On the complete bounds of Lp\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L_p$$\end{document}-Schur multipliers

We study the class Mp\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {M}_p$$\end{document} of Schur multipliers on the Schatten-von Neumann class Sp\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {S}_p$$\end{document} with 1≤p≤∞\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$1 \le p \le \infty $$\end{document} as well as the class of completely bounded Schur multipliers Mpcb\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {M}^{cb}_p$$\end{document}. We first show that for 2≤p<q≤∞\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$2 \le p < q \le \infty $$\end{document} there exists m∈Mpcb\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$m \in \mathcal {M}^{cb}_p$$\end{document} with m∉Mq\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$m \not \in \mathcal {M}_q$$\end{document}, so in particular the following inclusions that follow from interpolation are strict: Mq⊊Mp\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {M}_q \subsetneq \mathcal {M}_p$$\end{document} and Mqcb⊊Mpcb\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {M}^{cb}_q \subsetneq \mathcal {M}^{cb}_p$$\end{document}. In the remainder of the paper we collect computational evidence that for p≠1,2,∞\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p\not = 1,2, \infty $$\end{document} we have Mp=Mpcb\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {M}_p = \mathcal {M}^{cb}_p$$\end{document}, moreover with equality of bounds and complete bounds. This would suggest that a conjecture raised by Pisier (Astérisque 247:vi+131, 1998) is false.

Schur multipliers appear in several different contexts. They are widely applied in harmonic analysis because of their close connection with Fourier multipliers and transference techniques, see, e.g., [4,6,16]. In operator theory Schur multipliers of divided differences occur naturally in problems involving commutators of operators, see, e.g., [20] and references given there. Further, recently new applications of transference techniques have been found in approximation properties of Lie groups [8,9,15]. In each of these applications crucial new results were obtained on the (complete) bounds of Schur multipliers.
The boundedness properties of M m depend on the norm imposed on M n (C). If M n (C) is equipped with the operator norm, the bounds of M m can be described by Grothendieck's characterization [19,Theorem 5.1]. In particular the bounds and complete bounds of a Schur multiplier agree and in an infinite dimensional setting we see that every bounded multiplier is in particular automatically completely bounded. If M n (C) is equipped with the Schatten S p -norm, then finding bounds, or even optimal bounds, of M m becomes very complicated as there is no such charaterization as Grothendieck's available.
In the current paper we show two things. Let M p (resp. M cb p ) be the collection of symbols m that are bounded (resp. completely bounded) Schur multipliers of the Schatten-von Neumann classes S p associated with an infinite dimensional Hilbert space. We refer to Section 2 for exact definitions. Through complex interpolation we have that M q ⊆ M p in case 2 ≤ p < q ≤ ∞. We show that this inclusion is strict; in fact we get a slightly stronger result in particular yielding the parallel result on the complete bounds as well. This extends the results by Harcharras [14,Theorem 5.1] which proves this for even p and it settles the question of strict inclusions (the problem was also stated in [11, p. 51]).
Secondly, we study the question whether M p and M cb p are equal for 1 < p = 2 < ∞. In fact, the following conjecture is stated in [17]: If we replace M p and M cb p by the class of, respectively, the bounded and completely bounded Fourier multipliers on a locally compact abelian group, then Conjecture 1.1 is true as is proven in [17] in case of the torus and in [1] for arbitrary locally compact abelian groups. Pisier's argument relies on lacunary sets in Z (for the bounds) and transference to Schur multipliers and unconditionality of the matrix units as a basis for S p (for failure of the complete bounds). From this perspective it is very reasonable to state Conjecture 1.1.
In [15] it was proved that for continuous Schur multipliers on B(L 2 (R)) we have M p = M cb p with equal bounds and complete bounds as operators on S p (L 2 (R)). The continuity is essential in their proof and leaves open Conjecture 1.1. It deserves to be noted that Lafforgue and De la Salle find several other fundamental properties of Schur multipliers in the same paper [15].
In the current paper we approximate the norms of Schur multipliers by computer algorithms; they suggest that M p = M cb p with equality of norms and complete norms (just as in the case p = ∞). We show that this is true in case of the triangular truncation (Corollary 4.2).

Schatten classes S p .
Let H be a Hilbert space and let B(H) be the space of bounded operators on H. For 1 ≤ p < ∞ we let S p = S p (H) be the space of operators x ∈ B(H) such that The assignment p defines a norm on S p which turns it into a Banach space and which is moreover an ideal in B(H). We set S ∞ for the C * -algebra of compact operators with operator norm ∞ . In case H = C n we write S n for S p = S p (C n ). Fixing an orthonormal basis f j , j ∈ N ≥1 , we have that we may identify S n p (completely) isometrically as a subspace of S p by mapping the matrix unit e i,j ∈ S n p to the matrix unit e fi,fj ∈ S p given by e fi,fj f k = f k , f j f i . Let P n be the projection onto the span of f 1 , . . . , f n . Then this map is an isometric isomorphism S n p P n S p P n . Moreover, under this isomorphsim ∪ n S n p is dense in S p . In case 1 ≤ p ≤ q ≤ ∞ we have S p ⊆ S q and the inclusion is (completely) contractive. This in particular turns (S p , S q ) into a compatible couple of Banach spaces and for any p ≤ r ≤ q we have that S r is a complex interpolation space between (S p , S q ), see [3,18]. Any tensor product S n p ⊗ S p will be understood as a L p -tensor product, i.e. the p-norm closure as a subspace of S p (C n ⊗ H).

Operator space structure.
For the theory of operator spaces we refer to [12], [18]; we shall only need a result of Pisier on completely bounded maps on Schatten classes which we recall here. In [17] Pisier shows that S p have a natural operator space structure as interpolation spaces between S 1 and S ∞ . In [17] it was proved that a linear map M : S p → S p is completely bounded iff for every s ∈ N the amplification The reader may take (2.1) as a definition, other properties (besides interpolation) of the operator space structure of S p shall not be used in this text.

Schur multipliers. A symbol is a function
Here we view again S n p as a subspace of S p by fixing a basis. From the closed graph theorem, as M m is presumed to be defined on all of S p , the map M m is automatically bounded. The space of all L p -Fourier multipliers will be denoted by M p which carries the operator norm · Mp of B(S p ). This turns M p into a Banach space. We denote M cb p for the subset of m ∈ M p such that M m : S p → S p is completely bounded. We equip M cb p with the completely bounded norm · M cb p as completely bounded maps on S p . With slight abuse of terminology we shall refer to both the symbol m as well as the map M m as a Schur multiplier and usually write M m Mp for m Mp (and similarly for the completely bounded norms). Obviously M cb p ⊆ M p . The question whether this inclusion is strict remains open, see Conjecture 1.1.

Strict inclusions of the set of Schur multipliers.
Here we prove that for 2 ≤ p < q ≤ ∞ there exists a symbol m : Z × Z → C that is a completely bounded L p -Schur multiplier but which fails to be a bounded L q -Schur multiplier. The following lemma is based on [7, Lemma 1]. For a finitely supported meausre μ on the torus T we write μ * for the convolution operator L p (T) → L p (T) : f → μ * f . We let μ be the norm of the measure.
There exists a finitely supported measure μ n , n ∈ N, on the torus T such that and note that the supports of μ n and s n * ν n have empty intersection. We claim that for every Indeed, this is clear for n = 0 and further by the parallellogram law So that μ n * CB(L2(T)) ≤ 2 (n+1)/2 . Also μ n * CB(L1(T)) ≤ μ n = 2 n and by duality also μ n * CB(L∞(T)) ≤ 2 n . By complex interpolation therefore μ n * CB(Lp(T)) ≤ √ 2 2 n/p . This proves the upperbound in (3.1). For the lower bounds let f ∈ C c (T) be a function with small support close to 1 ∈ T. If the support is small enough, then μ n * f consists of 2 n disjointly supported translates of f so that μ n * f p = 2 n/p f p . This yields the lower bound.
The following theorem shows in particular that the class of L p -Schur multipliers can be distinguished from the L q -Schur multipliers for 2 ≤ p < q ≤ ∞.
There exists a symbol m : Z 2 → C such that m ∈ M cb p but m ∈ M q . Proof. We first treat the case 2 ≤ p < q < ∞. Let μ n be the finitely supported measure on T of Lemma 3.1. Let m n : Z → C be its Fourier transform given by From these two estimates we are able to prove the theorem as follows.
Suppose that the theorem is false, so that we have an inclusion map i : M cb p → M q . By the closed graph theorem this inclusion is continuous. Indeed, if k j ∈ M cb p is a net in symbols such that k j → 0 in M cb p and such that k j converges to k in M q , then for every matrix x ∈ S n p ⊆ S p we find that M kj (x) → 0 in S n p and hence also in the norm of S n q . This shows that for x ∈ S n q ⊆ S q we have M k (x) = lim j M kj (x) = 0. But by density of ∪ n S n q in S q we get that M k (x) = 0. Hence the graph of i is closed indeed.
However, the estimates (3.5) show that which converges to infinity if n → ∞. This contradicts that i : M cb p → M q is bounded. Now, if 1 < q < p ≤ 2, then the statement follows from duality as M * m = M m ∨ , where m ∨ (k, l) = m(k, l) and duality preserves the (complete) bounds of linear maps. In case q = 1 or q = ∞ the counter example is given by triangular truncation, see [10].
In particular we get the weaker statements that give non-inclusions of bounded and completely bounded multipliers.

Corollary 3.3. Let either
We may in fact improve on this theorem in the following way. Proof. Let q n > p be a decreasing sequence with q n p. Let m n ∈ M cb p with m n M cb p = 1 be such that m n ∈ M qn . We copy-paste part of the symbols m n to diagonal blocks of a new symbol m as follows. Let k n ∈ N be such that there exist x n ∈ S kn qn with x n q = 1 and M mn (x n ) qn > n. Let m n : [−k n , k n ] × [−k n , k n ] → C be the restriction of m n to a discrete interval. As M m n (x n ) q ≥ n we see that M m n ≥ n. Then let m : Z × Z → C be the block symbol given by   Proof. The first statement is a consequence of the fact that the Schur multiplier (id s ⊗ M m ) commutes with the isometric action of ⊕ n i=1 U (s). Therefore, for The second statement follows from the polar decomposition. Indeed, take x ∈ C s m . We claim first that we may assume that x i,i is a positive semidefinite matrix. For each 1 ≤ i ≤ n consider the polar decomposition . We find that w * u * xw ∈ C s m and further (w * u * xw) i,i = d i .
Proposition 4.1 can be used to significatly speed up our computations in Section 5.1.

Vol. 113 (2019)
On the complete bounds of Lp-Schur multipliers 195 As a side remark we obtain the following corollary that shows that the bounds and complete bounds of an infinite dimensional triangular truncation agree. This result was already recorded in (the discussion before) [

Corollary 4.2.
Let h : Z 2 → C be the symbol of triangular truncation given by h(i, j) = δ ≥0 (i − j). Then for every 1 < p < ∞ we have h Mp = h M cb p . Proof. We use the notation of Proposition 4.1. Let π s : S s p ⊗ S p → S p : e i,j ⊗ e k,l → e sk+i,sl+j be the isometric isomorphism that re-indexes matrix units. Let x ∈ C s h , s ∈ N ≥2 . By Proposition 4.1 we may assume that each ) and therefore,

Approximation.
In this section we argue that if Conjecture 1.1 is true, then we should be able to find computer based evidence for it, which we make precise in the following way. Consider the following three statements: 1. For every 1 < p = 2 < ∞ there exists a bounded Schur multiplier that is not completely bounded. 2 is weaker than 1, and 3 is just the negative of 2. If 2 is already true, then it is possible to show this by sampling dense sets of Schur multipliers on finite dimensional Schatten classes and by approximating their norms with finite sets. The problem however is that it is not clear how much computations and computational power is needed in order to obtain a symbol m that witnesses statement 2 above. We state a quantitative statement in this direction in the next proposition.
Further, for every m ∈ A ε we have,  Table 2. n = dimension of the symbol, N = number of random sample multipliers m, second and third column = approximation of the 2nd and 3rd amplification of m is a gradient descent algorithm to find local minima/maxima of a function. We apply it here to find local maxima of In case p ∈ 2N ≥1 , so that the p/2-powers are integer powers, this expression is faster to compute as it avoids determining eigenvalues of x * x and M m (x) * M m (x). The precise algorithm is available on [5] and it makes use of the reductions in Section 4. Note that the sample sets A ε in Proposition 5.1 scale exponentially with the dimension and therefore we are bound to use faster algorithms that only allow us to compute local maxima.

Approximation for a fixed Schur multiplier.
In order to illustrate our larger computations we start with the example (fixed) Schur multiplier, 3) The following table shows the approximation of the norm of where [r] is the largest integer k with k ≤ r (Table 1).

Approximations for random Schur multipliers.
Next we sample random symbols m of Schur multipliers. In the next table N is the number of random samples m ∈ M n (C) and to each of these we approximate its norm in essentially the same way as we did to the single example of Section 5.2. We then take the maximum over all samples m over the difference of the norms (Table 2).  3 10 −2.616529215515584 · 10 −10 −2.0181634141636096 · 10 −10 n N p = 4.5 p = 5 2 100 −1.9551027463649007 · 10 −13 −3.397282455352979 · 10 −13 3 10 −3.2847524700230224 · 10 −10 −5.515231604746873 · 10 −9 Note that the values in this table are negative because it is harder to approximate M is smaller than the approximation of M m B(S n 4 ) .

Different values of 2 ≤ p < ∞.
For arbitrary p we may still approximate the norm by the same algorithm except that x p is computed by determining the eigenvalues λ 1 , . . . , λ n of x * x so that then x p p = n i=1 |λ i | p/2 . Though that this is computationally more involved we can still carry out our approximations which are displayed in the following figure (Table 3).