In this paper we study the theory of operators on complex Hilbert spaces, which achieve the norm in the unit sphere. We prove important results concerning the characterization of the AN operators, see Definition 1.2. The class of the AN operators contains the algebra of the compact ones.
Mathematics Subject Classification (2010). Primary 47B07;Secondary 47A58, 47B65, 47A12, 47A15.
We consider the stochastic divergence-free continuity equations with Ladyzhenskaya-Prodi-Serrin condition. Wellposedness is proved meanwhile uniqueness may fail for the deterministic PDE. The main issue of strong uniqueness, in the probabilistic sense, relies on stochastic characteristic method and the generalized Itô-Wentzell-Kunita formula. The stability property for the unique solution is proved with respect to the initial data. Moreover, a persistence result is established by a representation formula.
Following Ben-Artzi and LeFloch, we consider nonlinear hyperbolic conservation laws posed on a Riemannian manifold, and we establish an L 1 -error estimate for a class of finite volume schemes allowing for the approximation of entropy solutions to the initial value problem. The error in the L 1 norm is of order h 1/4 at most, where h represents the maximal diameter of elements in the family of geodesic triangulations. The proof relies on a suitable generalization of Cockburn, Coquel, and LeFloch's theory which was originally developed in the Euclidian setting. We extent the arguments to curved manifolds, by taking into account the effects to the geometry and overcoming several new technical difficulties.
In this paper we study the theory of operators on complex Hilbert spaces, which attain their minima in the unit sphere. We prove some important results concerning the characterization of the N * , and also AN * operators, see respectively Definition 1.1 and Definition 1.3. The injective property plays an important role in these operators, and shall be established by these classes.
a b s t r a c tWe consider a hyperbolic conservation law posed on an (N + 1)-dimensional spacetime, whose flux is a field of differential forms of degree N. Generalizing the classical Kuznetsov's method, we derive an L 1 error estimate which applies to a large class of approximate solutions. In particular, we apply our main theorem and deal with two entropy solutions associated with distinct flux fields, as well as with an entropy solution and an approximate solution. Our framework encompasses, for instance, equations posed on a globally hyperbolic Lorentzian manifold.
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