Operator pencil passing through a given operator

Biggs, A.; Khudaverdian, Hovhannes M.

doi:10.1063/1.4839418

Cited by 6 publications

(18 citation statements)

References 13 publications

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“…Proof. Due to the previous lemma and Theorem 1.1 About the recent investigations of operator pencils see the interesting papers [1,4,6,7,17,18] (see also [10,11,12]). …”

Section: Applications To Pencils Let B and C Be Operators Acting In mentioning

confidence: 96%

On inequalities for eigenvalues of 2 × 2 matrices with Schatten–von Neumann entries

Gil’

2015

Quaestiones Mathematicae

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“…Proof. Due to the previous lemma and Theorem 1.1 About the recent investigations of operator pencils see the interesting papers [1,4,6,7,17,18] (see also [10,11,12]). …”

Section: Applications To Pencils Let B and C Be Operators Acting In mentioning

confidence: 96%

On inequalities for eigenvalues of 2 × 2 matrices with Schatten–von Neumann entries

Gil’

2015

Quaestiones Mathematicae

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“…It is illuminating to consider the following example. Let X, Y be two vector fields on a manifold M. Consider the second order operator ∆ = L X L Y ∈ D (2) λ where L X L Y are Lie derivatives of densities of weight λ. We wish to calculate the image of this operator under the isomorphism Π µ λ .…”

Section: Pencil Liftings Of Second Order Operatorsmentioning

confidence: 99%

“…To make the problem well-defined we put restrictions on the pencil lifting map. We consider regular pencil lifting maps which are equivariant with respect to some Lie subalgebra of vector fields [2]. Let us briefly recall what this means.…”

Section: Introductionmentioning

confidence: 99%

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Operator pencils on the algebra of densities

Biggs

Khudaverdian

2014

Proc. Steklov Inst. Math.

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Abstract. In this paper we continue to study equivariant pencil liftings and differential operators on the algebra of densities. We emphasize the role that the geometry of the extended manifold plays where the algebra of densities is a special class of functions. Firstly we consider basic examples. We give a projective line of diff (M )-equivariant pencil liftings for first order operators, and the canonical second order self-adjoint lifting. Secondly we study pencil liftings equivariant with respect to volume preserving transformations. This helps to understand the role of self-adjointness for the canonical pencils. Then we introduce the Duval-Lecomte-Ovsienko (DLO)-pencil lifting which is derived from the full symbol calculus of projective quantisation. We use the DLO-pencil lifting to describe all regular proj -equivariant pencil liftings. In particular the comparison of these pencils with the canonical pencil for second order operators leads to objects related to the Schwarzian.

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“…The geometry of operator (5) was studied in detail in articles [2,3,4]. Here we present and analyze these results, using Kaluza-Klein-like mechanism.…”

Section: Second Order Operators and Kaluza-klein Reductionmentioning

confidence: 99%

Kaluza-Klein theory revisited: projective structures and differential operators on algebra of densities

Khudaverdian

2014

J. Phys.: Conf. Ser.

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Abstract. We consider differential operators acting on densities of arbitrary weights on manifold M identifying pencils of such operators with operators on algebra of densities of all weights. This algebra can be identified with the special subalgebra of functions on extended manifold M . On one hand there is a canonical lift of projective structures on M to affine structures on extended manifold M . On the other hand the restriction of algebra of all functions on extended manifold to this special subalgebra of functions implies the canonical scalar product. This leads in particular to classification of second order operators with use of Kaluza-Klein-like mechanisms. Algebra of densitiesIn mathematical physics it is very useful to consider differential operators acting on densities of various weights on a manifold M (see [1] and citations there). We say that s = s(x)|Dx| λ is a density of weight λ on M if under change of local coordinates(λ is an arbitrary real number). We denote by F λ (M ) the space of densities of weight λ on manifold M . (The space of functions on M is F 0 (M ), densities of weight λ = 0.) Densities can be multiplied. If s 1 = s 1 (x)|Dx| λ 1 and s 2 = s 2 (x)|Dx| λ 2 are densities of weights λ 1 , λ 2 respectively then s = s 1 · s 2 = s 1 (x)s 2 (x)|Dx| λ 1 +λ 2 is a density of weight λ 1 + λ 2 . We come to the algebra F(M ) = ⊕ λ F λ (M ) of finite linear combinations of densities of arbitrary weights. Use a formal variable t instead volume form |Dx|. Thus an arbitrary density s = s 1 (x)|Dx| λ 1 + · · · + s k |Dx| λ k can be written as a function on x, t which is quasipolynomial on t, s(x, t) = s 1 (x)t λ 1 + · · · + s k (x)t λ k . An arbitrary density s ∈ F(M ) can be identified with function s r (x)t λr on the extended manifold M , which is quasipolynomial on 'vertical' variable t. There is a natural fibre bundle structure M → M . Extended manifold M is the frame bundle of the determinant bundle of M , (x i , t) are local coordinates on M . Changing of local coordinates is:The fibre bundle M → M can be used for studying projective geometry on M since there is a canonical construction which assigns to an arbitrary projective connection on manifold M an

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Operator pencil passing through a given operator

Cited by 6 publications

References 13 publications

On inequalities for eigenvalues of 2 × 2 matrices with Schatten–von Neumann entries

On inequalities for eigenvalues of 2 × 2 matrices with Schatten–von Neumann entries

Operator pencils on the algebra of densities

Kaluza-Klein theory revisited: projective structures and differential operators on algebra of densities

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