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Summary  

Let X be a complex Hilbert 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, which is closed under the adjoint operation. Suppose there exists a linear mapping D: A(X) → L(X) satisfying the relation D(AA*A) = D(A) A*A + AD(A*)A + AA*D(A), for all A ∈ A(X). In this case D is of the form D(A) = AB-BA, for all AA(X) and some B L(X), which means that D is a derivation. We apply this result to semisimple H*-algebras.

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Summary  

Let Abe a semisimple H*-algebra and let T: AAbe an additive mapping such that T(x n +1)=T(x)x n+x n T(x) holds for all xAand some integer n≥1. In this case Tis a left and a right centralizer.

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Abstract  

We investigate the extension to Banach-space-valued functions of the classical inequalities due to Paley for the Fourier coefficients with respect to a general orthonormal system Φ. This leads us to introduce the notions of Paley Φ-type and Φ-cotype for a Banach space and some related concepts. We study the relations between these notions of type and cotype and those previously defined. We also analyze how the interpolation spaces inherit these characteristics from the original spaces, and use them to obtain sharp coefficient estimates for functions taking values in Lorentz spaces.

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We obtain a necessary and sufficient condition for the lacunary polynomials to be dense in weighted Banach spaces of functions continuous on the rays emerging from the origin. This generalizes the solution to the classical Bernstein problem given by S. Izumi, T. Kawata and T. Hall.

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Let X be a Banach space and α ∈ (0, 1]. We find equivalent conditions for a function f: [0,1] → X to admit an equivalent parametrization, which is C 1,α (i.e., has α-Hölder derivative). For X = ℝ, a characterization is well-known. However, even in the case X = ℝ2 several new ideas are needed.

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Weak atomic decompositions of B-valued martingales with two-parameters in weak Hardy spaces w pΣα and w p H α are established and the boundedness of sublinear operators on these spaces are proved. By using them, some characterizations of the smoothness of Banach spaces are obtained.

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Abstract  

We obtain some new existence theorems for the maximal and minimal fixed points of discontinuous increasing operators. As applications, we consider the existence of the maximal and minimal solutions of nonlinear integro-differential equations with discontinuous terms in Banach spaces. Our results generalize and improve many well-known conclusions.

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Stochastic evolution equations with monotone operators in Banach spaces are considered. The solutions are characterized as minimizers of certain convex functionals. The method of monotonicity is interpreted as a method of constructing minimizers to these functionals, and in this way solutions are constructed via Euler-Galerkin approximations.

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In this paper we deal with elliptic systems with discontinuous nonlinearities. The discontinuous nonlinearities are assumed to satisfy quasimonotone conditions. We shall use the method of upper and lower solutions with fixed point theorems on increasing operators in ordered Banach spaces to show some existence theorems.

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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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