scholarly journals The Bourguignon Laplacian and Harmonic Symmetric Bilinear Forms

Mathematics ◽  
2020 ◽  
Vol 8 (1) ◽  
pp. 83
Author(s):  
Vladimir Rovenski ◽  
Sergey Stepanov ◽  
Irina Tsyganok
Keyword(s):  
Riemannian Manifold ◽  
Vector Space ◽  
Spectral Properties ◽  
Cotangent Bundle ◽  
Bilinear Forms ◽  
Dimensional Vector ◽  
Surface Geometry ◽  
Dimensional Vector Space ◽  
Infinite Dimensional ◽  
Negative Sectional Curvature

In this paper, we study the kernel and spectral properties of the Bourguignon Laplacian on a closed Riemannian manifold, which acts on the space of symmetric bilinear forms (considered as one-forms with values in the cotangent bundle of this manifold). We prove that the kernel of this Laplacian is an infinite-dimensional vector space of harmonic symmetric bilinear forms, in particular, such forms on a closed manifold with quasi-negative sectional curvature are zero. We apply these results to the description of surface geometry.

scholarly journals Baer–Levi semigroups of linear transformations

2004 ◽  
Vol 134 (3) ◽  
pp. 477-499 ◽  
Author(s):  
Suzana Mendes-Gonçalves ◽  
R. P. Sullivan
Keyword(s):  
Vector Space ◽  
Dimensional Vector ◽  
Linear Transformations ◽  
Dimensional Vector Space ◽  
Infinite Dimensional ◽  
Basic Properties ◽  
Maximal Subsemigroups ◽  
Linear Version ◽  
Infinite Set

Given an infinite-dimensional vector space V, we consider the semigroup GS (m, n) consisting of all injective linear α: V → V for which codim ran α = n, where dim V = m ≥ n ≥ ℵ0. This is a linear version of the well-known Baer–Levi semigroup BL (p, q) defined on an infinite set X, where |X| = p ≥ q ≥ ℵ0. We show that, although the basic properties of GS (m, n) are the same as those of BL (p, q), the two semigroups are never isomorphic. We also determine all left ideals of GS (m, n) and some of its maximal subsemigroups; in this, we follow previous work on BL (p, q) by Sutov and Sullivan as well as Levi and Wood.

Automorphisms of supermaximal subspaces

1985 ◽  
Vol 50 (1) ◽  
pp. 1-9 ◽  
Author(s):  
R. G. Downey ◽  
G. R. Hird
Keyword(s):  
Vector Space ◽  
Recursive Relation ◽  
Effective Procedure ◽  
Dimensional Vector ◽  
Algebraic Systems ◽  
Dimensional Vector Space ◽  
Infinite Dimensional ◽  
Vector Addition ◽  
State Construction ◽  
Recursive Set

An infinite-dimensional vector space V∞ over a recursive field F is called fully effective if V∞ is a recursive set identified with ω upon which the operations of vector addition and scalar multiplication are recursive functions, identity is a recursive relation, and V∞ has a dependence algorithm, that is a uniformly effective procedure which when applied to x, a1,…,an, ∈ V∞ determines whether or not x is an element of {a1,…,an}* (the subspace generated by {a1,…,an}). The study of V∞, and of its lattice of r.e. subspaces L(V∞), was introduced in Metakides and Nerode [15]. Since then both V∞ and L(V∞) (and many other effective algebraic systems) have been studied quite intensively. The reader is directed to [5] and [17] for a good bibliography in this area, and to [15] for any unexplained notation and terminology.In [15] Metakides and Nerode observed that a study of L(V∞) may in some ways be modelled upon a study of L(ω), the lattice of r.e. sets. For example, they showed how an e-state construction could be modified to produce an r.e. maximal subspace, where M ∈ L(V∞) is maximal if dim(V∞/M) = ∞ and, for all W ∈ L(V∞), if W ⊃ M then either dim(W/M) < ∞ or dim(V∞/W) < ∞.However, some of the most interesting features of L(V∞) are those which do not have analogues in L(ω). Our concern here, which is probably one of the most striking characteristics of L(V∞), falls into this category. We say M ∈ L(V∞) is supermaximal if dim(V∞/M) = ∞ and for all W ∈ L(V∞), if W ⊃ M then dim(W/M) < ∞ or W = V∞. These subspaces were discovered by Kalantari and Retzlaff [13].

Sums of Quadratic Endomorphisms of an Infinite-Dimensional Vector Space

2018 ◽  
Vol 61 (2) ◽  
pp. 437-447 ◽  
Author(s):  
Clément de Seguins Pazzis
Keyword(s):  
Vector Space ◽  
Dimensional Vector ◽  
Dimensional Vector Space ◽  
Infinite Dimensional ◽  
Four Square

AbstractWe prove that every endomorphism of an infinite-dimensional vector space over a field splits into the sum of four idempotents and into the sum of four square-zero endomorphisms, a result that is optimal in general.

Products of Involutions of an Infinite-dimensional Vector Space

2019 ◽  
pp. 1-26
Author(s):  
Clément de Seguins Pazzis
Keyword(s):  
Vector Space ◽  
Dimensional Vector ◽  
Dimensional Vector Space ◽  
Infinite Dimensional

Abstract We prove that every automorphism of an infinite-dimensional vector space over a field is the product of four involutions, a result that is optimal in the general case. We also characterize the automorphisms that are the product of three involutions. More generally, we study decompositions of automorphisms into three or four factors with prescribed split annihilating polynomials of degree  $2$ .

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