We establish an analogue of the Grothendieck inequality where the rectangular matrix is replaced by a symmetric/Hermitian matrix and the bilinear form by a quadratic form. We call this the symmetric Grothendieck inequality; despite its name, it is a generalization -- the original Grothendieck inequality is a special case. While there are other proposals for such an inequality, ours differs in two important ways: (i) we have no additional requirement like positive semidefiniteness; (ii) our symmetric Grothendieck constant is universal, i.e., independent of the matrix and its dimensions. A consequence of our symmetric Grothendieck inequality is a "conic Grothendieck inequality" for any family of cones of symmetric matrices: The original Grothendieck inequality is a special case; as is the Nesterov $π/2$-Theorem, which corresponds to the cones of positive semidefinite matrices; as well as the Goemans-Williamson inequality, which corresponds to the cones of Laplacians. For yet other cones, e.g., of diagonally dominant matrices, we get new Grothendieck-like inequalities. A slight extension leads to a unified framework that treats any Grothendieck-like inequality as an inequality between two norms within a family of "Grothendieck norms" restricted to a family of cones. This allows us to place on equal footing the Goemans-Williamson inequality, Nesterov $π/2$-Theorem, Ben-Tal-Nemirovski-Roos $4/π$-Theorem, generalized Grothendieck inequality, order-$p$ Grothendieck inequality, rank-constrained positive semidefinite Grothendieck inequality; and in turn allows us to simplify proofs, extend results from real to complex, obtain new bounds or establish sharpness of existing ones. The symmetric Grothendieck inequality may also be applied to obtain polynomial-time approximation bounds for NP-hard combinatorial, integer, and nonconvex optimization problems.
In 1955, A.~Grothendieck proved a basic inequality which shows that any bounded linear operator between $L^1(μ)$-spaces maps (Lebesgue-) dominated sequences to dominated sequences. An elementary proof of this inequality is obtained via a new decomposition principle for the lattice of measurable functions. An exposition is also given of the M.~Lévy extension theorem for operators defined on subspaces of $L^1(μ)$-spaces.
We present here several versions of the Grothendieck inequality over the skew field of quaternions: The first one is the standard Grothendieck inequality for rectangular matrices, and two additional inequalities for self-adjoint matrices, as introduced by the first and the last authors in a recent paper. We give several results on ``conic Grothendieck inequality'': as Nesterov $π/2$-Theorem, which corresponds to the cones of positive semidefinite matrices; the Goemans--Williamson inequality, which corresponds to the cones of weighted Laplacians; the diagonally dominant matrices. The most chall…
We present an elementary, self-contained proof of Grothendieck's inequality that unifies the real and complex cases and yields both the Krivine and Haagerup bounds, the current best-known explicit bounds for the real and complex Grothendieck constants respectively. This article is intended to be pedagogical, combining and streamlining known ideas of Lindenstrauss--Pełczyński, Krivine, and Haagerup into a proof that need only univariate calculus, basic complex variables, and a modicum of linear algebra as prerequisites.
As part of the search for the value of the smallest upper bound of the best constant for the famous Grothendieck inequality, the so-called Grothendieck constant (a hard open problem - unsolved since 1953), we provide a further approach, primarily built on functions which map correlation matrices entrywise to correlation matrices by means of the Schur product, multivariate Gaussian analysis, copulas and inversion of suitable Taylor series. We summarise first results and point towards related open problems and topics for future research.
Recently in symplectic geometry there arose an interest in bounding various functionals on spaces of matrices. It appears that Grothendieck's theorems about factorization are a useful tool for proving such bounds. In this note we present two such applications.
In the spirit of Grothendieck's famous inequality from the theory of Banach spaces, we study a sequence of inequalities for the noncommutative Schwartz space, a Fréchet algebra of smooth operators. These hold in non-optimal form by a simple nuclearity argument. We obtain optimal versions and reformulate the inequalities in several different ways.
Using free random varaibles we find an embedding of the operator space $OH$ in the predual of a von Neumann algebra. The properties of this embedding allow us to determined the projection constant of $OH_n$, i.e. there exists a projection $P:B(\ell_2)\to OH_n$ whose completely bounded norm behaves as n^{1/2}/(1+ln n)^{1/2}. According to recent results of Pisier/Shlyahtenko, the lower bound holds for every projection. Improving a previous estimate of order $(1+ ln n)$ of the author, Pisier/Shlyahtenko obtained a `logarithmic little Grothendieck inequality'. We find a second proof of this inequa…
The classical Grothendieck inequality is viewed as a statement about representations of functions of two variables over discrete domains by integrals of two-fold products of functions of one variable. An analogous statement is proved, concerning continuous functions of two variables over general topological domains. The main result is a construction of a continuous map $Φ$ from $l^2(A)$ into $L^2(Ω_A, P_A)$, where $A$ is a set, $Ω_A = {-1,1}^A$, and $P_A$ is the uniform probability measure on $Ω_A$, such that $$\sum_{α\in A} x(α) \bar{y}(α)} = \int_{Ω_A} Φ(x)Φ(\bar{y})dP_A, x \in l^2(A), y \in…
Probably the most famous of Grothendieck's contributions to Banach space theory is the result that he himself described as "the fundamental theorem in the metric theory of tensor products". That is now commonly referred to as "Grothendieck's theorem" (GT in short), or sometimes as "Grothendieck's inequality". This had a major impact first in Banach space theory (roughly after 1968), then, later on, in $C^*$-algebra theory, (roughly after 1978). More recently, in this millennium, a new version of GT has been successfully developed in the framework of "operator spaces" or non-commutative Banach …
We prove the little Grothendieck theorem for any 2-convex noncommutative symmetric space. Let $\M$ be a von Neumann algebra equipped with a normal faithful semifinite trace $\t$, and let $E$ be an r.i. space on $(0, \8)$. Let $E(\M)$ be the associated symmetric space of measurable operators. Then to any bounded linear map $T$ from $E(\M)$ into a Hilbert space $\mathcal H$ corresponds a positive norm one functional $f\in E_{(2)}(\M)^*$ such that $$\forall x\in E(\M)\quad \|T(x)\|^2\le K^2 \|T\|^2 f(x^*x+xx^*),$$ where $E_{(2)}$ denotes the 2-concavification of $E$ and $K$ is a universal constan…
The goal of this short note is to point out three observations around the Grothendieck norm and semidefinite programming. The first is that the Grothendieck norm captures the difficulty of relating the off-diagonal entries of a real, symmetric matrix to a probabilistic correlation, the second is that there is an interesting ``Fourier''-type duality between the Schur and Grothendieck norms of a real matrix, and the third and last centers around the difficulty of finding an efficiently computable noncommutative analog to the Grothendieck norm.
We prove the operator space Grothendieck inequality for bilinear forms on subspaces of noncommutative $L_p$-spaces with $2<p<\infty$. One of our results states that given a map $u: E\to F^*$, where $E, F\subset L_p(M)$ ($2<p<\infty$, $M$ being a von Neumann algebra), $u$ is completely bounded iff $u$ factors through a direct sum of a $p$-column space and a $p$-row space. We also obtain several operator space versions of the classical little Grothendieck inequality for maps defined on a subspace of a noncommutative $L_p$-space ($2<p<\infty$) with values in a $q$-column space for every $q\in [p'…
We prove a version of Holder's inequality with a constant for p-th roots of symmetric operator spaces of operators affiliated to a semifinite von Neumann algebra factor, and with constant equal to 1 for strongly symmetric operator spaces.
In this paper we study an extension problem for the Laplace-Beltrami operator on Riemannian symmetric spaces of noncompact type and use the solution to prove Hardy-type inequalities for fractional powers of the Laplace-Beltrami operator. Next, we study the mapping properties of the extension operator. In the last part we prove Poincaré-Sobolev inequalities on these spaces.
Well-known subadditivity results for positive operators (of Brown-Kosaki and Rotfeld/Ando-Zhan types) are extended to Hermitian and normal ones. Applications to Cartesian decomposition and block-matrices are given.
