Functional Analysis and Its Applications最新文献

It is shown how certain recent results in the theory of determinants and traces can be applied to obtain new theorems on the distribution of eigenvalues of nuclear operators on Banach spaces and to prove the equality of the spectral and nuclear traces of such operators. As an example, we consider a new class of operators: the class of generalized Lapresté nuclear operators.

We consider a third-order non-self-adjoint operator which is an (L)-operator in the Lax pair for the Boussinesq equation on the circle. We construct a mapping from the set of operator coefficients to the set of spectral data, similar to the corresponding mapping for the Hill operator constructed by E. Korotyaev. We prove that, in a neighborhood of zero, our mapping is analytic and one-to-one.

In this paper we study the everywhere Hölder continuity of the minima of the following class of vectorial integral funcionals:

with some general conditions on the density (G).

We make the following assumptions about the function (G). Let (Omega) be a bounded open subset of (mathbb{R}^{n}), with (ngeq 2), and let (G colon Omega timesmathbb{R}^{m}timesmathbb{R}_{0,+}^{m}to mathbb{R}) be a Carathéodory function, where (mathbb{R}_{0,+}=[0,+infty)) and (mathbb{R} _{0,+}^{m}=mathbb{R}_{0,+}times dots timesmathbb{R}_{0,+}) with (mgeq 1). We make the following growth conditions on (G): there exists a constant (L>1) such that

for (mathcal{L}^{n}) a.e. (xin Omega ), for every (s^{alpha}in mathbb{R}) and every (xi^{alpha}inmathbb{R}) with (alpha=1,dots,m), (mgeq 1) and with (a(x) in L^{sigma}(Omega)), (a(x)geq 0) for (mathcal{L}^{n}) a.e. (xin Omega), (sigma >{n}/{p}), (1leq q<{p^{2}}/{n}) and (1<p<n).

Assuming that the previous growth hypothesis holds, we prove the following regularity result. If (u,{in}, W^{1,p}(Omega,mathbb{R}^{m})) is a local minimizer of the previous functional, then (u^{alpha}in C_{mathrm{loc}}^{o,beta_{0}}(Omega) ) for every (alpha=1,dots,m), with (beta_{0}in (0,1) ). The regularity of minimizers is obtained by proving that each component stays in a suitable De Giorgi class and, from this, we conclude Hölder continuity.

Quasiderivations of the universal enveloping algebra (Umathfrak{gl}_n) were first introduced by D. Gurevich, P. Pyatov, and P. Saponov in their study of reflection equation algebras; they are linear operators on (Umathfrak{gl}_n) that satisfy certain algebraic relations, which generalise the usual Leibniz rule. In this note, we show that the iterated action of the operator equal to a linear combination of the quasiderivations on a certain set of generators of the center of (Umathfrak{gl}_n) (namely on the symmetrised coefficients of the characteristic polynomial) produces commuting elements. The resulting algebra coincides with the quantum Mischenko–Fomenko algebra in (Umathfrak{gl}_n), introduced earlier by Tarasov, Rybnikov, Molev, and others.

We consider the problem of the motion of a dynamically symmetric heavy rigid body about a fixed point and give a detailed proof that the problem has no additional real-analytic first integral in all but the well-known classical cases.

Let (mathbb H) be a quaternion algebra generated by (I,J) and (K). We say that a hypercomplex nilpotent Lie algebra (mathfrak g) is (mathbb H)-solvable if there exists a sequence of (mathbb H)-invariant subalgebras containing (mathfrak g_{i+1}=[mathfrak g_i,mathfrak g_i]),

such that ([mathfrak g_i^{mathbb H},mathfrak g_i^{mathbb H}]subsetmathfrak g^{mathbb H}_{i+1}) and (mathfrak g_{i+1}^{mathbb H}=mathbb H[mathfrak g_i^{mathbb H},mathfrak g_i^{mathbb H}] ). Let (N=Gammasetminus G) be a hypercomplex nilmanifold with the flat Obata connection and (mathfrak g=operatorname{Lie}(G)). We prove that the Lie algebra (mathfrak g) is (mathbb H)-solvable.

Wang and Ma introduced the notion of asymmetric (L_p)-difference bodies. They further gave the extrema of volumes for the asymmetric (L_p)-difference body and its polar. Thereafter, Shi and Wang obtained their versions of quermassintegrals and dual quermassintegrals. In this paper, we determine the extrema of the (q)-quermassintegrals and dual (q)-quermassintegrals for the asymmetric (L_p)-difference bodies.

In this article, we present generalized improvements of certain Cauchy–Bunyakovsky–Schwarz type inequalities. As applications of our results, we provide improvements of some numerical radius inequalities for Hilbert space operators. Finally, we obtain certain numerical radii of Hilbert space operators involving geometrically convex functions.

An explicit expression for the expected value of a regularized multiplicative functional under the sine-process is obtained by passing to the scaling limit in the Borodin–Okounkov–Geronimo–Case formula.

We consider the Kantorovich optimal transportation problem in the case where the cost function and marginal distributions continuously depend on a parameter with values in a metric space. We prove the existence of approximate optimal Monge mappings continuous with respect to the parameter.