Multipliers and operator space structure of weak product spaces
Raphaël Clouâtre Michael Hartz



  In the theory of reproducing kernel Hilbert spaces, weak product spaces generalize
  the notion of the Hardy space H^1. For complete Nevanlinna–Pick spaces ℋ, we characterize
  all multipliers of the weak product space ℋ⊙ℋ. In particular,
  we show that if  has the so-called column-row property, then the multipliers of 
  and of ⊙ coincide. This result applies in particular to the classical Dirichlet space
  and to the Drury–Arveson space on a finite dimensional ball.
  As a key device, we exhibit a natural operator space structure on ⊙, which enables
  the use of dilations of completely bounded maps.


Support Vector Spectrum Approximations: Efficient Calculations of Light Scattering Lineshapes
    Gregory J. Hunt^1,*, 
  Cody R. Ground^2, 
  and Robin L. Hunt^2

  ^1William & Mary, PO Box 8763 Williamsburg, VA 23187

^2Hypersonic Airbreathing Propulsion Branch, NASA Langley Research Center, 12 Langley Blvd. Hampton, VA 23681.

^*ghunt@wm.edu

    December 30, 2023
=============================================================================================================================================================================================================================================================





§ INTRODUCTION


Let  be a reproducing kernel Hilbert space of functions on a set X.
The weak product space is defined by

    ⊙ =
      { h = ∑_i=1^∞ f_i g_i : ∑_i=1^∞f_i g_i < ∞},

with norm

    h_⊙ = inf{∑_i=1^∞f_i g_i : h = ∑_i=1^∞ f_i g_i  }.

This is a Banach function space on X, meaning in particular that the functionals of evaluation at points in X
are continuous on ⊙. If = H^2(), the Hardy space on the unit disc ,
then ⊙ = H^1() with equality of norms. In fact, every function in H^1()
is a product of two functions in H^2().
The notion of a weak product space has its origins in a famous paper of Coifman, Rochberg
and Weiss <cit.>. There, it is shown that the Hardy spaces and Bergman spaces on the unit ball _d
satisfy H^1( ∂_d) = H^2(∂_d) ⊙ H^2 (∂_d)
and L^1_a(_d) = L^2_a(_d) ⊙ L^2_a(_d), with equivalence of norms.
In general, one can regard weak product spaces as a replacement for the Hardy space H^1 in the context
of an arbitrary reproducing kernel Hilbert space. In this setting,
weak product spaces are closely related to boundedness of Hankel forms; see e.g.the introduction of <cit.>. Weak product spaces have been concretely
studied for instance for the classical Dirichlet  space <cit.>, the Drury–Arveson
space <cit.> and more generally for complete Nevanlinna–Pick spaces <cit.>.

If  is a Banach function space on X, the multiplier algebra of  is defined by

    () = {φ: X →: φ· f ∈ for all  f ∈}.

The closed graph theorem implies that for each φ∈(), the associated
multiplication operator M_φ on  is bounded, so we may define φ_()
= M_φ_B(). It is immediate from the definition of the weak product space
that () ⊂(⊙), with φ_(⊙)≤φ_() for all φ∈().
On the other hand, for the classical Hardy space, it is not hard to see that both
(H^1()) and (H^2()) agree with H^∞(), with equality of norms.
This naturally raises the following question.


   Is (⊙) = ()?


This question was studied by Richter and Wick <cit.>, who provided a positive answer
for first order weighted Besov spaces on the unit ball  using function theoretic estimates.
In particular, their
result applies to the classical Dirichlet space and to the Drury–Arveson space
H^2_d for d ≤ 3, but not to H^2_d for d ≥ 4.

In this article, we characterize multipliers of ⊙ for
normalized complete Nevanlinna–Pick spaces
and are thus able to give a positive answer to Question <ref> in many instances.
The prototypical example of a normalized complete Nevanlinna–Pick space is the Hardy space H^2(),
but there are many
other examples such as the classical Dirichlet space or the Drury-Arveson space H^2_d
for d ∈∪{∞}.
Normalized complete Nevanlinna–Pick spaces
can be defined in terms of the validity of a suitable
version of the Nevanlinna–Pick interpolation theorem. Equivalently,
by results of McCullough, Quiggin and Agler–, a reproducing kernel Hilbert space
is a normalized complete Nevanlinna–Pick space if and only if its reproducing kernel
K is of the form

    K(z,w) = 1/1 - ⟨ b(z),b(w) ⟩   (z,w ∈ X),

where b maps X into the open unit ball of an auxiliary Hilbert space and satisfies b(z_0) = 0
for some distinguished point z_0 ∈ X. This characterization and more background information
can be found in <cit.>.

A key ingredient in our analysis of (⊙) is the observation
that ⊙ can be equipped with a natural operator space structure, and we will use
this additional structure crucially.
Briefly,
⊙ can be identified as the dual of the space of compact Hankel operators
on  (see <cit.>), hence ⊙ is the dual of an operator
space and thus itself an operator space.
For more details, see Section <ref>.
In particular, for each n ∈, the space
M_n(⊙) of n × n matrices with entries in ⊙ carries a natural norm.
As is customary in operator space theory, for each n∈ one associates with a linear map A: ⊙→⊙
its amplification A^(n): M_n(⊙) → M_n(⊙), defined by applying A entrywise.
The linear map A is then said to be a complete contraction if each map A^(n) is a contraction.
In particular, given a function θ: X →, we say that θ is a contractive multiplier
of ⊙
if the multiplication operator M_θ: ⊙→⊙ is a contraction,
and that θ is a completely contractive multiplier of ⊙ if M_θ
is a complete contraction.

Let M^C_1() be the space of sequences of multipliers of  that are contractive when viewed as a column operator on .
In other words, a sequence (φ_n) belongs to M^C_1() if and only if the operator

    [ M_φ_1; M_φ_2;     ⋮ ]: →⊗ℓ^2

is contractive.
Our main result can now be stated as follows.


  
  Let  be a normalized complete Nevanlinna–Pick space on X.
  The following assertions are equivalent for a function θ: X →.
  
    
  * The function θ is a contractive multiplier of ⊙.
    
  * The function θ is a completely contractive multiplier of ⊙.
    
  * There exist sequences (φ_n)_n=1^∞, (ψ_n)_n=1^∞ in M^C_1() such that
      θ = ∑_n=1^∞φ_n ψ_n.
  


In particular, it follows that the norm of an element θ∈(⊙) is given by

    θ_(⊙)
      = inf{[ φ_1; φ_2;   ⋮ ][ ψ_1; ψ_2;   ⋮ ] : θ = ∑_n=1^∞φ_n ψ_n },

where the norms of the columns in the infimum are taken in (, ⊗ℓ^2). Moreover, the infimum is attained.

The proof of this theorem will be separated in several steps.
The implication (iii) ⇒ (i) easily follows from the definition
of ⊙ and already appears in <cit.>.
We provide the short argument in Proposition <ref>.
The proof of the implication (i) ⇒ (ii)
uses a recent result of Jury and Martin <cit.> and is presented in Proposition <ref>.
The majority of the work occurs in the proof of the implication (ii) ⇒ (iii),
which uses dilation theory and is done in Theorem <ref>.

While not logically necessary, we also provide a direct proof of the implication (iii) ⇒ (ii);
this is done in Proposition <ref>. This proof shows how the operator
space structure of ⊙ enters the picture and motivates our approach to the harder
implication (ii) ⇒ (iii).

In many instances, Theorem <ref> implies an affirmative answer to Question <ref>.
The key property is the following.
A normalized complete Nevanlinna–Pick space  is said to satisfy the column-row property (with constant κ) if whenever (φ_n)_n=1^∞ is a sequence in () with

    [ φ_1; φ_2;   ⋮ ]_(, ⊗ℓ^2)≤ 1,

then

    [ φ_1 φ_2   ⋯ ]_(⊗ℓ^2, )≤κ.

The classical Hardy space H^2() satisfies the column-row property with constant 1,
as the norm of a row and of a column of multipliers are both given by the supremum norm
over .
It is a result of Trent <cit.>
that the classical Dirichlet space satisfies the column-row property with constant at most √(18).
The Drury–Arveson space H^2_d with d < ∞ also satisfies the column-row property
with some finite constant, which possibly depends on d <cit.>.
More generally, it was shown in <cit.> that radially weighted Besov spaces on the unit ball in finite dimensions
satisfy the column-row property with a finite constant, which again possibly depends on the dimension and on the particular
space.
It is an open question whether every normalized complete Nevanlinna–Pick space satisfies
the column-row property with a finite constant (see <cit.> for some work on this question).
Recently, it has become clear that the column-row property is a very useful technical property
when studying weak product spaces, see for instance <cit.>. A striking example
of this is the result of Jury and Martin <cit.>, according to which every function
in ⊙ factors as a product of precisely two functions in , provided that  satisfies
the column-row property.

Theorem <ref> combined with an argument from  <cit.> implies a positive answer to Question <ref> in the
presence of the column-row property.


  
  Let  be a normalized complete Nevanlinna–Pick space that satisfies the column row-property
  with constant κ. Then () = (⊙) and
  
    θ_(⊙)≤θ_()≤κθ_(⊙)

  for every θ∈().



  It is elementary to check that every contractive multiplier of  is also a contractive
  multiplier of ⊙; alternatively, this also follows from the implication (iii) ⇒ (i)
  of Theorem <ref>.
  
  Conversely, if θ is a contractive multiplier of ⊙,
  then the implication (i) ⇒ (iii) of Theorem <ref> shows that
  there are (φ_n), (ψ_n) ∈ M^C_1() so that θ = ∑_n=1^∞φ_n ψ_n. In other words, we have
  
    θ =
        [ φ_1 φ_2   ⋯ ][ ψ_1; ψ_2;   ⋮ ].

  The column is a contractive multiplier of  as (ψ_n) ∈ M^C_1(), and the row has norm at most κ
  by the column-row property since (φ_n) ∈ M^C_1(). Hence θ∈() and
  θ_()≤κ.


Matrix-valued multipliers play an important role in the theory of complete Nevan­linna–Pick spaces, and hence so does the operator space structure of (). For instance, this operator space structure 
encodes the difference between the Nevanlinna–Pick property and the complete Nevanlinna–Pick property.
In the final section of the paper, we show how to equip (⊙) with an operator space structure,
i.e. we define norms for matrices of multipliers of ⊙. We establish a version
of Theorem <ref> for matrices of multipliers (Theorem <ref>) and then use this result
to compare the operator space structures of () and of (⊙).
In the case of the Drury–Arveson space and of the classical Dirichlet space,
we show that while the inclusion () ↪(⊙) is a Banach space isomorphism
(by Corollary <ref>), it is not an isomorphism of operator spaces.




§ PRELIMINARIES



Let  be a normalized complete Nevanlinna–Pick space of functions on a set X.
We assume throughout that  is separable.



 §.§ Hankel operators



Nehari's theorem identifies the dual space of H^1() with the space of
symbols of bounded Hankel operators on H^2() via the standard integral
pairing on the unit circle.
When studying multipliers of ⊙, we will heavily use a generalization of
this fact, namely the duality between ⊙ and the space () of symbols of Hankel operators
on . We now recall the necessary basics from <cit.>.

The conjugate Hilbert space of  is

    = {f: f ∈}.

This space can be linearly and isometrically identified with the dual space ^* of 
by means of the inner product of . Notice that  is again a normalized complete Nevanlinna–Pick
space on X whose reproducing kernel is the complex conjugate of that of .
The map f↦f yields an anti-unitary operator between  and  which conjugates () to ().

It was shown in <cit.> that the dual space of ⊙ can
be linearly and isometrically identified with a subspace () ⊂ B(, );
operators in () are called Hankel operators on . Every operator T ∈()
is uniquely determined by its symbol T 1 ∈. Let

    () = {T 1∈: T ∈() }

denote the space of symbols of Hankel operators.
Given b ∈(), we write H_b for the unique operator in () that satisfies
H_b 1 = b. Notice that the map b ↦ H_b is conjugate linear.
The operator H_b is uniquely determined by the requirement that

    ⟨ H_b f, φ⟩_ = ⟨φ f,b ⟩_  for all  f ∈, φ∈().

(Since  is a normalized complete Nevanlinna–Pick space, the kernel functions
are contained in () and hence () is densely contained in .)


  If  satisfies the column-row property, then the space ()
  can be more concretely described as the space of all those b ∈
  for which the densely defined bilinear form on ×,
  
    (φ ,f) ↦⟨φ f,b ⟩   (φ∈(), f ∈),

  is bounded; see <cit.>.


The version of Nehari's theorem obtained in <cit.> asserts that
there is a conjugate linear isometric isomorphism
() ≅ (⊙)^*, b ↦ L_b, satisfying

    L_b( φ f) = ⟨φ f,b ⟩   for all  b ∈(), f ∈, φ∈().

Here, the norm on () is given by b_() = H_b_B(,).
We write [f,H_b] = L_b(f) for f ∈⊙ and b ∈().
Thus, combining (<ref>) and (<ref>), we see that
the linear duality between () and ⊙ is given by

    [f g, H_b] = ⟨ H_b f, g⟩_   (f,g ∈, b ∈()).

This duality also endows () with a weak-* topology.
It follows from the construction in <cit.>, or one checks directly,
that this weak-* topology agrees with the one inherited from B(,) as the dual of
the space T(,) of trace class operators.


In this article, we find it convenient to distinguish between a Hankel operator and its symbol,
as the correspondence between them is conjugate linear.
Ultimately, we will work with the operators directly and only use Equation (<ref>),
which in principle could be understood without explicitly mentioning symbol functions.


Given a bounded linear operator A: ⊙→⊙,
let A^†: () →() be the Banach space adjoint
of A modulo the linear isometric isomorphism () ≅ (⊙)^*.
Equation (<ref>) shows that the action of A^† is characterized by
the formula

    ⟨ A^† (H_b) f, g⟩_ = [ A( f g ), H_b ]
         (f,g ∈, b ∈()).


It follows from a theorem of Hartman that H^1() can be identified with the dual
of the space of all symbols of compact Hankel operators.
In a similar fashion, the weak product ⊙ is also a dual space.
More precisely, let

    _0() = () ∩ K(,)

be the space  of all compact Hankel operators on .
According to Theorems 2.1 and 2.5 of <cit.>, the duality
between ⊙ and () also yields an isometric isomorphism
_0()^* ≅⊙.

If k_z ∈ denotes the kernel function associated with z ∈ X, then
boundedness of point evaluations on ⊙ and the duality
between ⊙ and () (see Equation (<ref>)) imply that k_z ∈(). Moreover,
Equation (<ref>) shows that H_k_z f = f(z) k_z for f∈, so that H_k_z
is a rank-one operator and H_k_z∈_0().
In particular, point evaluations on ⊙ are continuous in the weak-* topology
given by the duality ⊙ = _0()^*. On bounded subsets of ⊙,
the weak-* topology agrees with the topology of pointwise
convergence on X; see <cit.> and the remarks preceding it.
It also follows from the Hahn–Banach theorem that _0() is weak-* dense in ().




 §.§ Multiplication operators and duality


To distinguish multiplication operators on  and on ⊙, we will use the following
notation. Given θ∈(⊙), the corresponding multiplication operator
on ⊙ is denoted by

    M_θ: ⊙→⊙,    f ↦θ· f.

Given φ∈(), we denote the associated multiplication operator on  by

    T_φ: →,    f ↦φ· f.

(If = H^2(), then T_φ is an analytic Toeplitz operator, which motivates our choice of notation.)

Since every multiplier of  is also a multiplier of ⊙, if b∈() and ψ∈() then L_b∘ M_ψ∈ (⊙)^*. This element in the dual is verified to correspond to T_ψ^*b∈(). Moreover,
the defining equation of a Hankel operator, Equation (<ref>), easily implies the
intertwining relation

    H_b T_ψ = T_ψ^* H_b = H_ T^*_ψ b   for all ψ∈(), b ∈();

see also <cit.>. In particular,
() is a ()^*-()-bimodule.
The following lemma shows that if θ∈(⊙), then
M_θ^† respects this bimodule structure.


  
  Let θ∈(⊙), let φ∈() and let b ∈().
  Then:
  
    
  * M_φ^†(H_b) = H_b T_φ = T_φ^* H_b.
    
  * M_θ^†( H_b T_φ) = M_θ^†(H_b) T_φ = M_θ^†( T_φ^* H_b) = T_φ^* M_θ^†(H_b).
    
  * M_θ^†(H_k_z) = θ(z) H_k_z for all z ∈ X.
  



  (a) For f,g ∈, we find using (<ref>) and (<ref>) that
  
    ⟨ M_φ^† (H_b) f, g⟩_
        = [ φ f g, H_b]
        = ⟨ H_b T_φ f, g⟩_,

  from which the first half of (a) follows. The second half follows from (<ref>).

  (b)
  Dualizing the commutation relation M_φ M_θ = M_θ M_φ and using (a), we
  see that
  
    M_θ^†( H_b T_φ) = (M_θ^† M_φ^†) (H_b)
        = (M_φ^† M_θ^†) (H_b) = M_θ^† (H_b) T_φ.

  The remaining parts of (b) follow from this and from (<ref>).

  (c) For z ∈ X we have that [h,H_k_z]=h(z) for every h∈⊙. Using (<ref>) and  (<ref>), we obtain
  for f,g ∈ the identity
  
    ⟨ M_θ^† (H_k_z) f, g⟩_ℋ
        = [θ f g, H_k_z]
        = θ(z) [f g, H_k_z]
        = θ(z) ⟨ H_k_z f, g⟩_ℋ,

  from which (c) follows.






 §.§ H dot H as an operator space



As mentioned in the introduction, a key device in our analysis of multipliers of ⊙
is the observation that the weak product carries a natural operator space structure. We therefore
recall the necessary background from the theory of operator spaces. For precise definitions and more information,
the reader is referred to the books <cit.>.

Given a vector space V, let M_n(V) be the vector space of all n × n matrices
with entries in V.
An (abstract) operator space is a normed space V, together with a norm on each M_n(V) satisfying certain axioms. We will not require the precise form of the axioms and thus
simply refer to <cit.>.
Perhaps the most important example of an operator space is the space B(,),
where  and  are Hilbert spaces. In this case, we can identify M_n(B(,))
with B(^n,^n), which equips M_n(B(,)) with a norm. In the same
vein, any subspace of B(,) becomes an operator space in this way.

If V,W are operator spaces, each linear map A: V → W induces for each n∈ a linear map
A^(n): M_n(V) → M_n(W) defined by applying A entrywise. The map A is said to be
completely bounded if

    A_c b = sup_n ∈ A^(n) < ∞,

and completely contractive if A_cb≤ 1.
Similarly,
A is said to be a complete isometry if each A^(n) is an isometry.
We write (V,W) for the space of all completely bounded linear maps from V to W,
endowed with the cb norm.
It is a well-known phenomenon in operator space
theory that completely bounded maps exhibit much better behavior than maps that are merely bounded.

If V is an abstract operator space, then its dual space V^* carries a natural
operator space structure, called the dual operator space structure.
It is defined via the identification M_n(V^*) = (V,M_n).
The dual operator space structure has the property
that if A: V → W is a completely bounded map between operator spaces, then the adjoint
A^*: W^* → V^* is completely bounded with A^*_c b = A_cb;
see <cit.>.

We now apply these abstract considerations to our setting of weak products. Since
_0() ⊂() ⊂ B(,), the spaces
_0() and () carry a natural operator space structure.
Since ⊙ is isometrically isomorphic
to _0()^*, we may endow ⊙ with the corresponding dual operator space structure.
Taking duals again, the resulting dual
operator space structure on (⊙)^* ≅() agrees with the
operator space structure of () inherited from B(,),
as the identification of K(,)^** with B(,) is a complete isometry;
see Theorem 1.4.11 in <cit.> and the discussion preceding it.
In particular, it follows that ⊙ is endowed with the unique operator
space structure that makes () the operator space dual of ⊙ with
respect to the given duality; this is also known as the predual operator space structure,
see <cit.> for further discussion.

In particular, we see that a linear map A: ⊙→⊙
is completely contractive if and only if its adjoint A^†: () →()
is completely contractive. We will frequently use this fact. Indeed, for our purposes
it will be more convenient to study properties of A through A^†, because
the latter acts on a concrete space of operators as opposed to the space ⊙,
whose operator space structure is less explicit.

We will only use the description of the operator space structure on ⊙
in terms of the duality given above.
Nevertheless, we will provide a concrete description of the norm on M_n(⊙)
in Lemma <ref>.



§ PROOF OF THE MAIN RESULT



We continue to assume throughout that  is a normalized complete Nevanlinna–Pick space of functions on X.



 §.§ Factorization implies complete contractivity


We first show that (iii) ⇒ (i) in Theorem <ref>,
that is, that every function that can be factored using a pair of elements in M_1^C() is a contractive
multiplier of ⊙. As mentioned in the introduction, this result is
known <cit.>, but we include the short argument for the convenience of the reader.


  
  Let (φ_n), (ψ_n) ∈ M^C_1() and define θ = ∑_n=1^∞φ_n ψ_n.
  Then θ is a contractive
  multiplier of ⊙.



  Let h = ∑_n=1^∞ f_n g_n ∈⊙ with ∑_n=1^∞f_ng_n < ∞.
  By continuity of point evaluations on  and by the Cauchy–Schwarz inequality, the sums defining h and θ converge pointwise absolutely on X,
  hence
  
    θ h = ∑_k,n=1^∞ (φ_k f_n) (ψ_k g_n).

  Since (φ_n)_n=1^∞ and (ψ_n)_n=1^∞∈ M_1^C(), we find that
  
    ∑_n,k=1^∞φ_k f_n ψ_k g_n≤∑_n=1^∞( ∑_k=1^∞φ_kf_n^2)^1/2(∑_k=1^∞ψ_kg_n^2 )^1/2
    ≤∑_n=1^∞f_n g_n,

  so taking the infimum over all representations h = ∑_n=1^∞ f_n g_n, it follows that
  θ h_⊙≤h_⊙, so that θ is a contractive multiplier
  of ⊙.


While not logically necessary, we improve the preceding result by showing
that functions that factor as above are actually completely contractive multipliers
of ⊙;  this is the implication (iii) ⇒ (ii) of Theorem
<ref>.
We provide this proof as it shows how the operator space structure of ⊙
and the duality between ⊙ and () enter the picture, and it
foreshadows
the dilation theoretic proof of the reverse
implication (ii) ⇒ (iii) of Theorem <ref>.


  
  Let (φ_n), (ψ_n) ∈ M^C_1() and define θ = ∑_n=1^∞φ_n ψ_n.
  Then θ is a completely contractive
  multiplier of ⊙.



  Observe that it suffices to show that for each N ∈, the
  function θ_N = ∑_n=1^N φ_n ψ_n is a completely contractive multiplier
  of ⊙. Indeed, θ_N converges pointwise to θ. Hence,
  assuming that each θ_N is a completely contractive multiplier of ⊙,
  we see that for all f ∈⊙, the sequence (θ_N f) converges to θ f in the weak-* topology
  of ⊙. Thus, θ is completely contractive if each θ_N is.

  Therefore, we may assume that θ = ∑_n=1^N φ_n ψ_n for some N ∈. In particular, θ∈(). We will show that, equivalently, the adjoint map M_θ^†: () →() is completely contractive.
  To this end, we apply part (a) of Lemma <ref> to conclude that
  
    M_θ^†(H_b) = H_b T_θ =
        [ T_ψ_1^*       ⋯ T_ψ_N^* ]( H_b ⊕⋯⊕ H_b )
        [ T_φ_1;     ⋮; T_φ_N ]

  for every b ∈().
   This formula implies that M_θ^† is completely contractive
   once we know that the row and the column are contractive,
   which in turn follows from the assumption (φ_n),(ψ_n) ∈ M^C_1()
  (see also the remarks about the conjugate Hilbert space in Subsection <ref>).




 §.§ Contractive multipliers are completely contractive


The goal of this subsection is to show that
every contractive multiplier of ⊙ is completely contractive,
that is, we prove the implication (i) ⇒ (ii) of Theorem <ref>.

The key tool is the following lemma, which uses a recent result
of Jury and Martin <cit.>.
For notational convenience, we regard finite sequences of multipliers
as infinite sequences that are eventually zero.


  
  Let [A_i j] ∈  M_n(B(,)). Then
  
    [A_i j]  = sup{∑_i,j=1^n T_ψ_i^* A_i j T_φ_j:
        (φ_i)_i=1^n, (ψ_i)_i=1^n ∈ M^C_1() }.




  If (φ_i),(ψ_i) ∈ M_1^C(), then
  
    ∑_i,j=1^n T_ψ_i^* A_i j T_φ_j =
        [ T_ψ_1^*       ⋯ T_ψ_n^* ]
        [A_i j]
        [ T_φ_1;     ⋮; T_φ_n ]≤ [A_i j] ,

  as the row and the column are contractions, hence the inequality “≥” holds
  in the statement of the lemma.

  To prove the reverse inequality, it suffices to show that for every pair of sequences
  (f_i)_i=1^n, (g_i)_i=1^n of elements of  with ∑_i=1^n f_i^2 = ∑_j=1^n g_i^2 = 1,
  there exist (φ_i),(ψ_i) ∈ M^C_1() so that
  
    | ∑_i,j=1^n ⟨ A_i j f_j, g_i⟩_|
        ≤∑_i,j=1^n T_ψ_i^* A_i j T_φ_j.

  To this end, we apply Theorem 1.1 of <cit.>, which yields (φ_i),(ψ_i) ∈ M^C_1() and
  F,G ∈ with F≤ 1, G≤ 1 such that f_i = φ_i F, g_i = ψ_i G for all i.
  Then
  
    | ∑_i,j=1^n ⟨ A_i j f_j, g_i⟩_|
        =| ∑_i,j=1^n ⟨ A_i j T_φ_j F, T_ψ_iG⟩_|
        ≤∑_i,j=1^n T_ψ_i^* A_i j T_φ_j,

  as desired.



  In the language of operator bimodules, Lemma <ref> says
  that the pair (()^*,()) is matricially norming for B(,),
  and in particular for ().
  This property is most commonly studied for C^*-bimodules, see for example <cit.>.


Given the matricial norming property of Lemma <ref>,
it is now routine to finish the proof of the implication (i) ⇒ (ii) of Theorem
<ref>; cf. <cit.>.


  
  Every contractive multiplier of ⊙ is completely contractive.



  Let θ∈(⊙) be a contractive multiplier.
  By duality, it suffices to show that the contractive map M_θ^†: () →()
  is completely contractive.
  To this end, we use Lemma <ref> and part (b)
  of Lemma <ref> to see that
  for [H_i j] ∈ M_n(()),
  
    [ M_θ^† (H_i j) ] 
        = sup∑_i,j=1^n T_ψ_i^* M_θ^† (H_i j)  T_φ_j   = sup M_θ^†(∑_i,j=1^n T_ψ_i^* H_i j T_φ_j) 
       ≤sup∑_i,j=1^n T_ψ_i^* H_i j T_φ_j = [H_i j],

  where all suprema are taken over (φ_i), (ψ_i) ∈ M^C_1().




 §.§ Completely contractive multipliers admit a factorization


In this subsection, we prove the remaining implication (ii) ⇒ (iii) of
Theorem <ref>,
that is, the factorization for completely contractive multipliers θ of ⊙.
To this end, we use dilation theory to obtain a representation for the adjoint
M_θ^† as in the proof of Proposition <ref>.
We emphasize that it is complete contractivity that enables this use of dilation theory.
The first step is the following consequence of the Haagerup–Paulsen–Wittstock
dilation theorem.


  
  Let A: () →() be a completely contractive linear map
  that is (weak-*,weak-*) continuous. Then there exist linear contractions
  V: →⊗ℓ^2 and W: →⊗ℓ^2 such that
  
    A(H_b) = W^* (H_b ⊗ I_ℓ^2) V

  for all b ∈().



  Recall that () ⊂ B(,) and _0() ⊂ K(,).
  To be in the more familiar setting
  of spaces of operators on a single Hilbert space,
  we fix a (non-canonical) linear unitary U: → and define
  _0() = U _0() ⊂ K()
  and
  
    A: _0() → B(),   A(U H_b)
        = U A(H_b).

  Then A is completely contractive, so by the Haagerup–Paulsen–Wittstock
  dilation theorem (Theorems 8.2 and 8.4 in <cit.>), there exist a Hilbert
  space ⊃, a *-representation
  π: K() → B() and contractions X,Y: → so that
  
    A(U H_b) = X^* π(U H_b) Y    (H_b ∈_0()).

  Since  is separable,  can be chosen to be separable as well.
  Every *-representation of K() is unitarily equivalent to a multiple
  of the identity representation, hence there exist contractions V_0,W_0: →⊗ℓ^2
  so that
  
    U A(H_b) = A(U H_b) = W_0^* (U H_b ⊗ I) V_0    (H_b ∈_0()).

  Thus, defining V = V_0 and
  W = (U^* ⊗ I) W_0 U, we see that
  
    A(H_b) = W^* ( H_b ⊗ I) V    (H_b ∈_0()).

  Recall from Subsection <ref> that _0()
  is weak-* dense in () and that the inclusion
  () ⊂ B(,) is (weak-*,weak-*) continuous,
  so the (weak-*,weak-*) continuity of A therefore implies
  that (<ref>) holds whenever H_b ∈().



  The use in the previous proof of the somewhat unnatural operator U: →
  can be avoided by using “rectangular” dilation theory, see for example <cit.>.
  In this setting, A dilates to a triple representation of the TRO K(,),
  and every triple representation of K(,) is unitarily equivalent
  to a multiple of the identity representation.


We are ready to prove the remaining implication (ii) ⇒ (iii) of Theorem <ref>.


  
  Let  be a normalized complete Nevanlinna–Pick space and let θ
  be a completely contractive multiplier of ⊙.
  Then there exist (φ_n), (ψ_n) ∈ M_1^C() such that
      θ = ∑_n=1^∞φ_n ψ_n.



  Since θ is a completely contractive multiplier of ⊙,
  the adjoint map M_θ^†: () →() is a (weak-*,weak-*) continuous complete contraction.
  Hence, the dilation theoretic Lemma <ref> implies that there
  exist contractions V: →⊗ℓ^2
  and W: →⊗ℓ^2 such that
  
    M_θ^†(H_b) = W^* (H_b ⊗ I) V

  for all b ∈(). We will show that V and W can be replaced with suitable
  multiplication operators, thus obtaining
  a representation as in the proof of Proposition <ref>.

  To this end, let
  
    = ( ⋂_b ∈() (W^* (H_b ⊗ I)) )^⊂⊗ℓ^2.

  Since H_b T_φ is a Hankel operator for all φ∈(), we find that 
is invariant under T_φ^* ⊗ I for all φ∈().
  Let X = P_ V.
  Then Equation (<ref>) implies that
  
    M_θ^†(H_b) = W^* (H_b ⊗ I) X

  for all b ∈(). Thus, by part (b) of Lemma <ref>,
  we obtain for all φ∈() and all b ∈()
  the identity
  
    W^* (H_b ⊗ I) X T_φ = M_θ^†(H_b) T_φ =
        M_θ^†(H_b T_φ)= W^* (H_b ⊗ I) (T_φ⊗ I) X.

  Therefore, for all φ∈(), we find that
  
    W^* (H_b ⊗ I) [ X T_φ - (T_φ⊗ I) X] = 0

  for all b ∈(), so the definition of  shows that
  
    X T_φ = P_ (T_φ⊗ I) X

  for all φ∈(). In this setting, the Ball–Trent–Vinnikov commutant lifting
  theorem (see <cit.>) implies that there exists a contractive multiplier
  Φ∈(, ⊗ℓ^2) with X = P_ T_Φ, and hence (<ref>)
  shows that
  
    M_θ^†(H_b) = W^*(H_b ⊗ I) P_ T_Φ = W^* (H_b ⊗ I) T_Φ

  for all b ∈().

  A similar argument, applied to the space
  
    = ⋁_b ∈() ( (H_b ⊗ I) T_Φ) ⊂⊗ℓ^2,

  shows that there exists a contractive multiplier Ψ∈(, ⊗ℓ^2) such that
  
    M_θ^†(H_b) = T_Ψ^* (H_b ⊗ I) T_Φ

  for all b ∈().

  To finish the proof, we write Φ = (φ_n) and Ψ = (ψ_n) with (φ_n),(ψ_n) ∈ M^C_1(),
  so that
  
    ⟨ M_θ^†(H_b) 1, 1⟩_
        = ⟨ T_Ψ^* (H_b ⊗ I) T_Φ 1 ,1⟩_
        = ∑_n=1^∞⟨ H_b φ_n, ψ_n⟩_
        = ∑_n=1^∞⟨φ_n ψ_n, b ⟩_.

  Choosing b = k_z and using part (c) of Lemma <ref>, we see that
  
    θ(z) = ∑_n=1^∞φ_n(z) ψ_n(z)
 as desired.


The proof above shows that Theorem <ref> can be regarded as a dilation
theorem for the completely bounded bimodule map M_θ^†: () →().
In different settings, dilation theorems for completely bounded bimodule maps were obtained by several
authors, see for instance <cit.> and the references given there.



§ MULT(H DOT H) AS AN OPERATOR SPACE




In this section, we endow (⊙) with an operator space structure.
Recall that if V and W are operator spaces, then (V,W) is the space of all completely bounded maps
from V into W. This space becomes itself an operator space, via the identification
M_n((V,W)) = (V,M_n(W)); see <cit.>.
It follows from Theorem <ref> that every multiplier of ⊙
defines a completely bounded map on ⊙, so we can regard (⊙) ⊂(⊙, ⊙) and we endow (⊙) with the resulting operator space structure.



 §.§ Factoring elements of Mn(Mult(H dot H))


First, we establish a generalization of the equivalence of (ii) and (iii) of Theorem <ref>
for elements of M_n((⊙)).

Given Φ,Ψ∈(⊗^n, ⊗ℓ^2), say

    Φ =
      [ φ_1 1 φ_1 2     … φ_1 n; φ_2 1 φ_2 2     … φ_2 n;     ⋮     ⋮     ⋱     ⋮ ]   and   Ψ =
      [ ψ_1 1 ψ_1 2     … ψ_1 n; ψ_2 1 ψ_2 2     … ψ_2 n;     ⋮     ⋮     ⋱     ⋮ ],

let Ψ^T denote the transpose of the matrix Ψ and define an n × n matrix
Ψ^T Φ of functions on X by

    (Ψ^T Φ)_i j = ∑_k=1^∞φ_k jψ_k i   (1≤ i,j≤ n).

Note that the sum converges pointwise absolutely by the Cauchy–Schwarz inequality.

With this notation, the norm on M_n((⊙)) can be described as follows.

  
  Let  be a normalized complete Nevanlinna–Pick space on X and let Θ be an n × n
  matrix of functions on X. The following statements are equivalent.
  
    
  * The matrix Θ belongs to the closed unit ball of M_n((⊙)).
    
  * There exist Φ,Ψ in the closed unit ball of (⊗^n, ⊗ℓ^2) so that Θ = Ψ^T Φ.
  


The proof is closely modeled after those of Proposition <ref>
and Theorem <ref>. To use duality, we require
the following result refining the fact that A^*_cb = A_c b
for a completely bounded map A: V → W.
This result is undoubtedly known, but we were not able to find an explicit reference.


  
  Let V and W be operator spaces. Then, the map
  
    (V,W) →(W^*,V^*),    A ↦ A^*

  is a complete isometry.



  Let
  
    [A_i j] ∈ M_n((V,W)) = (V,M_n(W)).

  An elementary computation shows that the norm of [A_i j^*] in M_n((W^*,V^*)) = (W^*,M_n(V^*))
  is at most that of [A_i j] in M_n((V,W)) = (V,M_n(W)). Thus, the map A ↦ A^*
  is a complete contraction. Applying this map again, using that A^** agrees with A
  on V and the fact that the inclusion of an operator space into its bidual is a complete isometry (see
  <cit.>), we conclude that A ↦ A^*
  is a complete isometry.



  (ii) ⇒ (i) As in the proof of Proposition <ref>, an approximation
  argument allows us to assume that Φ,Ψ∈(⊗^n, ⊗^N) for some N ∈. In particular, Θ∈(⊗^n). To compute
  the norm of
  
    Θ = [θ_i j] ∈ M_n((⊙)) ⊂ M_n(( (⊙), (⊙))),

  we apply Lemma <ref> and instead compute the norm of
  
    [ M_θ_i j^†] ∈ M_n( ( (), ()))
        = ( (), M_n(())).

  So let b ∈(). An application of part (a) of Lemma <ref> shows that, using notation as in the discussion
  preceding Theorem <ref>,
  
    [ M_θ_i j^† (H_b)]
        = ∑_r=1^N [ T_ψ_r i^* H_b T_φ_r j ]
        = T_Ψ^* ( H_b ⊗ I_^N) T_Φ.

  Since T_Φ and T_Ψ have norm at most 1, this formula implies that [M_θ_ij^†]
  is a completely contractive map, so (i) holds.

  (i) ⇒ (ii) We merely sketch the main steps, as the proof closely follows that of Theorem <ref>. Let
  Θ = [θ_i j] be an element of the unit ball of M_n((⊙)).
  Using duality, more precisely Lemma <ref>, it follows that the map
  
    () → M_n(()),    H_b ↦ [ M_θ_i j^† (H_b)],

  is completely contractive. With minor changes, the dilation theoretic argument in the proof of Lemma <ref>
  yields linear contractions V: ⊗^n →⊗ℓ^2 and W: ⊗^n →⊗ℓ^2 such that
  
    [M_θ_ij^† (H_b)] = W^* (H_b ⊗ I) V    (b ∈()).

  As in the proof of Theorem <ref>, the commutant lifting theorem allows us
  to replace V and W with multiplication operators. More precisely, there are contractive multipliers Φ, Ψ∈(⊗^n, ⊗ℓ^2)
  so that
  
    [M_θ_ij^† (H_b)] = T_Ψ^*  (H_b ⊗ I) T_Φ   (b ∈()).

  Somewhat more explicitly, to find Φ, define ⊂⊗ℓ^2 and X = P_ V verbatim
  as in the proof of Theorem <ref>. The bimodule property of M_θ_i j^† (part (b) of Lemma <ref>)
  implies that X (T_φ⊗ I_^n) = P_ (T_φ⊗ I_ℓ^2) X for all φ∈(),
  hence the commutant lifting theorem applies. Finally, testing (<ref>) for b = k_z yields
  Θ(z) = Ψ^T(z) Φ(z), so we have found the desired factorization.


The ideas used to prove Theorems <ref> and <ref>
also yield a more concrete description of the norm on M_n(⊙).
If n = 1 and h ∈⊙, then

    h_⊙ = inf{ (f_k) _⊗ℓ^2(g_k)_⊗ℓ^2 : h = ∑_k=1^∞ f_k g_k }.

Indeed, this follows from the definition of the norm on ⊙ by trading constant factors between f_k and g_k.
This last formula can be generalized.
The column operator space structure on  is defined by the identification = B(,),
and the resulting operator space is denoted by _c; see
<cit.>. Thus, M_n(_c) = B(^n,^n).
We also require matrices with infinitely many rows. Let M_∞,n(_c) be the space of all matrices
with entries in  of the form

    f =
      [ f_1 1 f_1 2     ⋯ f_1 n; f_2 1 f_2 2     ⋯ f_2 n;     ⋮     ⋮     ⋱     ⋮ ]

satisfying ∑_i=1^∞f_i j^2 < ∞ for 1 ≤ j ≤ n.
As we did for finite matrices, we regard such a matrix as a bounded
linear operator from ^n to ⊗ℓ^2, and we set

    f_M_∞,n(_c) = f_B(^n, ⊗ℓ^2).

Notice that if n = 1, then M_∞,1(_c) = ⊗ℓ^2 with equality of norms.
Given f,g ∈ M_∞,n(_c), we define as above an n × n matrix g^T f of functions on X by

    (g^T f)_i j  = ∑_k=1^∞ f_k j g_k i   (1 ≤ i, j ≤ n).

By the Cauchy–Schwarz inequality, the sum converges pointwise absolutely on X. 


  
  The following assertions are equivalent for an n × n matrix h of functions on X.
  
    
  * The matrix h belongs to the closed unit ball of M_n(⊙).
    
  * There exist f and g in the closed unit ball of M_∞,n(_c)   so that h = g^T f.
  
  Thus, if h ∈ M_n(⊙), then
  
    h_M_n(⊙) = inf{f_M_∞,n(_c)g_M_∞,n(_c) : h = g^T f },

  and the infimum is attained.



  (ii) ⇒ (i)
  By the Cauchy–Schwarz inequality, the sum defining each entry of h converges
  absolutely in the Banach space ⊙. In particular, each entry of h belongs to ⊙.
  We will show that h belongs to the unit ball of M_n(⊙).
  By definition of the operator space structure on ⊙ as the dual of _0(),
  we have to show that the map
  
    A: _0() → M_n,    H_b ↦[ [h_i j, H_b ] ]_i,j,

  is completely contractive, where
  the inner brackets denote the duality between ⊙ and ().
  To this end, notice that for 1 ≤ i,j ≤ n,
  Equation (<ref>) implies that
  
    [h_i j, H_b] = ∑_k=1^∞ [ f_k j g_k i, H_b]
        = ∑_k=1^∞⟨ H_b f_k j, g_k i⟩_

  for all b ∈().
  Let g denote the entry-wise complex conjugate of g, regarded as a contractive
  operator from ^n to ⊗ℓ^2, and let g^*: ⊗ℓ^2 →^n
  be the Hilbert space adjoint of g. Then
  
    A(H_b) = [ [h_i j, H_b ] ]_i,j = g^* (H_b ⊗ I_ℓ^2) f    (H_b ∈_0()),

  which implies that the map A is completely contractive.

  (i) ⇒ (ii) If h belongs to the unit ball of M_n(⊙),
  then by definition of the operator space structure on ⊙, the map A
  defined in the first part of the proof is completely contractive.
  Applying the Haagerup–Paulsen–Wittstock dilation theorem
  as in the proof of Lemma <ref> and using the
  fact that every *-representation of K()
  is unitarily equivalent to a multiple of the identity representation, we obtain
  linear contractions V: ^n →⊗ℓ^2 and W: ^n →⊗ℓ^2
  so that
  
    A(H_b) = W^* (H_b ⊗ I_ℓ^2) V    (H_b ∈_0()).

  Define f,g ∈ M_∞,n() by f = V
  and g = W. We see that f and g have norm 1 and
  
    [h_i j, H_b] = ∑_k=1^∞⟨ H_b f_k j, g_k i⟩_

  for 1 ≤ i,j ≤ n and H_b ∈_0(). Testing this equation for b = k_z, we conclude
  that h(z) = g^T(z) f(z), as desired.



Lemma <ref> and Theorem <ref> can be restated
  in terms of the Haagerup tensor product ⊗_h of operator spaces (see <cit.>, <cit.> or <cit.>), its weak-* version ⊗_w*  h (see <cit.>) and the
  opposite operator space structure V^op of an operator space V (see <cit.>).
  Concretely, Lemma <ref>
  implies that
  
    _c^op⊗_h _c →⊙,   ∑_n=1^∞ f_n ⊗ g_n ↦∑_n=1^∞ f_n g_n,

  is a complete quotient mapping. Theorem <ref> implies that
  
    ()^op⊗_w*   h() →(⊙),   ∑_n=1^∞φ_n ⊗ψ_n
        ↦∑_n=1^∞φ_n ψ_n,

  is a complete quotient mapping. We will not use these formulations.




 §.§ Comparing the operator space structures of Mult(H) and Mult(H dot H)


We saw in Corollary <ref> that if  satisfies the column-row property (which is the case for instance
for the Drury–Arveson space), then the inclusion

    ι: () ↪(⊙)

is an isomorphism of Banach spaces.
For any normalized complete Nevanlinna–Pick space, the implication (ii) ⇒ (i) of Theorem <ref> shows that ι is a complete contraction. If = H^2(), then the norm of M_n(())
is simply the supremum norm over , hence Theorem <ref> implies that the same
is true for M_n((⊙)). In other words, in the case of H^2(), the map ι
is a completely isometric isomorphism. Note however that the entire
space (⊙) is not completely boundedly isomorphic
  to an operator algebra unless ⊙ is isomorphic
to a Hilbert space by <cit.>.

We show that the phenomenon observed above is somewhat special to the univariate Hardy space.


  Let  be either the Drury–Arveson space H^2_d for d ≥ 2 or the classical Dirichlet space.
  Then the inclusion
  
    ι: () ↪(⊙)

  does not have a completely bounded inverse.



  It follows from Theorem <ref> that for each n ∈, the transpose map
  
    M_n((⊙)) → M_n((⊙)),   Θ↦Θ^T,

  is isometric. On the other hand, there exist sequences of multipliers
  (φ_n) in () that yield a bounded row multiplication operator, but an unbounded column multiplication
  operator. For the Dirichlet space, this can be seen from the discussion preceding Lemma 1 in <cit.>;
  for the Drury–Arveson space, see <cit.>. In particular,
  the norms of the transpose maps
  
    M_n(()) → M_n(()),   Φ↦Φ^T,

  are not uniformly bounded in n, so that the completely contractive map ι
  does not have a completely bounded inverse.


In fact, it is possible to determine explicitly the growth of the norms of (ι^-1)^(n)
in the case of the Drury–Arveson space.
We begin with the following easy estimate.


  
  Let  be a reproducing kernel Hilbert space that satisfies the column-row property
  with constant κ. Then
  
    Ψ^T_(⊗ℓ^2, ⊗^n)≤√(n)κΨ_(⊗^n,⊗ℓ^2)

  for all Ψ∈(⊗^n, ⊗ℓ^2).



  Suppose that
  
    Ψ =
      [ ψ_1 1 ψ_1 2     … ψ_1 n; ψ_2 1 ψ_2 2     … ψ_2 n;     ⋮     ⋮     ⋱     ⋮ ]

  has multiplier norm at most 1. Then each of the columns has multiplier norm at most 1, so
  the column-row property shows that each row
  
    R_i =
        [ T_ψ_1 i T_ψ_2 i T_ψ_3 i       … ]

  has norm at most κ, hence
  
    Ψ^T_(⊗ℓ^2, ⊗^n)^2
        =
        [ R_1;   ⋮; R_n ]^2
        = ∑_i=1^n R_i^* R_i ≤ n κ^2.



If A: V → W is a bounded map between operator spaces, then A^(n)≤ n A,
and this inequality is sharp in general; see for instance <cit.>.
In our setting, the preceding lemma, combined with the implication (i) ⇒ (ii) of Theorem <ref>, implies the following better upper bound.


  
 Let  be a normalized complete Nevanlinna–Pick space on X that satisfies the column-row property
  with constant κ and let
  
    ι : () →(⊙)

  be the completely contractive inclusion. Then, ι is a bijection,
  and
  
    (ι^-1)^(n)≤√(n)κ

  for all n ∈. 

In the Drury–Arveson space, the upper bound in the preceding corollary is essentially best possible.
To see this, we require a refinement of the construction in <cit.>.
Given {φ_1,…,φ_n }⊂(), the quantities

    [ φ_1 φ_2   ⋯ φ_n ]_(⊗^n, )   and   [ φ_1; φ_2;   ⋮; φ_n ]_(, ⊗^n)

are called the row norm and column norm, respectively.


  
  Let d ≥ 2. Then for all n ≥ 1, there exists
  {φ_1,…,φ_n }⊂(H^2_d) with row norm 1 and column norm √(n).



  For 0 ≤ k ≤ n, let
  
    ψ_k =
        nk^1/2
        z_1^k z_2^n-k.

 If α = (α_1,…,α_n) ∈{1,2}^n
  is an ordered n-tuple, let z_α = z_α_1… z_α_n. Since
  each monomial z_1^k  z_2^n-k occurs as one of the monomials
  z_α precisely nk times, we find that
  
    ∑_k=0^n T_ψ_k T_ψ_k^*
        = ∑_k=0^n nk T_z_1^k z_2^n-k T_z_1^k z_2^n-k^*
        = ∑_α∈{1,2}^n T_z_α T_z_α^*.

  It easily
  follows from the fact that the coordinate functions form a row contraction
  that ∑_α T_z_α T_z_α^* ≤ I, hence the row norm
of {ψ_0,…,ψ_n } is at most 1.

  On the other hand,
  
    [ T_ψ_0;     ⋮; T_ψ_n ] 1 ^2   =∑_k=0^n ψ_k^2= ∑_k=0^n nkz_1^k z_2^n-k^2 = n+1.

  Hence, the column norm of {ψ_0,…,ψ_n} is at least √(n+1). Since
  the column norm is also dominated by
  
    ∑_k=0^n T_ψ_k^* T_ψ_k^1/2≤√(n+1)max_0 ≤ k ≤ nT_ψ_k≤√(n+1),

  the estimates for both the column and the row norm are in fact equalities.


Thus, we obtain the exact behavior of (ι^-1)^(n), up to multiplicative constants,
in the case of the Drury–Arveson space.


  Let d ≥ 2 and consider the completely contractive inclusion
  
    ι : (H^2_d) →(H^2_d ⊙ H^2_d).

  Then, there exists a constant κ>0 depending only on d, so that
  
    √(n)≤ (ι^-1)^(n)≤κ√(n)

  for all n ≥ 1.



  The upper bound follows from Corollary <ref> and the column-row property for H^2_d; see <cit.>.

  To obtain the lower bound,
  we use Lemma <ref> to find a row multiplier
  
    Ψ =
        [ ψ_1 ψ_2   … ψ_n ]

  of norm 1 so that Ψ^T_(, ⊗^n) = √(n).
  Let
  
    Φ =
        [ 1 0 … 0 ].

  The implication (ii) ⇒ (i) of Theorem <ref> shows that
  Ψ^T Φ belongs to the closed unit ball of M_n((⊙)). On the other hand,
  
    Ψ^T Φ_M_n(()) = Ψ^T _(, ⊗^n) = √(n)

  thus showing that (ι^-1)^(n)≥√(n).




 §.§ Note added in proof
 The recent paper <cit.> shows that every complete Nevanlinna-Pick space satisfies the column-row property with constant 1. Hence the additional assumption in Corollary <ref> is automatically satisfied



amsplain


