Gelfand Numbers of Identity Operators Between Symmetric Sequence Spaces

Complementing and generalizing classical as well as recent results, we prove asymptotically optimal formulas for the Gelfand and approximation numbers of identities Eⁿ ↪ Fⁿ, where Eⁿ and Fⁿ denote the n-th sections of symmetric quasi-Banach sequence spaces E and F satisfying certain interpolation assumptions. We illustrate our results by considering classical spaces such as Lorentz and Orlicz sequence spaces. Supported by DFG grant Hi 584/2-2.     

Matrix elements of vertex operators of the deformed W A n -algebra and the Harish-Chandra solutions to Macdonald's difference equations

In this paper we prove that certain matrix elements of vertex operators of the deformed W A _n -algebra satisfy Macdonald's difference equations and form a natural (n + 1)!-dimensional space of solutions. These solutions are the analogues of the Harish-Chandra solutions of the radial parts of the Laplace-Casimir operators on noncompact Riemannian symmetric spaces G/K with prescribed asymptotic behavior. We obtain formulas for analytic continuation of our Harish-Chandra type solutions as a consequence of braiding properties (obtained earlier by Y. Asai, M. Jimbo, T. Miwa, and Y. Pugay) of certain vertex operators of the deformed W A _n -algebra.     

Basic sequences, embeddings, and the uniqueness of the symmetric structure in unitary matrix spaces

For every symmetric sequence space E we denote by C_E the associated unitary matrix space. We show that every basic sequence in C_E has a subsequence which embeds in l₂ E. This reduces many problems about C_E to the analogous problems on E. We also study the asymptotic behavior of the embeddings of one unitary sequence space into another, and the problem of the uniqueness of the symmetric structure of unitary matrix spaces.     

Products of operator ideals and extensions of Schatten classes

We study relations between Schatten classes and product operator ideals, where one of the factors is the Banach ideal Π_E,2 of (E, 2)‐summing operators, and where E is a Banach sequence space with ?₂ ? E. We show that for a large class of 2‐convex symmetric Banach sequence spaces the product ideal Π_E,2 ○ ??^a_q,s is an extension of the Schatten class ??_F with a suitable Lorentz space F. As an application, we obtain that if 2 ≤ p, q < ∞, 1/r = 1/p + 1/q and E is a 2‐convex symmetric space with fundamental function λ_E(n) ≈? n^1/p, then Π_E,2 ○ Π_q is an extension of the Schatten class ??r,q (© 2010 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Integrability of rotationally symmetric n-harmonic maps

A rotationally symmetric n-harmonic map is a rotationally symmetric p-harmonic map between two n-dimensional model spaces such that p=n. We show that rotationally symmetric n-harmonic maps can be integrated and are n-harmonic diffeomorphism, and apply such results to investigate the asymptotic behaviors of these maps. We also derive this integrability using Lie theory.     

Strictly singular and strictly co-singular inclusions between symmetric sequence spaces

Strict singularity and strict co-singularity of inclusions between symmetric sequence spaces are studied. Suitable conditions are provided involving the associated fundamental functions. The special case of Lorentz and Marcinkiewicz spaces is characterized. It is also proved that if E?F are symmetric sequence spaces with E≠?₁ and F≠c₀ and ?_∞ then there exist a intermediate symmetric sequence space G such that E?G?F and both inclusions are not strictly singular. As a consequence new characterizations of the spaces c₀ and ?₁ inside the class of all symmetric sequence spaces are given.     

On the vanishing of subspaces of alternating bilinear forms

Given a field F and integer n≥3, we introduce an invariant s_n (F) which is defined by examining the vanishing of subspaces of alternating bilinear forms on 2-dimensional subspaces of vector spaces. This invariant arises when we calculate the largest dimension of a subspace of n?×?n skew-symmetric matrices over F which contains no elements of rank 2. We show how to calculate s_n (F) for various families of field F, including finite fields. We also prove the existence of large subgroups of the commutator subgroup of certain p-groups of class 2 which contain no non-identity commutators.     

On some classes of algebraic surfaces with an infinite set of subspaces of skew symmetry

Special classes of (m – 1)-dimensional algebraic surfaces F ⁿ in a space E^m with inifinite set ofsubspaces of skew symmetry (in particular, orthogonal) are studied. It is assumed that directions of symmetry, as a rule, are asymptotic for F _n .Translated from Dinamicheskie Sistemy, No. 8, pp. 119–126, 1989.     

Projecting <Emphasis Type="Italic">l</Emphasis><Subscript>∞</Subscript> onto Classical Spaces

We describe an explicit construction of a linear projection of a symmetric conical section of the n-dimensional cube onto a (1+ε)-isomorphic version of the Euclidean ball of proportional dimension, or more generally onto a (1+ε)-isomorphic image of an l _p ^m -ball. Allowing non-linear projections (of logarithmic polynomial nonlinearity) we may even project the full n-dimensional cube onto the same images. This is done by gluing together explicit projections onto two-dimensional spaces, interpreting and modifying a paper of Ben-Tal and Nemirowski on polynomial reductions of conic quadratic programming problems to linear programming problems in terms of Banach spaces.      

Test elements for endomorphisms of free groups and algebras

LetE be a bounded Borel subset of ℝⁿ,n≥2, of positive Lebesgue measure andP _E the corresponding ‘Pompeiu transform”. We prove thatP _E is injective onL ^p(ℝⁿ) if 1≤p≤2n/(n-1). We explore the connection between this problem and a Wiener-Tauberian type theorem for theM(n) action onL ^q(ℝⁿ) for various values ofq. We also take up the question of whenP _E is injective in caseE is of finite, positive measure, but is not necessarily a bounded set. Finally, we briefly look at these questions in the contexts of symmetric spaces of compact and non-compact type.     

On ℤp-graded identities and cocharacters of the Grassmann algebra

Let p be an odd prime number, F a field of characteristic zero, and let Ebe the unitary Grassmann algebra generated by the infinite-dimensionalF-vector space L. We determine the bases of the ?_p-graded identities.Moreover we compute the ?_p-graded codimension and cocharacter sequences for the algebra E endowed with any ?_p-grading such that L is a homogeneous subspace.     

Banach ideals of operators with applications to the finite dimensional structure of Banach spaces

We present a few applications of the theory of Banach ideals of operators. In particular, we give operator characterizations of the ℒ _p spaces, compute the relative projection constant of isometric embeddings of Hilbert spaces inL _p -spaces, and show that Π₁ (E, F), the space of absolutely summing operators, is reflexive ifE andF are reflexive andE has the approximation property. Research supported by NSF-GP-34193 Research supported by NSF-Science Development Grant     

Differentiability points of functions in weighted Sobolev spaces

We consider weighted Sobolev spaces W _p ^l , l ∈ ?, with weighted L _p -norm of higher derivatives on an n-dimensional cube-type domain. The weight γ depends on the distance to an (n ? d)-dimensional face E of the cube. We establish the property of uniform L _p -differentiability of functions in these spaces on the face E of an appropriate dimension. This property consists in the possibility of L _p -approximation of the values of a function near E by a polynomial of degree l ? 1.     

POLYNOMIAL IDENTITIES ON SUPERALGEBRAS AND ALMOST POLYNOMIAL GROWTH

Let A be a superalgebra over a field of characteristic zero. In this paper we investigate the graded polynomial identities of A through the asymptotic behavior of a numerical sequence called the sequence of graded codimensions of A. Our main result says that such sequence is polynomially bounded if and only if the variety of superalgebras generated by A does not contain a list of five superalgebras consisting of a 2-dimensional algebra, the infinite dimensional Grassmann algebra and the algebra of 2 × 2 upper triangular matrices with trivial and nontrivial gradings. Our main tool is the representation theory of the symmetric group.     

Bounds for Relative Projection Constants in L 1 (-1,1)

For n -dimensional subspaces E _n , F _n of L ¹ (-1,1) with E _n spanned by Chebyshev polynomials of the second kind and F _n the set of Müntz polynomials with , , it is shown that the relative projection constants satisfy (E _n , L ¹ (-1,1)) C log n and (F _n , L ¹ (-1,1)) = O(1) , . The spaces L ¹ _w(α,β) , where w _α,β is the weight function of the Jacobi polynomials and , are also studied. The Jacobi partial sum projections, which are used in connection with E _n , are not minimal. September 26, 1996.     

Cyclic Tableaux and Symmetric Functions

We introduce the notion of cyclic tableaux and develop involutions for Waring's formulas expressing the power sum symmetric function p_n in terms of the elementary symmetric function e_n and the homogeneous symmetric function h_n. The coefficients appearing in Waring's formulas are shown to be a cyclic analog of the multinomial coefficients, a fact that seems to have been neglected before. Our involutions also spell out the duality between these two forms of Waring's formulas, which turns out to be exactly the “duality between sets and multisets.” We also present an involution for permutations in cycle notation, leading to probably the simplest combinatorial interpretation of the Möbius function of the partition lattice and a purely combinatorial treatment of the fundamental theorem on symmetric functions. This paper is motivated by Chebyshev polynomials in connection with Waring's formula in two variables.     

A recursive construction for universal cycles of 2-subspaces

We prove that universal cycles of 2-dimensional subspaces of vector spaces over any finite field F exist, i.e., if V is a finite-dimensional vector space over F, there is a cycle of vectors v₁,v₂,…,v_n such that each 2-dimensional subspace of V occurs exactly once as the span of consecutive vectors.     

The distance between certainn-dimensional Banach spaces

IfE andF aren-dimensional Banach spaces, ifE has cotype 2, and if the ball ofF* has a small number of extreme points, then the Banach-Mazur distanced(E, F)≦C √nlogn. The techniques lead to the formally stronger result: IfE andF* have type 2 constantsa andb, respectively, thend(E, F)≦√n(a+b). IfE isn-dimensional, the identity map onE, when restricted to a large subspace ofE, factors through with normC √n. The authors’ work was supported in part, respectively, by NSF grant numbers MCS 78-02194, MCS 79-02489, and MCS 77-04174.     

Differential Polynomial Identities of Upper Triangular Matrices Under the Action of the Two-Dimensional Metabelian Lie Algebra

We study the differential polynomial identities of the algebra UT_m(F) under the derivation action of the two dimensional metabelian Lie algebra, obtaining a generating set of the T_L-ideal they constitute. Then we determine the S_n-structure of their proper multilinear spaces and, for the minimal cases m =?2, 3, their exact differential codimension sequence.

     

The Banach-Mazur distance between symmetric spaces

We show that the Banach-Mazur distance betweenN-dimensional symmetric spacesE andF satisfies , wherec is a numerical constant. IfE is a symmetric space, then max (M ⁽²⁾(E),M ₍₂₎(E)), whereM ⁽²⁾(E) (resp.M ₍₂₎(E)) denotes the 2-convexity (resp. the 2-concavity) constant ofE. We also give an example of a spaceF with an 1-unconditional basis and enough symmetries that satisfiesd(F, l ₂ ^dimF )=M ⁽²⁾(F)M ₍₂₎(F). Partially supported by NSF Grant MCS-8201044.