Eigenvalues of Set-Valued Operators in Banach Spaces

This work deals with the spectral analysis of set-valued operators from a Banach space X into its dual space X^*. The main goal of the paper is to study semicontinuity properties of the spectrum operator.Mathematics Subject Classifications (2000) 47H04, 47H12, 58C40.UMR 2071 Universidad de Chile-CNRS.     

Hermite插值算子在Orlicz空间内的逼近

插值算子逼近是逼近论中一个非常有趣的问题,尤其是以一些特殊的点为结点的插值算子的逼近问题很受人们的关注.研究了以第一类Chebyshev多项式零点为插值结点的Hermite插值算子在Orlicz范数下的逼近.     

Interpolation of Bilinear Operators Between Quasi-Banach Spaces

We study interpolation, generated by an abstract method of means, of bilinear operators between quasi-Banach spaces. It is shown that under suitable conditions on the type of these spaces and the boundedness of the classical convolution operator between the corresponding quasi-Banach sequence spaces, bilinear interpolation is possible. Applications to the classical real method spaces, Calderón-Lozanovsky spaces, and Lorentz-Zygmund spaces are presented. The author is supported by the National Science Foundation under grant DMS 0099881. The author is supported by KBN Grant 1 P03A 013 26.     

加权Hardy空间与加权L^p空间之间的算子内插

利用Calderon-Zygmund分解,给出了当w∈A1时加权Hardy空间与加权L^p空间之间的算子内插。     

Lp空间有理插值型算子的逼近

考虑了两类有理插值型算子的Jackson型估计．当p〉1时．建立了Dilzian—Totik型定理，当p=1时．利用通常连续模给出了Jackson型估计．     

Interpolation of Operators on Scales of Generalized Lorentz-Zygmund Spaces

We define generalized Lorentz-Zygmund spaces and obtain interpolation theorems for quasilinear operators on such spaces, using weighted Hardy inequalities. In the limiting cases of interpolation, we discover certain scaling property of these spaces and use it to obtain fine interpolation theorems in which the source is a sum of spaces and the target is an intersection of spaces. This yields a considerable improvement of the known results which we demonstrate with examples. We prove sharpness of the interpolation theorems by showing that the constraints on parameters are necessary for the interpolation theorems.     

Entropy and Widths of Multiplier Operators on Two-Point Homogeneous Spaces

In this article we continue the development of methods of estimating n-widths and entropy of multiplier operators begun in 1992 by A. Kushpel (Fourier Series and Their Applications, pp. 49–53, 1992; Ukr. Math. J. 45(1):59–65, 1993). Our main aim is to give an unified treatment for a wide range of multiplier operators Λ on symmetric manifolds. Namely, we investigate entropy numbers and n-widths of decaying multiplier sequences of real numbers \(\varLambda=\{\lambda_{k}\}_{k=1}^{\infty}\), |λ ₁|≥|λ ₂|≥?, \(\varLambda:L_{p}(\mathbb{M}^{d}) \rightarrow L_{q}(\mathbb{M}^{d})\) on two-point homogeneous spaces \(\mathbb{M}^{d}\): \(\mathbb{S}^{d}\), ? ^d (?), ? ^d (?), ? ^d (?), ?¹⁶(Cay). In the first part of this article, general upper and lower bounds are established for entropy and n-widths of multiplier operators. In the second part, different applications of these results are presented. In particular, we show that these estimates are order sharp in various important situations. For example, sharp order estimates are found for function sets with finite and infinite smoothness. We show that in the case of finite smoothness (i.e., |λ _k |?k ^?γ (lnk)^?ζ, γ/d>1, ζ≥0, k→∞), we have \(e_{n}(\varLambda U_{p}(\mathbb{S}^{d}), L_{q}(\mathbb{S}^{d})) \ll d_{n}(\varLambda U_{p}(\mathbb{S}^{d}), L_{q}(\mathbb{S}^{d}))\), n→∞, but in the case of infinite smoothness (i.e., \(|\lambda_{k}| \asymp e^{-\gamma k^{r}}\), γ>0, 0<r≤1, k→∞), we have \(e_{n}(\varLambda U_{p}(\mathbb{S}^{d}), L_{q}(\mathbb{S}^{d})) \gg d_{n}(\varLambda U_{p}(\mathbb{S}^{d}), L_{q}(\mathbb{S}^{d}))\), n→∞ for different p and q, where \(U_{p}(\mathbb{S}^{d})\) denotes the closed unit ball of \(L_{p}(\mathbb{S}^{d})\).     

Entropy Numbers of Diagonal Operators between Vector-Valued Sequence Spaces

Upper and lower bounds are established for the entropy numbersof certain diagonal operators between Banach sequence spaces.These diagonal operators are isomorphisms between the spacesconsidered in the paper and weighted sequence spaces consideredby Leopold so that the entropy numbers in question coincidewith those considered by Leopold. The results in the paper improvethe previous results in at least two ways. The estimates inthe paper are ‘almost’ sharp in the sense that theupper and lower estimates differ only by logarithmic factorsfor a much wider range of parameters. Moreover, all the upperestimates are improvements on the previous ones, the improvementbeing quite significant in some cases.     

齐型空间上算子的加权内插及其应用

本文在齐型空间上建立了算子的加权情形下的实内插定理，运用该结果，立即可推导出齐型空间上Ｃａｌｄｅｒｏｎ－Ｚｙｇｍｕｎｄ算子的加权Ｌ^ｐ有界性（ｐ＞１）和弱Ｌ¹有界性。     

Pullback Moduli Spaces

Geometric invariant theory can be used to construct moduli spaces associated to representations of finite dimensional algebras. One difficulty which occurs in various natural cases is that nonisomorphic modules are sent to the same point in the moduli spaces which arise. In this article, we study how this collapsing phenomenon can sometimes be reduced by considering pullbacks of modules for an auxiliary algebra. One application is a geometric proof that the twisting action of an algebra automorphism induces an algebraic isomorphism between moduli spaces.     

On Finite Rank Perturbations of Selfadjoint Operators in Krein Spaces and Eigenvalues in Spectral Gaps

It is shown that the finiteness of eigenvalues in a spectral gap of a definitizable or locally definitizable selfadjoint operator in a Krein space is preserved under finite rank perturbations. This results is applied to a class of singular Sturm–Liouville operators with an indefinite weight function.     

Interpolation of Limited and Weakly Compact Operators on Families of Banach Spaces

In this paper, we study how the limited and weakly compact properties of operators are preserved by interpolation of the real method for infinite families of Banach spaces introduced by Carro in Studia Math. 109 (1994). We apply these results to the case of Sparr, Fernández and Cobos–Peetre methods of interpolation for finite families.     

Tropicalization of Classical Moduli Spaces

The image of the complement of a hyperplane arrangement under a monomial map can be tropicalized combinatorially using matroid theory. We apply this to classical moduli spaces that are associated with complex reflection arrangements. Starting from modular curves, we visit the Segre cubic, the Igusa quartic, and moduli of marked del Pezzo surfaces of degrees 2 and 3. Our primary example is the Burkhardt quartic, whose tropicalization is a 3-dimensional fan in 39-dimensional space. This effectuates a synthesis of concrete and abstract approaches to tropical moduli of genus 2 curves.     

Duality Construction of Moduli Spaces

The duality of moduli spaces of coherent sheaves on curves and surfaces gives a GIT free construction of these spaces. It is a suitable generalization for topics as: strange duality, jumping lines of vector bundles on projective space, and Faltings' construction of the moduli space of semistable vector bundles on a complex curve given in his article Stable G-bundles and projective connections. Using this duality we propose a construction for the moduli space of semistable sheaves on a surface.     

On Eigenvalues of Pseudo-Differential Operators

Conditions are given to ensure the existence of finitely manyeigenvalues on the left-hand side of the bottom of the essentialspectrum. Estimates for the number of such eigenvalues are alsodiscussed.     

Entropy Numbers in Weighted Function Spaces and Eigenvalue Distributions of Some Degenerate Pseudodifferential Operators I

In this paper we study weighted function spaces of type B(?ⁿ, Q(x)) and F(?ⁿ, Q(x)), where Q(x) is a weight function of at most polynomial growth. Of special interest are the weight functions Q(x) = (1 + |x|²)^α/2 with α ? ?. The main result deals with estimates for the entropy numbers of compact embeddings between spaces of this type.     

Entropy Numbers in Weighted Function Spaces and Eigenvalue Distributions of Some Degenerate Pseudodifferential Operators II

This paper is the continuation of [17]. We investigate mapping and spectral properties of pseudodifferential operators of type Ψ with χ χ ? ? and 0 ≤ γ ≤ 1 in the weighted function spaces B (?ⁿ, w(x)) and F (?ⁿ, w(x)) treated in [17]. Furthermore, we study the distribution of eigenvalues and the behaviour of corresponding root spaces for degenerate pseudodifferential operators preferably of type b₂(x) b(x, D) b₁(x), where b₁(x) and b₂(x) are appropriate functions and b(x, D) ? Ψ. Finally, on the basis of the Birman-Schwinger principle, we deal with the “negative spectrum” (bound states) of related symmetric operators in L₂.