Strongly elliptic operators and exponentiation of operator Lie algebras

2021 ◽  
Vol 127 (2) ◽  
pp. 264-286
Author(s):  
Rodrigo A. H. M. Cabral
Keyword(s):  
Lie Algebras ◽  
Elliptic Operators ◽  
Linear Operators ◽  
Enveloping Algebra ◽  
Arbitrary Operator ◽  
Unbounded Linear Operators

An intriguing feature which is often present in theorems regardingthe exponentiation of Lie algebras of unbounded linear operators onBanach spaces is the assumption of hypotheses on the Laplacianoperator associated with a basis of the operator Lie algebra.The main objective of this work is to show that one can substitutethe Laplacian by an arbitrary operator in the enveloping algebra andstill obtain exponentiation, as long as its closure generates astrongly continuous one-parameter semigroup satisfying certain normestimates, which are typical in the theory of strongly ellipticoperators.

Normed Lie Algebras

1972 ◽  
Vol 24 (4) ◽  
pp. 580-591 ◽  
Author(s):  
F.-H. Vasilescu
Keyword(s):  
Lie Algebra ◽  
Lie Algebras ◽  
Normed Space ◽  
Linear Operators ◽  
Continuous Linear ◽  
Normed Algebra ◽  
Normed Spaces ◽  
Enveloping Algebra ◽  
Algebraic Concept ◽  
Continuous Linear Operators

In this paper we attempt to define the notion of normed Lie algebra by endowing the corresponding algebraic concept with topologicalmetric properties. More precisely, we define normed Lie algebras as being normed spaces possessing a Lie product, the latter satisfying a compatibility relation. It turns out that any normed algebra, in particular the algebra of continuous linear operators on a normed space, is a normed Lie algebra in the sense denned below, with the usual Lie product given by the additive commutator.It seems that some algebraic features have within the framework of normed Lie algebras the natural topological extensions. We mention, for instance, the convergence of the well-known Campbell-Hausdorff formula. Let us also mention the occurence of a variant of the Kleinecke-Sirokov theorem, obtained via universal enveloping algebra, which might be unknown in this context.

Unbounded Linear Operators

2018 ◽  
Author(s):  
D. E. Edmunds ◽  
W. D. Evans
Keyword(s):  
Hilbert Spaces ◽  
Linear Operators ◽  
Von Neumann ◽  
Limit Circle ◽  
Sectorial Operators ◽  
Closed Operators ◽  
The Stability ◽  
Symmetric Operators ◽  
Unbounded Linear Operators ◽  
Special Case

This chapter is concerned with closable and closed operators in Hilbert spaces, especially with the special classes of symmetric, J-symmetric, accretive and sectorial operators. The Stone–von Neumann theory of extensions of symmetric operators is treated as a special case of results for compatible adjoint pairs of closed operators. Also discussed in detail is the stability of closedness and self-adjointness under perturbations. The abstract results are applied to operators defined by second-order differential expressions, and Sims’ generalization of the Weyl limit-point, limit-circle characterization for symmetric expressions to J-symmetric expressions is proved.

scholarly journals KONTSEVICH'S UNIVERSAL FORMULA FOR DEFORMATION QUANTIZATION AND THE CAMPBELL–BAKER–HAUSDORFF FORMULA

2000 ◽  
Vol 11 (04) ◽  
pp. 523-551 ◽  
Author(s):  
VINAY KATHOTIA
Keyword(s):  
Lie Algebras ◽  
Deformation Quantization ◽  
Differential Operators ◽  
Main Tool ◽  
Graphical Analysis ◽  
Poisson Structures ◽  
Nilpotent Lie Algebras ◽  
Enveloping Algebra ◽  
Universal Formula ◽  
The Given

We relate a universal formula for the deformation quantization of Poisson structures (⋆-products) on ℝd proposed by Maxim Kontsevich to the Campbell–Baker–Hausdorff (CBH) formula. We show that Kontsevich's formula can be viewed as exp (P) where P is a bi-differential operator that is a deformation of the given Poisson structure. For linear Poisson structures (duals of Lie algebras) his product takes the form exp (C+L) where exp (C) is the ⋆-product given by the universal enveloping algebra via symmetrization, essentially the CBH formula. This is established by showing that the two products are identical on duals of nilpotent Lie algebras where the operator L vanishes. This completely determines part of Kontsevich's formula and leads to a new scheme for computing the CBH formula. The main tool is a graphical analysis for bi-differential operators and the computation of certain iterated integrals that yield the Bernoulli numbers.

scholarly journals The numerical ranges of unbounded linear operators

1975 ◽  
Vol 12 (1) ◽  
pp. 23-25 ◽  
Author(s):  
Béla Bollobás ◽  
Stephan E. Eldridge
Keyword(s):  
Banach Space ◽  
Scalar Field ◽  
Banach Spaces ◽  
Unbounded Operator ◽  
Numerical Range ◽  
Linear Operators ◽  
Density Property ◽  
Unbounded Linear Operators

Giles and Joseph (Bull. Austral. Math. Soc. 11 (1974), 31–36), proved that the numerical range of an unbounded operator on a Banach space has a certain density property. They showed, in particular, that the numerical range of an unbounded operator on certain Banach spaces is dense in the scalar field. We prove that the numerical range of an unbounded operator on a Banach space is always dense in the scalar field.

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