Search results for "Multilinear map"
showing 10 items of 35 documents
A Star-Variety With Almost Polynomial Growth
2000
Abstract Let F be a field of characteristic zero. In this paper we construct a finite dimensional F -algebra with involution M and we study its ∗ -polynomial identities; on one hand we determine a generator of the corresponding T -ideal of the free algebra with involution and on the other we give a complete description of the multilinear ∗ -identities through the representation theory of the hyperoctahedral group. As an outcome of this study we show that the ∗ -variety generated by M , var( M , ∗ ) has almost polynomial growth, i.e., the sequence of ∗ -codimensions of M cannot be bounded by any polynomial function but any proper ∗ -subvariety of var( M , ∗ ) has polynomial growth. If G 2 is…
Graded algebras with polynomial growth of their codimensions
2015
Abstract Let A be an algebra over a field of characteristic 0 and assume A is graded by a finite group G . We study combinatorial and asymptotic properties of the G -graded polynomial identities of A provided A is of polynomial growth of the sequence of its graded codimensions. Roughly speaking this means that the ideal of graded identities is “very large”. We relate the polynomial growth of the codimensions to the module structure of the multilinear elements in the relatively free G -graded algebra in the variety generated by A . We describe the irreducible modules that can appear in the decomposition, we show that their multiplicities are eventually constant depending on the shape obtaine…
A multilinear Phelps' Lemma
2007
We prove a multilinear version of Phelps' Lemma: if the zero sets of multilinear forms of norm one are 'close', then so are the multilinear forms.
On ideals of polynomials and multilinear mappings between Banach spaces
2003
It is shown that for every quasi-normed ideal ${\cal Q}$ of n-homogeneous continuous polynomials between Banach spaces there is a quasi-normed ideal ${\cal A}$ of n-linear continuous mappings ${\cal A}$ such that $q \in {\cal Q}$ if and only if the associated n-linear mapping $\check{q}$ of q is in ${\cal A}$.
Weakly compact multilinear mappings
1997
The notion of Arens regularity of a bilinear form on a Banach space E is extended to continuous m-linear forms, in such a way that the natural associated linear mappings, E→L (m−1E) and (m – l)-linear mappings E × … × E → E', are all weakly compact. Among other applications, polynomials whose first derivative is weakly compact are characterized.
A multilinear Lindenstrauss theorem
2006
Abstract We show that the set of N -linear mappings on a product of N Banach spaces such that all their Arens extensions attain their norms (at the same element) is norm dense in the space of all bounded N -linear mappings.
Domination spaces and factorization of linear and multilinear summing operators
2015
[EN] It is well known that not every summability property for multilinear operators leads to a factorization theorem. In this paper we undertake a detailed study of factorization schemes for summing linear and nonlinear operators. Our aim is to integrate under the same theory a wide family of classes of mappings for which a Pietsch type factorization theorem holds. Our construction includes the cases of absolutely p-summing linear operators, (p, sigma)-absolutely continuous linear operators, factorable strongly p-summing multilinear operators, (p(1), ... , p(n))-dominated multilinear operators and dominated (p(1), ... , p(n); sigma)-continuous multilinear operators.
Restricted weak type on maximal linear and multilinear integral maps
2006
It is shown that multilinear operators of the form T ( f 1 , . . . , f k ) ( x ) T(f_1,...,f_k)(x) = ∫ R n K ( x , y 1 , . . . , y k ) f 1 ( y 1 ) . . . f k ( y k ) d y 1 . . . d y k =\!\int _{\mathbb {R}^n}\!K(x,y_1,...,y_k)f_1(y_1)... f_k(y_k)dy_1...dy_k of restricted weak type ( 1 , . . . , 1 , q ) (1,...,1,q) are always of weak type ( 1 , . . . , 1 , q ) (1,...,1,q) whenever the map x → K x x\to K_x is a locally integrable L 1 ( R n ) L^1(\mathbb {R}^n) -valued function.
Summability and estimates for polynomials and multilinear mappings
2008
Abstract In this paper we extend and generalize several known estimates for homogeneous polynomials and multilinear mappings on Banach spaces. Applying the theory of absolutely summing nonlinear mappings, we prove that estimates which are known for mappings on l p spaces in fact hold true for mappings on arbitrary Banach spaces.
On the Bishop–Phelps–Bollobás theorem for multilinear mappings
2017
Abstract We study the Bishop–Phelps–Bollobas property and the Bishop–Phelps–Bollobas property for numerical radius. Our main aim is to extend some known results about norm or numerical radius attaining operators to multilinear and polynomial cases. We characterize the pair ( l 1 ( X ) , Y ) to have the BPBp for bilinear forms and prove that on L 1 ( μ ) the numerical radius and the norm of a multilinear mapping are the same. We also show that L 1 ( μ ) fails the BPBp-nu for multilinear mappings although L 1 ( μ ) satisfies it in the operator case for every measure μ.