operators on X, and let P : A(X) → Y be an orthogonally additive, continuous n-homogeneous polynomial. If X * has the bounded approximation property, then we show that there exists a unique continuous linear map Φ : A(X) → Y such that P (T ) = Φ(T n ) for each T ∈ A(X). P (a) = Φ(a n ) (a ∈ A) 2010 Mathematics Subject Classification. 47H60, 46H35, 47L10. Key words and phrases. Algebra of approximable operators; bounded approximation property; orthogonally additive polynomial.

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