Isometries between spaces of homogeneous polynomials

Boyd, Christopher; Lassalle, Silvia

doi:10.1016/j.jfa.2005.01.009

Cited by 9 publications

(14 citation statements)

References 41 publications

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“…This characterisation of the isometries of spaces of approximable polynomials extends to complex Banach space with an additional geometric assumption. Our work to date on the centraliser of the space of symmetric injective tensor products allows us to extend the results in Section 3 of [5] to the vector-valued case. We will use the fact that given Banach spaces E and F every P in P( n E; F ) has a norm-preserving extension to P in P( n E ; F ), see [2].…”

Section: ξ+η) (A Similar Argument Work If R(ξ+η) = −B(ξ+η)(ξ+η))mentioning

confidence: 98%

“…In [5] we studied the (surjective) isometries of spaces of scalar-valued homogeneous polynomials. We showed that if E and F are real Banach spaces, n is a positive integer and T is an isometric isomorphism from…”

Section: ξ+η) (A Similar Argument Work If R(ξ+η) = −B(ξ+η)(ξ+η))mentioning

confidence: 99%

“…The first possibility is ruled out and hence T must have the form given in (b). From [5,Theorem 7] it follows that each isometry T 1 of P A ( n E) has the form…”

Section: ξ+η) (A Similar Argument Work If R(ξ+η) = −B(ξ+η)(ξ+η))mentioning

confidence: 99%

“…We shall use P f * ( n E ) to denote all n-homogeneous polynomials P on E of the form k j=1 x n j where x j ∈ E. The closure of P f * ( n E ) in P( n E ) is denoted by P A * ( n E ) and is called the spaces of weak * approximable n-homogeneous polynomials. The space P A * ( n E ) is isometrically isomorphic to n,s, E. The following terminology was introduced in [5].…”

Section: The Centraliser Of P a ( N E) When E Is Complexmentioning

confidence: 99%

“…By [5,Proposition 14] it follows that T is a power-preserver. We claim that T (x n ) is a scalar multiple of x n for each x ∈ E. Suppose, that this is not true.…”

Section: (ϕ)/S(ϕ)mentioning

confidence: 99%

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Centralisers of spaces of symmetric tensor products and applications

Boyd

Lassalle

2006

Math. Z.

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We show that the centraliser of the space of n-fold symmetric injective tensors, n ≥ 2, on a real Banach space is trivial. With a geometric condition on the set of extreme points of its dual, the space of integral polynomials we obtain the same result for complex Banach spaces. We give some applications of this results to centralisers of spaces of homogeneous polynomials and complex Banach spaces. In addition, we derive a Banach-Stone Theorem for spaces of vector-valued approximable polynomials.

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Section: ξ+η) (A Similar Argument Work If R(ξ+η) = −B(ξ+η)(ξ+η))mentioning

confidence: 98%

Section: ξ+η) (A Similar Argument Work If R(ξ+η) = −B(ξ+η)(ξ+η))mentioning

confidence: 99%

“…The first possibility is ruled out and hence T must have the form given in (b). From [5,Theorem 7] it follows that each isometry T 1 of P A ( n E) has the form…”

Section: ξ+η) (A Similar Argument Work If R(ξ+η) = −B(ξ+η)(ξ+η))mentioning

confidence: 99%

Section: The Centraliser Of P a ( N E) When E Is Complexmentioning

confidence: 99%

“…By [5,Proposition 14] it follows that T is a power-preserver. We claim that T (x n ) is a scalar multiple of x n for each x ∈ E. Suppose, that this is not true.…”

Section: (ϕ)/S(ϕ)mentioning

confidence: 99%

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Centralisers of spaces of symmetric tensor products and applications

Boyd

Lassalle

2006

Math. Z.

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Decomposable symmetric mappings between infinite-dimensional spaces

Boyd

Lassalle

2008

Ark. Mat.

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Decomposable mappings from the space of symmetric k-fold tensors over E, s,k E, to the space of k-fold tensors over F , s,k F , are those linear operators which map nonzero decomposable elements to nonzero decomposable elements. We prove that any decomposable mapping is induced by an injective linear operator between the spaces on which the tensors are defined. Moreover, if the decomposable mapping belongs to a given operator ideal, then so does its inducing operator. This result allows us to classify injective linear operators between spaces of homogeneous approximable polynomials and between spaces of nuclear polynomials which map rank-1 polynomials to rank-1 polynomials.

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