Multimatrix Polyalgebra Representations of the Polar Composition of Polynomial Operators

Amson, J C

doi:10.1112/plms/s3-25.3.465

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“…[2], and the details in the proof of Fact (4.5) below, etc.). Provided, of course that the tensor χ i, χ (2) ijk is symmetric in j, k. In the case of χ (2) ijk this is in fact true. In the case of χ (3) ijk a stronger form of symmetry obtains, namely 'full symmetry' viz.…”

Section: =Pmentioning

confidence: 99%

“…in the case of P (2) , P (2) i (t) = 0 jk χ (2) ijk E j (t) E k (t) reveals that the polarisation P (2) is a homogeneous operator of degree 2, for which the above coordinate-wise expression in its multimatrix representation (apart from some minor notational differences) (see e.g. [2], and the details in the proof of Fact (4.5) below, etc.). Provided, of course that the tensor χ i, χ (2) ijk is symmetric in j, k. In the case of χ (2) ijk this is in fact true.…”

Section: =Pmentioning

confidence: 99%

“…[2,10]) that to each each homogeneous polynomial operator hp of degree α > 1 on a Hilbert space H, there corresponds a unique symmetric multilinear operator sm of degree α and a unique linear operator such that the following diagram is fully commutative.…”

Section: =Pmentioning

confidence: 99%

“…It is familiar (see e.g. [1], [2]) that the degree of the functional composition of two homogeneous polynomial operators is the product (not the sum) of their degrees.…”

Section: Contextmentioning

confidence: 99%

“…Whence, by the uniqueness of the Fourier representation with respect to the basis (2) , and [ A ] follows.…”

Section: We First Show That [mentioning

confidence: 99%

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Unital Homogeneous Polynomial Operators on Hilbert Space

Amson

2013

Scientific Essays in Honor of H Pierre Noyes on the Occasion of His 90th Birthday

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For Pierre Noyes on his 90-th birthday, in esteem and friendship.The familiar notion of the linear unit operator on a Hilbert space is extended to its homogeneous polynomial operator analogues of arbitrary degree : 'unital' homogeneous polynomial operators, whose multimatrix representations are identified with generalized Kronecker symbols (tensors). Some properties are lost, many other novel structural properties are established.The set of unital operators of all degrees on a Hilbert space constitutes a unique object, a 'Unital System'. It is unit-normed, closed and commutative under functional composition, and closed under functional polar composition. It is a multiplicatively and additively graded set, graded by the positive integer subset of the integer ring. It is the disjoint union of its homogeneous component 'grade' subsets each comprised of an equivalence class (under unitary similarity) of unital operators of same degree α ≥ 1, the first grade subset being the only one itself closed under composition.Some operator equations of Fredholm and Characteristic type involving unital homogeneous polynomial operators are illustrated, their solubility described, and connections with an emerging theory of the spectrum of tensors and homogeneous polynomial operators are noted.Possible applications to the theory of nonlinear phenomena in the presence of very high field strengths are indicated. Scientific Essays inHonor of H Pierre Noyes on the Occasion of His 90th Birthday Downloaded from www.worldscientific.com by NANYANG TECHNOLOGICAL UNIVERSITY on 08/25/15. For personal use only. (2) ijk (which itself is the representation of a multilinear operator (see [12]) obeys a certain symmetry. The required symmetry is what I have called [2] 'pen-symmetry' viz. for each index Scientific Essays in Honor of H Pierre Noyes on the Occasion of His 90th Birthday Downloaded from www.worldscientific.com by NANYANG TECHNOLOGICAL UNIVERSITY on 08/25/15. For personal use only.

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Section: =Pmentioning

confidence: 99%

Section: =Pmentioning

confidence: 99%

Section: =Pmentioning

confidence: 99%

“…It is familiar (see e.g. [1], [2]) that the degree of the functional composition of two homogeneous polynomial operators is the product (not the sum) of their degrees.…”

Section: Contextmentioning

confidence: 99%

“…Whence, by the uniqueness of the Fourier representation with respect to the basis (2) , and [ A ] follows.…”

Section: We First Show That [mentioning

confidence: 99%

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Unital Homogeneous Polynomial Operators on Hilbert Space

Amson

2013

Scientific Essays in Honor of H Pierre Noyes on the Occasion of His 90th Birthday

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A Hilbert algebra of Hilbert-Schmidt quadratic operators

Amson

Reddy

1990

Bull. Austral. Math. Soc.

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A quadratic operator Q of Hilbert-Schmidt class on a real separable Hilbert space H is shown to be uniquely representable as a sequence \L ) of self-adjoint linear operators of Hilbert-Schmidt class on H, such that Q(x) -£) fc (L k x, x)ui with respect to a Hilbert basis (wk)t 6 j, {I Q N). It is shown that with the normand inner-product {((Q , P))) = £ t {{L k , M l » , together with a multiplication denned componentwise through the composition of the linear components, the vector space of all Hilbert-Schmidt quadratic operators Q on H becomes a linear H*-algebra containing an ideal of nuclear (trace class) quadratic operators. In the finite dimensional case, each Q is also shown to have another representation as a block-diagonal matrix of Hilbert-Schmidt class which simplifies the practical computation and manipulation of quadratic operators. INTRODUCTIONThe purpose of this paper is to show how a space of quadratic operators (that is homogeneous polynomial operators of degree 2, (see for example [3,4,7,8]) can be given a multiplication under which it becomes a linear algebra. The construction is new in that it does not make use of the composition of quadratic operators, nor of the composition of the linear maps associated with the tensor product of the quadratic operator's domain. The former attempt fails because the composition of two quadratic operators is generally of degree 4. The latter fails because even though a quadratic operator from one vector space E to another F is associated with a unique symmetric bilinear operator on the cartesian product E x E^*F and hence with a unique linear operator on the tensor product E®E-*F, those linear operators may only be composed in the extremely unusual case where F = E® E. More elaborate nonlinear algebras of homogeneous polynomial operators do exist of course [2], but these form a very different topic from the one studied here.

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Linear Ω-Algebras

Baranovich¹,

Burgin²

1975

Russ. Math. Surv.

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The light emitted when pion irradiated glucose is dissolved in luminol solution has been found to be proportional to the pion beam depth dose distribution in water as determined by a TE ionisation chamber. The lyoluminescence of glucose overlaps very closely with the response profile of the ionisation chamber to the 170 MeV/c Ir--mesons giving a Bragg peak to plateau ratio of 3 : 1. In comparison, the thermoluminescence response of LiF (TLD-700) to pions has been found to deviate significantly from this ratio. The close tissue equivalence of glucose, non-toxicity and its excellent lyoluminescent retention properties are important advantages over currently used dosemeters in clinical pion therapy, especially when direct in v i m measurements are considered.Pion doses ranging between 0.5 Gy (50 rad) and 30 Gy (3000 rad) were measured with an accuracy *s0/o and reproducibility 3-5%.

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Multimatrix Polyalgebra Representations of the Polar Composition of Polynomial Operators

Cited by 5 publications

References 1 publication

Unital Homogeneous Polynomial Operators on Hilbert Space

Unital Homogeneous Polynomial Operators on Hilbert Space

A Hilbert algebra of Hilbert-Schmidt quadratic operators

Linear Ω-Algebras

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