Tensor Norms and Operator Ideals

By Andreas Defant, K. Floret. 2011

722 citations recorded by the source corpus.

View Paper: https://www.semanticscholar.org/paper/83ba8724b8bccefbb346437232a9824dce64b8b3

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In Mathematical Physics

Citations in the corpus

On tensor products of quasi-Banach spaces (2010, background): “We refer to [14] for an introductory survey and to the monograph [3] for a detailed treatment of this theory.”
Wedderburn Rank Reduction and Krylov Subspace Method for Tensor Approximation. Part 1: Tucker Case (2010, background): “Keywords: Multidimensional arrays, sparse tensors, structured tensors, Tucker approximation, Krylov subspace methods, Wedderburn rank reduction, fast compression AMS classification: 15A23, 15A69, 65F99”
Summation of coefficients of polynomials on $\ell_{p}$ spaces (2013, methodology): “We will use the normed theory of tensor products as presented in [9].”
A new multilinear insight on Littlewood's 4/3-inequality (2009, background): “We refer to [12] for all needed background on the metric theory of tensor products, and [21,22] for whatever is used on polynomials and symmetric tensor products.”
Best rank one approximation of real symmetric tensors can be chosen symmetric (2011, background): “See for example [1] for a modern exposition on tensor norms and [3] for simple geometrical properties of cross norms.”
Low-rank methods for high-dimensional approximation and model order reduction (2015, neutral): “ resulting tensor Banach space Xk·k p = k·k p Nd ν=1 L p µν (Ξν) is such that Xk·k p = Lp µ(Ξ) for 1 ≤ p<∞, and Xk·k ∞ ⊂ L∞ µ (Ξ) for p= ∞, with equality Xk·k ∞ = L∞ µ (Ξ) if Ξ is a ﬁnite set (see [31]). For any p, the norm k · kp is a reasonable crossnorm. In the case p= 2, L2 µ(Ξ) is a Hilbert ”
Lipschitz operator ideals and the approximation property (2016, methodology): “We will follow the spirit of the definition of linear operator ideal explained in the excellent monographs [17, 33].”
Fast truncation of mode ranks for bilinear tensor operations (2010, positive): “In the paper we use Frobenius norm of tensors, that is deﬁned as follows and spectral norm of tensor (cf. [9]) $ We always assume”
Grothendieck's Theorem, past and present (2011, background): “We refer the reader to [93, 117, 153, 28, 29] for more on this development.”
Proper generalized decomposition for nonlinear convex problems in tensor Banach spaces (2011, positive): “For 1≤ p<∞, the space L p μ(I ; X) can be identified with V‖·‖Δp (see Section 7, Chapter 1 in [11]).”
Crossed products of $L^p$ operator algebras and the K-theory of Cuntz algebras on $L^p$ spaces (2013, background): “16 of [19], but is taken from Chapter 7 of [8].”
Large Violation of Bell Inequalities with Low Entanglement (2010, background): “The following corollary is a direct consequence of the metric mapping property of the norms π and ∧ in their corresponding categories (see [ 23 ,58]).”
p-Operator spaces and Figà-Talamanca-Herz algebras (2006, background): “See [33] or [7] for further details on these ideas.”
Some techniques on nonlinear analysis and applications (2010, background): “The same task was brilliantly done, one year later, by J. Lindenstrauss and A. Pe lczyński’s paper [37] which reformulated Grothendieck’s tensorial arguments giving birth to a comprehensible theory with broad applications in Banach Space Theory.”