ISSN 1439-8516 CN 11-2039/O1
Denote a semisimple Banach algebra with an identity e by A. This paper studies the Fredholm, Weyl and Browder spectral theories in a semisimple Banach algebra, and meanwhile considers the properties of the Fredholm element, the Weyl element and the Browder element. Further, for a ∈ A, we give the Weyl’s theorem and the Browder’s theorem for a, and characterize necessary and sufficient conditions that both a and f(a) satisfy the Weyl’s theorem or the Browder’s theorem, where f is a complex-valued function analytic on a neighborhood of σ(a). In addition, the perturbations of the Weyl’s theorem and the Browder’s theorem are investigated.
In this paper, distributed estimation of high-dimensional sparse precision matrix is proposed based on the debiased D-trace loss penalized lasso and the hard threshold method when samples are distributed into different machines for transelliptical graphical models. At a certain level of sparseness, this method not only achieves the correct selection of non-zero elements of sparse precision matrix, but the error rate can be comparable to the estimator in a non-distributed setting. The numerical results further prove that the proposed distributed method is more effective than the usual average method.
Under the framework of the complex column-vector loop algebra Cp, we propose a scheme for generating nonisospectral integrable hierarchies of evolution equations which generalizes the applicable scope of the Tu scheme. As applications of the scheme, we work out a nonisospectral integrable Schrödinger hierarchy and its expanding integrable model. The latter can be reduced to some nonisospectral generalized integrable Schrödinger systems, including the derivative nonlinear Schrödinger equation once obtained by Kaup and Newell. Specially, we obtain the famous Fokker–Plank equation and its generalized form, which has extensive applications in the stochastic dynamic systems. Finally, we investigate the Lie group symmetries, fundamental solutions and group-invariant solutions as well as the representation of the tensor of the Heisenberg group H3 and the matrix linear group SL(2, R) for the generalized Fokker–Plank equation (GFPE).
In this paper, we show that every super weakly compact convex subset of a Banach space is an absolute uniform retract, and that it also admits the uniform compact approximation property. These can be regarded as extensions of Lindenstrauss and Kalton’s corresponding results.
Let G be a classical group over an arbitrary field F, acting on an n-dimensional vector space V = V (n, F) over a field F. In this paper, we classify the maximal subgroups of G, which normalizes a solvable subgroup N of GL(n, F) not lying in F*1V.
In this paper, the mixed Pólya–Szegö principle is established. By the mixed Pólya–Szegö principle, the mixed Morrey–Sobolev inequality and some new analytic inequalities are obtained.
A subgroup A of a finite group G is called a local covering subgroup of G if AG = AB for all maximal G-invariant subgroup B of AG = <Ag, g ∈ G>. Let p be a prime and d be a positive integer. Assume that all subgroups of pd, and all cyclic subgroups of order 4 when pd = 2 and a Sylow 2-subgroup of G is nonabelian, of G are local covering subgroups. Then G is p-supersolvable whenever pd = p or pd ≤ √|G|p or pd ≤ |Op'p(G)|p/p.
This paper is devoted to studying Bergman spaces Aω1,2p(M) (1 < p < ∞) induced by regular-weight ω1,2 on annulus M. We characterize the function f in Lω1,21(M) for which the induced Hankel operator Hf is bounded (or compact) from Aω1,2p(M) to Lω1,2q(M) with 1 < p, q < ∞.
Let H2(D2) be the Hardy space over the bidisk D2, and let M?= [(z - ?(w))2] be the submodule generated by (z-?(w))2, where ?(w) is a function in H∞(w). The related quotient module is denoted by N? = H2(D2) ? M?. In the present paper, we study the Fredholmness of compression operators Sz, Sw on N?. When ?(w) is a nonconstant inner function, we prove that the Beurling type theorem holds for the fringe operator Fw on [(z - w)2] ? z[(z - w)2] and the Beurling type theorem holds for the fringe operator Fz on M? wM?if ?(0) = 0. Lastly, we study some properties of Fw on [(z - w2)2] ? z[(z - w2)2].
Recently, Gehér and Šemrl have characterized the general form of surjective isometries of the set of all projections on an infinite-dimensional separable Hilbert space using unitaries and antiunitaries. In this paper, we study the surjective L2-isometries of the projection lattice of an infinite dimensional Hilbert space and show that every such isometry can also be described by unitaries and antiunitaries.
