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 A∈ A(X) and some B ∈ L(X), which means that D is a derivation. We apply this result to semisimple H*-algebras.
Authors:J. Garcia-Cuerva, K. Kazarian, and V. Kolyada
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.
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.
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 C1,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.
Weak atomic decompositions of B-valued martingales with two-parameters in weak Hardy spaces wpΣα and wpHα 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.
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.
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.
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.