Upper Triangular Operator Matrices, SVEP and Browder, Weyl Theorems

A Banach space operator T ∈ B(χ) is polaroid if points λ ∈ iso σ(T) are poles of the resolvent of T. Let denote, respectively, the approximate point, the Weyl, the Weyl essential approximate, the upper semi–Fredholm and lower semi–Fredholm spectrum of T. For A, B and C ∈ B(χ), let M _C denote the operator matrix . If A is polaroid on , M ₀ satisfies Weyl’s theorem, and A and B satisfy either of the hypotheses (i) A has SVEP at points and B has SVEP at points , or, (ii) both A and A* have SVEP at points , or, (iii) A^* has SVEP at points and B ^* has SVEP at points , then . Here the hypothesis that λ ∈ π₀(M _C ) are poles of the resolvent of A can not be replaced by the hypothesis are poles of the resolvent of A. For an operator , let . We prove that if A* and B* have SVEP, A is polaroid on π ^a ₀(M _C) and B is polaroid on π ^a ₀(B), then .      

Beurling-type Representation of Invariant Subspaces in Reproducing Kernel Hilbert Spaces

By Beurling’s theorem, the orthogonal projection onto an invariant subspace M of the Hardy space on the unit disk can be represented as where Φ is an inner multiplier of . This concept can be carried over to arbitrary Nevanlinna-Pick spaces but fails in more general settings. This paper introduces the notion of Beurling decomposable subspaces. An invariant subspace M of a reproducing kernel Hilbert space will be called Beurling decomposable if there exist (operator-valued) multipliers such that and . We characterize the finite-codimensional and the finite-rank Beurling decomposable subspaces by means of their core function and core operator. As an application, we show that in many analytic Hilbert modules , every finite-codimensional submodule M can be written as with suitable polynomials p _i .      

Symmetric groups and expander graphs

We construct explicit generating sets S _n and of the alternating and the symmetric groups, which turn the Cayley graphs and into a family of bounded degree expanders for all n.     

A generalization of Meshulam’s theorem on subsets of finite abelian groups with no 3-term arithmetic progression

Let r ₁, …, r _s be non-zero integers satisfying r ₁ + ⋯ + r _s = 0. Let G be a finite abelian group with k _i |k _i-1(2 ≤ i ≤ n), and suppose that (r _i , k ₁) = 1(1 ≤ i ≤ s). Let denote the maximal cardinality of a set which contains no non-trivial solution of r ₁ x ₁ + ⋯ + r _s x _s = 0 with . We prove that . We also apply this result to study problems in finite projective spaces.      

<Emphasis Type="Italic">r</Emphasis>-Minimal submanifolds in space forms

Let be an n-dimensional compact, possibly with boundary, submanifold in an (n + p)-dimensional space form R ^n+p (c). Assume that r is even and , in this paper we introduce rth mean curvature function S _r and (r + 1)-th mean curvature vector field . We call M to be an r-minimal submanifold if on M, we note that the concept of 0-minimal submanifold is the concept of minimal submanifold. In this paper, we define a functional of , by calculation of the first variational formula of J _r we show that x is a critical point of J _r if and only if x is r-minimal. Besides, we give many examples of r-minimal submanifolds in space forms. We calculate the second variational formula of J _r and prove that there exists no compact without boundary stable r-minimal submanifold with in the unit sphere S ^n+p . When r = 0, noting S ₀ = 1, our result reduces to Simons’ result: there exists no compact without boundary stable minimal submanifold in the unit sphere S ^n+p .      

Boundedness of pluricanonical maps of varieties of general type

Using the techniques of [20] and [10], we prove that certain log forms may be lifted from a divisor to the ambient variety. As a consequence of this result, following [22], we show that: For any positive integer n there exists an integer r _n such that if X is a smooth projective variety of general type and dimension n, then is birational for all r≥r _n .     

Quantitative Néron theory for torsion bundles

Let R be a discrete valuation ring with algebraically closed residue field, and consider a smooth, geometrically connected, and projective curve C _K over the field of fractions K. For any positive integer r prime to the residual characteristic, we consider the finite K-group scheme of r-torsion line bundles on C _K . We determine when there exists a finite R-group scheme, which is a model of over R; in other words, we establish when the Néron model of is finite. The obvious idea would be to study the points of the Néron model over R, but in general these do not represent r-torsion line bundles on a semistable reduction of C _K . Instead, we recast the notion of models on a stack-theoretic base: there, we find finite Néron models, which represent r-torsion line bundles on a stack-theoretic semistable reduction of C _K . This allows us to quantify the lack of finiteness of the classical Néron models and finally to provide an efficient criterion for it. A. Chiodo was financially supported by the Marie Curie Intra-European Fellowship within the 6th European Community Framework Programme, MEIF-CT-2003-501940.     

The curve selection lemma and the Morse–Sard theorem

We use an inequality due to Bochnak and Lojasiewicz, which follows from the Curve Selection Lemma of real algebraic geometry in order to prove that, given a C ^r function , we have

Discretization of Compact Riemannian Manifolds Applied to the Spectrum of Laplacian

For κ ⩾ 0 and r₀ > 0 let ℳ(n, κ, r₀) be the set of all connected, compact n-dimensional Riemannian manifolds (Mⁿ, g) with Ricci (M, g) ⩾ −(n−1) κ g and Inj (M) ⩾ r₀. We study the relation between the kth eigenvalue λ_k(M) of the Laplacian associated to (Mⁿ,g), Δ = −div(grad), and the kth eigenvalue λ_k(X) of a combinatorial Laplacian associated to a discretization X of M. We show that there exist constants c, C > 0 (depending only on n, κ and r₀) such that for all M ∈ ℳ(n, κ, r₀) and X a discretization of for all k < |X|. Then, we obtain the same kind of result for two compact manifolds M and N ∈ ℳ(n, κ, r₀) such that the Gromov–Hausdorff distance between M and N is smaller than some η > 0. We show that there exist constants c, C > 0 depending on η, n, κ and r₀ such that for all . Mathematics Subject Classification (2000): 58J50, 53C20 Supported by Swiss National Science Foundation, grant No. 20-101 469     

On the Exact Constant in the Jackson–Stechkin Inequality for the Uniform Metric

The classical Jackson–Stechkin inequality estimates the value of the best uniform approximation of a 2π-periodic function f by trigonometric polynomials of degree ≤n−1 in terms of its r-th modulus of smoothness ω _r (f,δ). It reads

Sharp Lipschitz constant of bi-Lipschitz automorphism on Cantor set

Suppose C _r = (r C _r ) ∪ (r C _r + 1 − r) is a self-similar set with r ∈ (0, 1/2), and Aut(C _r ) is the set of all bi-Lipschitz automorphisms on C _r . This paper proves that there exists f* ∈ Aut(C _r ) such that

On the Affine Schur Algebra of Type A II

By studying certain centralizer subalgebras of the affine Schur algebra we show that is Noetherian and we determine its center. Assuming n ≥ r, we show that is Morita equivalent to , and the Schur functor is an equivalence under certain conditions. The author acknowledges support by National Natural Science Foundation of China No.10131010.     

An inequality between multipoint Seshadri constants

Let X be a projective variety of dimension n and L be a nef divisor on X. Denote by the d-dimensional Seshadri constant of r very general points in X. We prove that

Notes on the nuclearity of random subalgebras of O2{\mathcal {O}_2}

There exists a separable exact C*-algebra A which contains all separable exact C*-algebras as subalgebras, and for each norm-dense measure μ on A and independent μ-distributed random elements x ₁, x ₂, ... we have . Further, there exists a norm-dense non-atomic probability measure μ on the Cuntz algebra such that for an independent sequence x ₁, x ₂, ... of μ-distributed random elements x _i we have . We introduce the notion of the stochastic rank for a unital C*-algebra and prove that the stochastic rank of C([0, 1] ^d ) is d. B. Burgstaller was supported by the Austrian Schr?dinger stipend J2471-N12.     

Symmetric Tensors And Geometry of {\mathbb{P}} ^N Subvarieties

This paper using a geometric approach produces vanishing and nonvanishing results concerning the spaces of twisted symmetric differentials on subvarieties , with k ≤ m. Emphasis is given to the case of k = m which is special and whose nonvanishing results on the dimensional range dim X > 2/3(N − 1) are related to the space of quadrics containing X and the variety of all tangent trisecant lines of X. The paper ends with an application showing that the twisted symmetric plurigenera, along smooth families of projective varieties X_t are not invariant even for α arbitrarily large. Received: September 2006, Revision: May 2007, Accepted: June 2007     

Topological dimensions and multifractal Rényi dimensions of Polish spaces: a multifractal Szpilrajn type theorem

In this paper we consider the relationship between the topological dimension and the lower and upper q-Rényi dimensions and of a Polish space X for q ∈ [1, ∞]. Let and denote the Hausdorff dimension and the packing dimension, respectively. We prove that for all analytic metric spaces X (whose upper box dimension is finite) and all q ∈ (1, ∞); of course, trivially, for all q ∈ [1, ∞]. As a corollary to this we obtain the following result relating the topological dimension and the lower and upper q-Rényi dimensions: for all Polish spaces X and all q ∈ [1, ∞]; in (1) and (2) we have used the following notation, namely, for two metric spaces X and Y, we write X ∼ Y if and only if X is homeomorphic to Y. Equality (1) has recently been proved for q = ∞ by Myjak et al. Author’s address: Department of Mathematics, University of St. Andrews, St. Andrews, Fife KY16 9SS, Scotland     

Representation of certain homogeneous Hilbertian operator spaces and applications

Following Grothendieck’s characterization of Hilbert spaces we consider operator spaces F such that both F and F ^* completely embed into the dual of a C*-algebra. Due to Haagerup/Musat’s improved version of Pisier/Shlyakhtenko’s Grothendieck inequality for operator spaces, these spaces are quotients of subspaces of the direct sum C ⊕ R of the column and row spaces (the corresponding class being denoted by QS(C ⊕ R)). We first prove a representation theorem for homogeneous F∈QS(C ⊕ R) starting from the fundamental sequences $\Phi _{c}(n)=\Bigg\|\sum_{k=1}^ne_{k1}\otimes e_k\Bigg\|_{C\otimes _{\min}F}^2\quad\mbox{and}\quad \Phi _{r}(n)=\Bigg\|\sum_{k=1}^ne_{1k}\otimes e_k\Bigg\|_{R\otimes _{\min}F}^2$ given by an orthonormal basis (e _k ) of F. Under a mild regularity assumption on these sequences we show that they completely determine the operator space structure of F and find a canonical representation of this important class of homogeneous Hilbertian operator spaces in terms of weighted row and column spaces. This canonical representation allows us to get an explicit formula for the exactness constant of an n-dimensional subspace F _n of F: $\mathit{ex}(F_n)\sim\biggl[\frac{n}{ \Phi _{c}(n)}\Phi _{r}\bigg(\frac{ \Phi _{c}(n)}{\Phi _{r}(n)}\bigg)+\frac{n}{ \Phi _{r}(n)}\Phi _{c}\bigg(\frac{ \Phi _{r}(n)}{\Phi _{c}(n)}\bigg)\biggr]^{1/2}.$ In the same way, the projection (=injectivity) constant of F _n is explicitly expressed in terms of Φ _c and Φ _r too. Orlicz space techniques play a crucial role in our arguments. They also permit us to determine the completely 1-summing maps in Effros and Ruan’s sense between two homogeneous spaces E and F in QS(C ⊕ R). The resulting space Π ₁ ^o (E,?F) isomorphically coincides with a Schatten-Orlicz class S _φ . Moreover, the underlying Orlicz function φ is uniquely determined by the fundamental sequences of E and F. In particular, applying these results to the column subspace C _p of the Schatten p-class, we find the projection and exactness constants of C _p ⁿ , and determine the completely 1-summing maps from C _p to C _q for any 1≤p,?q≤∞.     

Generalized <Emphasis Type="BoldItalic">r</Emphasis>-Lah numbers

In this paper, we consider a two-parameter polynomial generalization, denoted by \(\mathcal {G}_{a,b}(n,k;r)\), of the r-Lah numbers which reduces to these recently introduced numbers when a = b = 1. We present several identities for \(\mathcal {G}_{a,b}(n,k;r)\) that generalize earlier identities given for the r-Lah and r-Stirling numbers. We also provide combinatorial proofs of some earlier identities involving the r-Lah numbers by defining appropriate sign-changing involutions. Generalizing these arguments yields orthogonality-type relations that are satisfied by \(\mathcal {G}_{a,b}(n,k;r)\).     

On Similarity of Multiplication Operator on Weighted Bergman Space

Let D be the unit disk and be the weighted Bergman space. In this paper, we prove that the multiplication operator is similar to M _z on . The author was supported in part by NSF Grant (10571041, L2007B05).