Computing Norms of Free Operators with Matrix Coefficients

Franz Lehner*

Abstract. Let g_{1}, g_{2}, …, g_{n} be the generators of the free group F_{n} and denote by λ its left regular representation. A formula for norms of operators of the form ∑ a_{i} ⊗ λ(g_{i}) where a_{i} are complex matrices or operators is derived similar to the formula of Akemann and Ostrand for the scalar case. This formula can be used to compute norms of arbitrary finitely supported convolution operators on the free group. Essentially the same technique yields a formula for the norms of sums of matrix valued creation and annihilation operators on full Fock space. Finally some results about the numerical evaluation of the formula are proved.

1. Introduction and main results

1.1. Free group. Let Γ_{n} = Z_{2} ∗ Z_{2} ∗ ⋯ ∗ Z_{2} be the free product of n copies of Z_{2} and denote its free generators by g_{1}, g_{2}, …, g_{n}. C. Akemann and P. Ostrand proved in their classical paper [AO76] a formula for the norm of free operators, that is operators of the form ∑ α_{i}λ(g_{i}). This formula improved estimates of M. Leinert [Lei74], M. Bożejko [Boż75] and M. Flory [Flo76]. It was previously known in the case of equal coefficients [Kes59] and simpler proofs were found in [Woe86b], [Woe86a] and [PP88]. In this paper we continue the investigation started in [Leh97] and improve the method to obtain a formula similar to the Akemann-Ostrand formula in the matrix case. The main result about the free group, proved in section 3, is the following theorem.

Theorem 1.1. Let n ≥ 2 and g_{1}, …, g_{n} be the generators of the free product group Γ_{n}, let a_{0}, a_{1}, a_{2}, …, a_{n} be hermitian m×m matrices and assume moreover that a_{0} is positive semidefinite. Then

∥a_{0} ⊗ I + ∑_{i=1}^{n} a_{i} ⊗ λ(g_{i})∥_{min} = inf_{z>0} ∥2 z + a_{0} + ∑_{i=1}^{n} z^{\frac{1}{2}} ((I_{m} + (z^{− \frac{1}{2}} a_{i}z^{− \frac{1}{2}} )^{2})^{\frac{1}{2}} − I_{m}) z^{\frac{1}{2}} ∥, (1.1)

where the infimum is taken over all positive definite invertible m×m matrices z. Moreover, the infimum can be restricted to those z, for which the expression inside the norm signs equals a positive scalar multiple of the identity matrix I_{m}.

A similar formula holds for operators on a general Hilbert space (see remark 4.3) and an elementary matrix trick allows us to deduce a formula for free operators with nonselfadjoint coefficient matrices. Consider first the case a_{0} = 0. Here the relevant property of g_{1}, g_{2}, …, g_{n} is the so called Leinert property (see [AO76] for a definition) and formula (1.1) holds for any set with this property. Other examples of sets with the Leinert property are the set {x_{1}, x_{2}, …, x_{n}, x_{1}^{−1}, x_{2}^{−1}, …, x_{n}^{−1}} of generators of the free group together with their inverses or any translate of the subset {e, x_{1}, x_{2}, …, x_{n}} of the free group. The latter actually is the generic example of a set with the Leinert property. The set of generators of Γ_{n} (which are sometimes called free Rademacher variables) is chosen here mainly because the induced unitaries are selfadjoint; the free group F_{n} can be embedded into Γ_{n+1} in a natural way by mapping x_{i} to g_{0}g_{i} and the remarks in section 5 can be applied to the free group as well. Given non-selfadjoint coefficient matrices a_{1}, a_{2}, …, a_{n} we have

Date: December 14, 1998.

1991 Mathematics Subject Classification. Primary 22D25; Secondary 60J15.

Key words and phrases. Norm of convolution operator, free group, Leinert property, random walk, semicircular system, Fock space.

*This work was supported by the EU-network “Non-commutative geometry” ERB FMRX CT960073

∥∑ a_{i} ⊗ λ(g_{i})∥ = ∥∑ a_{i}^{∗} ⊗ λ(g_{i})∥

= ∥∑ [\begin{matrix}a_{i} & 0 \\ 0 & a_{i}^{∗} \end{matrix}] ⊗ λ(g_{i})∥

= ∥∑ [\begin{matrix}0 & a_{i}^{∗} \\ a_{i} & 0 \end{matrix}] ⊗ λ(g_{i})∥ .

The latter has selfadjoint coefficient matrices and can be fed into formula (1.1). The last assertion in theorem 1.1 allows us to restrict the infimum to matrices of the form z = [ \begin{matrix}z_{1} & 0 \\ 0 & z_{2} \end{matrix} ] and we get the following formula.

Corollary 1.2. Let Γ be a discrete group and suppose that the subset {t_{1}, t_{2}, …, t_{n}} ⊂ Γ satisfies the Leinert property. Then for any set of m × m matrices a_{1}, a_{2}, …, a_{n} we have

∥∑_{i=1}^{n} a_{i} ⊗ λ(t_{i})∥_{min} =

inf_{z_{1},z_{2}>0} max \{∥2 z_{1} + ∑_{i=1}^{n} z1^{\frac{1}{2}} ((I_{m} + z_{1}^{− \frac{1}{2}} a_{i}^{∗}z_{2}^{−1}a_{i}z_{1}^{− \frac{1}{2}} )^{\frac{1}{2}} − I_{m}) z1^{\frac{1}{2}} ∥,

∥2 z_{2} + ∑_{i=1}^{n} z2^{\frac{1}{2}} ((I_{m} + z_{2}^{− \frac{1}{2}} a_{i}z_{1}^{−1}a_{i}^{∗}z_{2}^{− \frac{1}{2}} )^{\frac{1}{2}} − I_{m}) z2^{\frac{1}{2}} ∥\}.

Restricting the infimum to matrices of the form z_{1} = sI_{m} and z_{2} = tI_{m} yields the inequality proved in [Leh97]. The case where a_{0} is not positive definite can also be handled, using identity (5.1) below. The proof of theorem 1.1 splits into two parts. We show first that the norm of the operator in question is dominated by the expression on the right hand side of (1.1), using a technique which one might call a matrix-valued Gabber’s Lemma. A formulation and proof of Gabber’s lemma can be found in [GdlH97] and it was implicitly applied to the free group in [PP88]. In [Leh97] we observed how this technique can be adapted to the matrix valued case to get an inequality, which is sharp enough to derive the optimal constant for a Khinchin-type inequality on the free group which was proved in [HP93] with a different method. However the range of the infimum was not exhausting and examples showed that strict inequality can occur. In the present paper we overcome this limitation by extending the infimum over the whole range of positive definite invertible operators. There are two ways to show that the infimum of (1.1) is equal to the norm which we want to compute: the hard way exploits the fact that the derivative of the function on the right hand side of (1.1) becomes singular when approaching the infimum and gives rise to a family of “asymptotic eigenvectors” giving a lower bound for the norm, approaching the upper bound from below. These eigenvectors are also useful to give error estimates when evaluating the infimum numerically, as explained section 6. Alternatively, relying on the fact that free operators with matrix coefficients are free with amalgamation over the matrix algebra, one can use results about operator-valued free convolution from [Voi95], see lemma 2.3. Although not indispensable, this section sheds some light on how such formulas can be found and also exhibits the analogy to the scalar case.

1.2. Fock space. Another “canonical” example of free operators, which plays a central role in free probability theory and also has some relevance to random matrix theory, is given by semicircular operators. Let us recall the definition of the full Fock space over a Hilbert space, see e.g. [VDN92]. For a Hilbert space H with orthonormal basis {e_{i} : i ∈ I} the full Fock space is defined as the Hilbert space

T (H) = CΩ ⊕ ⨁_{n=1}^{∞} H^{⊗n}

with orthonormal basis

{e_{i_{1},i_{2},... ,i_{k}} = e_{i_{1}} ⊗ e_{i_{2}} ⊗ … ⊗ e_{i_{k}} : k ∈ N, i_{j} ∈ I} ∪ {Ω}. (1.2)

For h ∈ H define the creation operator on T (H) by

l(h) Ω = h l(h) h_{1} ⊗ h_{2} ⊗ ⋯ ⊗ h_{k} = h ⊗ h_{1} ⊗ h_{2} ⊗ ⋯ ⊗ h_{k};

its adjoint is the annihilation operator

l^{∗}(h) Ω = 0 l^{∗}(h) h_{1} ⊗ h_{2} ⊗ ⋯ ⊗ h_{k} = ⟨h_{1}, h⟩ h_{2} ⊗ h_{3} ⊗ ⋯ ⊗ h_{k}.

We shall abbreviate the notation by writing l_{i} = l(e_{i}). It is not difficult to show that the family of algebras A_{i} generated by {l_{i}, l_{i}^{∗}}, i ∈ I is free (see definition 2.1) with respect to the vacuum state ⟨ . Ω, Ω⟩. Full Fock space is the natural framework for free semicircular families (sometimes called semicircular systems), that is free families of self-adjoint operators x_{i} with distribution

τ(x_{i}^{n}) = \frac{1}{2π} ∫_{−2}^{2} t^{n}\sqrt{4 − t^{2}} dt (1.3)

which appears as the limit distribution of the free central limit theorem and is therefore sometimes called the free Gaussian distribution. It can be shown that the family {l_{i} + l_{i}^{∗} : i ∈ I} forms a semicircular system. The basis of T (H) given by (1.2) can be viewed as the vertex set of a tree with origin Ω and with edges [e_{i_{1},i_{2},... ,i_{k}}, e_{i_{1},i_{2},... ,i_{k+1}}] and it is not surprising that a formula similar to (1.1) holds in the context creation and annihilation operators on Fock space.

Theorem 1.3. For n ≥ 2 let l_{1}, l_{2}, …, l_{n} be the canonical set of creation operators on T (C^{n}) and let a_{0}, a_{1}, …, a_{n} be m×m matrices such that a_{0} is positive semidefinite. Then

∥a_{0} ⊗ I + ∑_{i=1}^{n} a_{i} ⊗ l_{i} + a_{i}^{∗} ⊗ l_{i}^{∗}∥_{min} = inf_{z>0} ∥z + a_{0} + ∑_{i=1}^{n} a_{i}^{∗}z^{−1}a_{i}∥, (1.4)

where the infimum is taken over all positive definite invertible m×m matrices z. Moreover, the infimum can be restricted to those z, for which the expression inside the norm signs equals a multiple of the identity matrix I_{m}.

In the case of the more general operator ∑ a_{i}⊗l_{i}+∑ b_{i}⊗l_{i}^{∗} we can replace the coefficients according to the rule

ã_{i} = [\begin{matrix}0 & a_{i} \\ b_{i}^{∗} & 0 \end{matrix}]  b̃_{i} = ã_{i}^{∗}

without changing the norm and we get the following formula.

Corollary 1.4. Let a_{1}, a_{2}, …, a_{n} and b_{1}, b_{2}, …, b_{n} be arbitrary m × m matrices. Then

∥∑_{i=1}^{n} a_{i} ⊗ l_{i} + b_{i} ⊗ l_{i}^{∗}∥_{min} = inf_{z_{1},z_{2}>0} max {∥z_{1} + ∑_{i=1}^{n} a_{i}^{∗}z_{2}^{−1}a_{i}∥, ∥z_{2} + ∑_{i=1}^{n} b_{i}z_{1}^{−1}b_{i}^{∗}∥}.

As a special case, whenever ∑ a_{i}^{∗}a_{i} = α^{2}I and ∑ b_{i}b_{i}^{∗} = β^{2}I, then

∥∑_{i=1}^{n} a_{i} ⊗ l_{i} + b_{i} ⊗ l_{i}^{∗}∥_{min} = α + β;

in particular, the Banach space spanned by l_{1}, l_{2}, …, l_{n} and their adjoints is isometric to the ℓ_{1}-direct sum ℓ_{2}^{n} ⊕_{1} ℓ_{2}^{n}.

As another special case, when the coefficient matrices a_{i} are selfadjoint, we get the following formula for semicircular operators.

Corollary 1.5. For n ≥ 2 let x_{1}, x_{2}, …, x_{n} be a free semicircular family and let a_{0}, a_{1}, …, a_{n} be hermitian m × m matrices such that a_{0} is positive semidefinite. Then

∥a_{0} ⊗ I + ∑_{i=1}^{n} a_{i} ⊗ x_{i}∥_{min} = inf_{z>0} ∥z + a_{0} + ∑_{i=1}^{n} a_{i}z^{−1}a_{i}∥, (1.5)

where the infimum is taken over all positive definite invertible m×m matrices z. Moreover, the infimum can be restricted to those z, for which the expression inside the norm signs equals a multiple of the identity matrix I_{m}.

Again remark 4.3 shows that a similar formula holds for coefficients taken from B(H) for an arbitrary Hilbert space H, and the same matrix trick as above can be used to get rid of the limitation to self-adjoint matrices. Also note that in the case where a_{0} = 0, formula (1.5) remains unchanged if we replace the semicircular system by a circular system, i.e. operators of the form \frac{1}{\sqrt{2}}(x_{i} + ix′_{i}), where x_{1}, x′_{1}, x_{2}, x′_{2}, …, x_{n}, x′_{n} form a free semicircular family. Such operators can be modeled on Fock space by free copies of l_{1} +l_{2}^{∗}. Following the lines before corollary 1.2 we get a formula for non-selfadjoint coefficient matrices.

Corollary 1.6. Let x_{1}, x_{2}, …, x_{n} be a free circular or semicircular family and let a_{1}, a_{2}, …, a_{n} be arbitrary m × m matrices. Then

∥∑_{i=1}^{n} a_{i} ⊗ x_{i}∥_{min} = inf_{z_{1},z_{2}>0} max {∥z_{1} + ∑_{i=1}^{n} a_{i}^{∗}z_{2}^{−1}a_{i}∥, ∥z_{2} + ∑_{i=1}^{n} a_{i}z_{1}^{−1}a_{i}^{∗}∥}

The upper bound in theorem 1.3 is again obtained by the use of a Gabber-type function. The proof of the lower bound is complicated by the fact that the vacuum state is neither tracial nor faithful, so that lemma 2.3 does not apply to give a lower bound for the norm. The alternative method of exhibiting a norming set of vectors is still possible and gives the lower bound. It is interesting to note here the following consequence of the formulae (1.1) and (1.5) together with the operator inequality (3.10).

Corollary 1.7. The map x_{i} ↦ λ(g_{i}) is a complete contraction, i.e. for any set of matrices a_{1}, …, a_{n} the inequality

∥∑ a_{i} ⊗ λ(g_{i})∥ ≤ ∥∑ a_{i} ⊗ x_{i}∥

holds.

In order to apply the formula to more general operators, we have to include the onedimensional projection p_{Ω}. A slight modification of the proof then yields the following formula.

Theorem 1.8. Let a_{0}, a_{1}, …, a_{n} and b be m × m matrices such that a_{0} and a_{0} + b are positive semidefinite. Then

∥a_{0} ⊗ I + b ⊗ p_{Ω} + ∑_{i=1}^{n} a_{i} ⊗ l_{i} + a_{i}^{∗} ⊗ l_{i}^{∗}∥_{min} = inf_{z>0,z+b>0} ∥z + a_{0} + b + ∑_{i=1}^{n} a_{i}^{∗}(z + b)^{−1}a_{i}∥, (1.6)

where the infimum is taken over all positive definite invertible m×m matrices z such that b + z is also positive definite and invertible. Moreover, the infimum can be restricted to those z, for which the expression inside the norm signs equals a multiple of the identity matrix I_{m}.

The paper is organized as follows. In section 2 we review some relevant facts about free probability. Sections 3 and 4 are devoted to the proofs of the main theorems on the free group and Fock space respectively. For the case of the free group only the upper inequality needs to be shown, the lower inequality follows from the general considerations in remark 2.4. However, we also include an alternative, more technical proof of the lower bound which involves asymptotic eigenvectors. This is also of some relevance for the numerical computations considered in section 6. This proof can be literally transferred to operators on Fock space, where the state is neither tracial nor faithful and remark 2.4 does not apply. Since the formulae are somewhat less complicated in the latter case, the reader may want to choose to read section 4 first. In section 5 we show how the matrix-valued formulae can be used to compute the norm of finitely supported convolution operators on free groups and on Fock space. Section 6 gives some theoretical background on how to perform numerical computations on a computer. Acknowledgements. I am grateful to Ken Dykema and Uffe Haagerup for some useful discussions.

2. Preliminaries about B-valued free convolution

In this section we give an overview about some results from [Voi95] which are needed in the proofs. Although the proofs can be done without explicitly relying on this theory, it enlightens the concepts involved. First of all let us recall the definition of freeness and freeness with amalgamation.

Definition 2.1 ([VDN92]). Let A be a unital algebra over C and let A_{i} ⊂ A, i ∈ I be unital subalgebras containing a common unital subalgebra B. Let further φ : A → B be a conditional expectation, i.e. a linear map such that φ(b) = b and φ(b′ab′′) = b′φ(a)b′′ for every b, b′, b′′ ∈ B and a ∈ A. The elements of A will also be called B-valued random variables and (A, φ) is called a B-valued noncommutative probability space. The family A_{i} is called B-free with respect to φ if

φ(a_{1}a_{2} … a_{k}) = 0

whenever a_{j} ∈ A_{i_{j}} with i_{1} ≠ i_{2} ≠ ⋯ ≠ i_{k} are such that φ(a_{j}) = 0. A family of subsets Ω_{i} ⊂ A (respectively elements a_{i} ∈ A) is called B-free if the family of subalgebras generated by Ω_{i} ∪ B (respectively, {a_{i}} ∪ B) is B-free. In the case where B = C and φ is a linear functional, the term free is used instead of C-free. The term C^{∗}-probability space is used when the involved algebras are C^{∗}-algebras and the conditional expectation is positive.

Typical examples of B-free families arise in amalgamated free products of C^{∗}-algebras. In what follows we shall be concerned mainly with the case of C^{∗}-algebras and tracial states, in particular the following situation. Suppose that A_{i} ⊂ A, i ∈ I are C-free with respect to some trace state τ on A. Let B be an arbitrary C^{∗}-algebra and form the minimal C^{∗} tensor product B ⊗ A. Then the algebras B ⊗ A_{i} are (B ⊗ I_{A})-free with respect to the conditional expectation Id_{B} ⊗ τ. We shall identify B with the subalgebra B ⊗ I of B ⊗ A. We shall also need some facts about B-valued additive free convolution. In the setting of definition 2.1 above, for any element a ∈ A we use the term G-transform to denote the function

G_{a}(b) = φ ((b − a)^{−1})

and the same term is used for the formal power series over B defined by

G̃_{a}(b) = G(b^{−1})

= ∑_{n=0}^{∞} φ (b(ab)^{n})

= φ (b(I − ab)^{−1}) .

This function is invertible for b ∈ B invertible and ∥b∥ sufficiently small, i.e. there exists some function L_{a} : B → B and some neighbourhood V of 0 such that

G̃_{a}(L_{a}(b)) = b = L_{a}(G̃_{a}(b))

for b ∈ GL(B) ∩ V. Moreover, the values of this function are invertible and have the form

K_{a}(b) = L_{a}(b)^{−1} = b^{−1} + R_{a}(b) (2.1)

so that

G(K(b)) = b

The functions K_{a} and R_{a} are called the K- and R-transforms of a. We will be working with slightly modified functions, namely K̃_{a}(z) = K_{a}(\frac{z^{−1}}{2} ) on the free group and K̃_{a}(z) = K_{a}(z^{−1}) on Fock space. The R-transform linearizes free convolution as shown in the following theorem.

Theorem 2.2 ([Voi95]). Let (A, φ) be B-valued noncommutative probability space and suppose a′, a′′ ∈ A are B-freely independent random variables with R-transforms R′ and R′′ respectively. Then the R-transform of a = a′ + a′′ is given by

R(b) = R′(b) + R′′(b).

We will need some more information about the range of invertibility of the G-transform in our special case, compare [Woe86b, Prop. 1] for the scalar case.

Lemma 2.3. Let (A, τ) be a C^{∗}-probability space with faithful trace state τ. Let further B ⊂ A be a finite dimensional unital ∗-subalgebra and denote by τ( . |B) the (faithful) conditional expectation from A to B. Let a ∈ A be a self-adjoint B-valued random variable in (A, τ( . |B)) with Cauchy transform G̃(b). Then G̃ as a function defined on some subset of the positive cone of B is invertible in an open set including the open line segment ]0, \frac{1}{∥a∥}[.

Proof. It suffices to show that the Fréchet derivative of G̃ is invertible along this line segment. Since B is finite dimensional, this is equivalent to it being injective. For h, b ∈ B the directional derivative of G̃ at b with respect to h can be computed as follows. It follows from the product rule that D_{h}b^{−1} = −b^{−1}hb^{−1} and hence

D_{h}G̃(b) = lim_{t→0} \frac{1}{t}(G̃(b + th) − G̃(b))
= τ (h(I − ab)^{−1} + b(I − ab)^{−1}ah(I − ab)^{−1}|B)
= τ ((I − ba)^{−1}h(I − ab)^{−1}|B)

Now for β ∈] − \frac{1}{∥a∥}, \frac{1}{∥a∥}[ the inverse of I − βa is positive definite and since the trace τ is faithful, if follows that

τ (D_{h}G̃(βI)h^{∗}) = τ (τ ((I − βa)^{−1}h(I − βa)^{−1}|B) h^{∗})
= τ (((I − βa)^{− \frac{1}{2}} h(I − βa)^{− \frac{1}{2}} ) ((I − βa)^{− \frac{1}{2}} h(I − βa)^{− \frac{1}{2}} )^{∗})

is different from zero, unless h = 0. In particular, D_{h}G̃(βI) ≠ 0.

Remark 2.4. From this lemma, one can deduce the following fact, analogous to the scalar case. Let (A, τ) be a C^{∗}-probability space with faithful finite trace state τ. Consider the M_{m}-valued noncommutative probability space M_{m}(A) with conditional expectation φ = Id M_{m} ⊗τ. Let a ∈ M_{m}(A) be a self-adjoint M_{m}-valued random variable. Then its Cauchy transform G is invertible in some open set containing ] − ∞, min σ(a)[ ∪ ] max σ(a), +∞[ and if K is an univalent inverse of G on some maximal open set Ω, then the norm of a can be computed from the following data:

min σ(a) = sup{s : ∃z < 0 : K(z) = sI_{m}}
max σ(a) = inf{s : ∃z > 0 : K(z) = sI_{m}}

The aim of the following two sections is to make this formula more explicit on the free group and to establish it also on Fock space, where the vacuum expectation is neither faithful nor tracial. We finish this section with an elementary Cauchy-Schwarz type lemma for Hilbert-Schmidt operators which was already used in [Leh97].

Lemma 2.5. Let H be a Hilbert space. Then for any sequence x_{1}, x_{2}, …, x_{n} ∈ B(H) and y_{1}, y_{2}, …, y_{n} ∈ HS(H) we have the inequality

∥∑_{i=1}^{n} x_{i}y_{i}∥_{HS} ≤ ∥∑_{i=1}^{n} x_{i}x_{i}^{∗}∥^{\frac{1}{2}} tr (∑_{i=1}^{n} y_{i}^{∗}y_{i})^{\frac{1}{2}} . (2.2)

3. Matrix valued operators on the free group

Throughout this section let T denote the operator

T = a_{0} ⊗ I + ∑ a_{i} ⊗ λ(g_{i}). (3.1)

Its K-transform can be computed using theorem 2.2.

Proposition 3.1. The K-transform of T at a positive invertible matrix z is given by

K(z) = z^{−1} + a_{0} + \frac{1}{2} ∑_{i=1}^{n} z^{− \frac{1}{2}} ((I + 4(z^{\frac{1}{2}} a_{i}z^{\frac{1}{2}} )^{2})^{\frac{1}{2}} − I) z^{− \frac{1}{2}} . (3.2)

Proof. Let g be the generator of Z_{2} and τ be the canonical trace on A = vN(Z_{2}). Then for a ∈ B the B-valued random variable a ⊗ λ(g) ∈ B ⊗ A has Cauchy-transform

G̃(b) = b(Id_{B} ⊗ τ) ((I_{B} ⊗ I − ab ⊗ λ(g))^{−1})

= b ∑_{k=0}^{∞}(ab)^{k}τ(λ(g)^{k})

= b(I − (ab)^{2})^{−1}

Finding the inverse of this function is accomplished by solving the operator equation G̃(b) = z i.e.

b = z(I − (ab)^{2}). (3.3)

In order to find the right formula we will assume that all the operators involved are invertible; once the formula is found, it can easily be verified for non-invertible arguments by a density argument. Assuming that a, b, z satisfy equation (3.3), it follows that az = ab(I − (ab)^{2})^{−1} commutes with ab and we can apply commutative algebra to solve the equivalent equation

(ab)^{2}az + ab = az;

multiplying by az and completing the square on the left hand side yields

azab = ± (\frac{1}{4}I + (az)^{2})^{\frac{1}{2}} − \frac{1}{2}I.

It follows that ab = f^{±}(az) where f^{±}(x) = \frac{±\sqrt{1+4x^{2}}−1}{2x} satisfies

\frac{1}{f^{±}(x)} = \frac{2x}{±\sqrt{1 + 4x^{2}} − 1} \frac{∓\sqrt{1 + 4x^{2}} − 1}{∓\sqrt{1 + 4x^{2}} − 1} = −f^{∓}(x) (3.4)

and the K-transform maps z to one of

b^{−1} = −f^{∓}(az)a = −\frac{1}{2} (∓ (I + 4(az)^{2})^{\frac{1}{2}} − I) z^{−1}

and since K(z) must have the form (2.1), we must choose the f^{+}-branch of the solution, i.e.

L(z) = a^{−1}f^{+}(az)
K(z) = −f^{−}(az)a = \frac{1}{2} ((I + 4(az)^{2})^{\frac{1}{2}} + I) z^{−1}.

For z positive definite we get

\begin{aligned}R(z) & = K(z) − z^{−1} \\ & = \frac{1}{2} ((I + 4(az)^{2})^{\frac{1}{2}} − I) z^{−1} \\ & = \frac{1}{2}z^{− \frac{1}{2}} ((I + 4(z^{\frac{1}{2}} az^{\frac{1}{2}} )^{2})^{\frac{1}{2}} − I) z^{− \frac{1}{2}} \end{aligned} (3.5)

It follows by theorem 2.2 that the K-transform of the operator (3.1) at z > 0 is given by (3.2).

It will be convenient to work with

K̃(z) = K(\frac{z^{−1}}{2} ) = 2z + a_{0} + ∑ R̃_{i}(z) (3.6)

instead; the modified R-transform R̃_{i}(z) = R_{i}(\frac{z^{−1}}{2} ) satisfies the following relations. Let

S_{i}^{±}(z) = ((I + (a_{i}z^{−1})^{2})^{\frac{1}{2}} ± I) z = z^{\frac{1}{2}} ((I + (z^{− \frac{1}{2}} a_{i}z^{− \frac{1}{2}} )^{2})^{\frac{1}{2}} ± I) z^{\frac{1}{2}} ; (3.7)

then it follows from (3.4) that

R̃_{i}(z) = R_{i}(\frac{z^{−1}}{2} ) = S_{i}^{−}(z) = a_{i}S_{i}^{+}(z)^{−1}a_{i} (3.8)

and

S_{i}^{+} − S_{i}^{−} = 2z. (3.9)

In particular, since S_{i}^{−}(z) is positive this implies S_{i}^{+}(z) ≥ 2z and from (3.8) we get that

R̃_{i}(z) ≤ \frac{1}{2}a_{i}z^{−1}a_{i} (3.10)

which is needed for corollary 1.7.

Proof of Theorem 1.1. The lower inequality follows from lemma 2.3 which ensures that the open line segment ]∥T ∥, ∞[ I_{m} is in the range of K and hence in the range of K̃. To prove the upper inequality we proceed as in [Leh97] with the new Gabber-function and the inclusion of the identity operator with coefficient a_{0}. Let us first assume that the matrices a_{i} are invertible. We represent M_{m}⊗vN(Γ_{n}) on the Hilbert space L_{2}(M_{m}, tr_{m})⊗_{2} l_{2}(Γ_{n}), where tr_{m} is the usual trace on M_{m} and the noncommutative L_{2}-norm on M_{2} coincides with the Hilbert-Schmidt norm. In the first part of the proof we show that the right hand side of (1.1) provides an upper bound for the norm of T. Given any finitely supported h = ∑ h(g) ⊗ δ_{g} ∈ l_{2}(Γ_{n}; HS(M_{m})) we have to estimate ∥T h∥_{2}^{2} = ∑_{g∈Γ_{n}} ∥T h(g)∥_{HS}^{2}. Let z ∈ M_{m} be a positive definite invertible matrix and denote by R̃_{i} the M_{m}-valued R-transform given by formula (3.8). Define for i ∈ {1, 2, …, n} the functions

P̃_{i}(z, g)) = \begin{cases}a_{i}^{−1}R̃_{i}(z) = a_{i}^{−1}S_{i}^{−}(z) & if |g_{i}g| = |g| + 1 \\ R̃_{i}(z)^{−1}a_{i} = a_{i}^{−1}S_{i}^{+}(z) & if |g_{i}g| = |g| − 1 \end{cases}

For fixed g ∈ Γ_{n} we have

∥T h(g)∥_{HS}^{2} = ∥∑_{i=1}^{n} a_{0}h(g) + a_{i} P̃_{i}(z, g) P̃_{i}(z, g)^{−1} h(g_{i}g)∥_{HS}^{2}

and since the operator a_{i}P̃_{i}(z, g) = S_{i}^{±}(z) is positive, we can apply lemma 2.5 to the sequences

x_{0} = a0^{\frac{1}{2}} x_{i} = (a_{i}P̃_{i}(z, g))^{\frac{1}{2}}

and

y_{0} = a0^{\frac{1}{2}} h(g) y_{i} = (a_{i}P̃_{i}(z, g))^{\frac{1}{2}} P̃_{i}(z, g)^{−1} h(g_{i}g)

to obtain

∥T h(g)∥_{HS}^{2} ≤ ∥a_{0} + ∑_{i=1}^{n} a_{i}P̃_{i}(z, g)∥ tr (h(g)^{∗}a_{0}h(g) + ∑_{i=1}^{n} h(g_{i}g)^{∗}P̃_{i}(z, g)^{∗}^{−1}a_{i}h(g_{i}g)) .

Note that among the words { g_{1}g, g_{2}g, …, g_{n}g } there is at most one cancellation so that there is always the “−” sign in P̃_{i}(z, g) except maybe one case and since all the summands a_{i} P̃_{i}(z, g) are positive operators, we get from (3.9) the operator inequality

a_{0} + ∑_{i=1}^{n} a_{i} P̃_{i}(z, g) ≤ 2z + a_{0} + ∑ R̃_{i}(z) = K̃(z)

This upper bound does not depend on the group element g and we can estimate the norm of T h as follows. Noting that P̃_{i}(z, g)^{∗}^{−1}a_{i} = S_{i}^{±}(z) is positive we get as before

∥T h∥_{2}^{2} ≤ ∥K̃(z)∥ ∑_{g∈Γ_{n}} tr (h(g)^{∗}a_{0}h(g) + ∑_{i=1}^{n} h(g)^{∗}P̃_{i}(z, g_{i}g)^{∗}^{−1} a_{i} h(g) )

≤ ∥K̃(z)∥ ∑_{g∈Γ_{n}}∥a_{0} + ∑_{i=1}^{n} P̃_{i}(z, g_{i}g)^{∗}^{−1} a_{i}∥ ∥h(g)∥_{HS}^{2};

now P̃_{i}(z, g_{i}g) has always the “+” sign with at most one exception possible in the case when there is already cancellation in g_{i}g. Thus P̃_{i}(z, g_{i}g)^{−1} has the “−” signs and with the formula for the inverse we get

a_{0} + ∑_{i=1}^{n} P̃_{i}(z, g_{i}g)^{∗}^{−1} a_{i} ≤ K̃(z)

Again the bound is independent of g and finally we get the inequality we wanted:

∥T h∥_{2}^{2} ≤ ∥K̃(z)∥^{2}∥h∥^{2}.

This finishes the proof for invertible coefficients a_{i}. If they are not, they can be approximated by such and the inequality still holds.

To give an alternative proof of the lower inequality we need to compute the derivative of the K-transform. This will involve the following variant of R̃_{i}, namely

P̃_{i}(z) = a_{i}^{−1}S_{i}^{−}(z) = S_{i}^{+}(z)^{−1}a_{i}. (3.11)

Denote for x ∈ M_{m} by L_{x} and R_{x} the representations of x on L_{2}(M_{m}, tr) by left and right multiplication.

Proposition 3.2. The derivative of the modified K-transform (3.6) is given by

D_{h}K̃(z) = 2(Id − ∑_{i=1}^{n} (Id − (Id + L_{P̃_{i}(z)^{∗}}R_{P̃_{i}(z)})^{−1}))[h] (3.12)

Proof. As a consequence of (3.8) we have R̃_{i}(z)^{−1}a_{i} = a_{i}^{−1}R̃_{i}(z) + 2a_{i}^{−1}z and taking the derivative of this equation we get

−R̃_{i}(z)^{−1}(D_{h}R̃_{i}(z))R̃_{i}(z)^{−1}a_{i} = a_{i}^{−1}D_{h}R̃_{i}(z) + 2a_{i}^{−1}h

which after some elementary manipulations leads to

D_{h}R̃_{i}(z) + P̃_{i}(z)^{∗}(D_{h}R̃_{i}(z) + 2h)P̃_{i}(z) = 0.

In terms of left and right multiplication operators we can rewrite this as

2h = (Id + L_{P̃_{i}(z)^{∗}}R_{P̃_{i}(z)})[D_{h}R̃_{i}(z) + 2h]

and hence

D_{h}R̃_{i}(z) = 2((Id + L_{P̃_{i}(z)^{∗}}R_{P̃_{i}(z)})^{−1}[h] − h), (3.13) which implies (3.12).

In order to relate formula (3.12) to asymptotic eigenvectors as in [PP88], we need the following noncommutative version of [FTS94, Ch. 2, Lemma 1.5]

Proposition 3.3. Let φ : Γ_{n} → B(H) be a function such that φ(e) = I and φ(xy) = φ(x)φ(y) whenever |xy| = |x| + |y|, i.e. whenever there is no cancellation in the product xy. Then

∑_{x∈Γ} φ(x) = (I − ∑_{i=1}^{n} φ(g_{i})(I + φ(g_{i}))^{−1})^{−1}

if the sum on the left hand side converges absolutely.

Proof. The proof is done by induction on the number n of generators of the group Γ_{n}. To simplify notation we shall denote a_{i} = φ(g_{i}). This determines the function completely and for any reduced word g = g_{i_{1}}g_{i_{2}} ⋯ g_{i_{k}} we have φ(g) = a_{i_{1}}a_{i_{2}} ⋯ a_{i_{k}}. For n = 1 the sum is finite and we have

φ(e) + φ(g_{1}) = I + a_{1}
= (I − a_{1}(I + a_{1})^{−1})^{−1}.

Assuming that the claim is true for 1, 2, …, n − 1, we split off the last free factor and set A = a_{n} and B = ∑_{x∈Γ_{n−1}\{e}} φ(x). Then any element of Γ_{n} has one of the forms

(g_{n}x_{1})(g_{n}x_{2}) ⋯ (g_{n}x_{m})
(g_{n}x_{1})(g_{n}x_{2}) ⋯ (g_{n}x_{m})g_{n}
(x_{1}g_{n})(x_{2}g_{n}) ⋯ (x_{m}g_{n})
(x_{1}g_{n})(x_{2}g_{n}) ⋯ (x_{m}g_{n})x_{m+1}

with x_{i} ∈ Γ_{n−1} \ {e}. Hence we can write

∑_{x∈Γ_{n}} φ(x) = ∑_{k=0}^{∞}(AB)^{k} + ∑_{k=0}^{∞}(AB)^{k}A + ∑_{k=1}^{∞}(BA)^{k} + ∑_{k=0}^{∞}(BA)^{k}B
= (I + B)(I − AB)^{−1}(I + A)
= ((I + A)^{−1}(I − AB)(I + B)^{−1})^{−1}
= ((I + A)^{−1}((I + A)(I + B) − A − B − 2AB)(I + B)^{−1})^{−1}
= (I − (I + A)^{−1}(A(I + B) + (I + A)B)(I + B)^{−1})^{−1}
= (I − A(I + A)^{−1} − B(I + B)^{−1})^{−1}

By assumption we have

B = (I − ∑_{i=1}^{n−1} a_{i}(I + a_{i})^{−1})^{−1} − I

and hence

B(I + B)^{−1} = ∑_{i=1}^{n−1} a_{i}(I + a_{i})^{−1}

which yields the claim.

Proposition 3.4. The multiplicative function

ω_{z,y} : Γ_{n} → M_{m}
g_{i_{1}}g_{i_{2}} ⋯ g_{i_{k}} ↦ P̃_{i_{1}}(z)P̃_{i_{2}}(z) ⋯ P̃_{i_{k}}(z)y.

satisfies

T ω_{z,y} = K̃(z)ω_{z,y} − 2zy ⊗ δ_{e} (3.14)

and the norm of ω̃_{z} = ω_{z,I_{m}} is given by

∥ω̃_{z}∥^{2} = 2 tr (DK̃(z)^{−1}[I_{m}]) (3.15)

Proof. The proof is straightforward calculation, namely

T ω_{z,y}(e) = a_{0}y + ∑_{i=1}^{n} a_{i}ω_{z,y}(g_{i})

= (a_{0} + ∑_{i=1}^{n} a_{i}P̃_{i}(z))y

= (K̃(z) − 2z) y

and

T ω_{z,y}(g_{i_{1}}g_{i_{2}} ⋯ g_{i_{k}}) = a_{i_{1}}ω_{z,y}(g_{i_{2}} ⋯ g_{i_{k}}) + ∑_{i≠i_{1}} a_{i}ω_{z,y}(g_{i}g_{i_{1}} ⋯ g_{i_{k}})

= (a_{i_{1}} P̃_{i_{1}}(z)^{−1} + ∑_{i≠i_{1}} a_{i}P̃_{i}(z))ω_{z,y}(g_{i_{1}}g_{i_{2}} ⋯ g_{i_{k}})

= K̃(z)ω_{z,y}(g_{i_{1}}g_{i_{2}} ⋯ g_{i_{k}})

which implies (3.14). We can compute the norm of ω̃_{z} = ω_{z,I_{m}} as element of l_{2}(Γ; L_{2}(M_{m})) using proposition 3.3,

∥ω̃_{z}∥^{2} = ∑_{x∈Γ} tr(ω̃_{z}(x)^{∗}ω̃_{z}(x))

= ∑_{i_{1},i_{2},... ,i_{k}} tr(P̃_{i_{1}}(z)^{∗}P̃_{i_{2}}(z)^{∗} ⋯ P̃_{i_{k}}(z)^{∗}P̃_{i_{k}}(z)P̃_{i_{k−1}}(z) ⋯ P̃_{i_{1}}(z))

= tr((Id − ∑_{i=1}^{n} L_{P̃_{i}(z)^{∗}}R_{P̃_{i}(z)}(Id + L_{P̃_{i}(z)^{∗}}R_{P̃_{i}(z)})^{−1})^{−1}[I_{m}])

and this together with (3.12) implies (3.15).

The following fix point iteration gives a method to compute numerically the matrix valued Cauchy transform of T, i.e. the inverse of the K-transform.

Proposition 3.5. Let a_{i}, i ∈ \{0, 1, …, n\} be matrices as in theorem 1.1, let c > \frac{3}{2} \sqrt{2} and let μ > \frac{1}{\sqrt{2}} be such that 2μ + \frac{1}{2μ} = c. Then for any selfadjoint b ∈ M_{m} such that b − a_{0} ≥ c∥ ∑_{i=1}^{n} a_{i}^{2}∥^{\frac{1}{2}} I_{m} the sequence defined recursively by

z_{0} ∈ M_{μ} = \{z ∈ M_{m} : z ≥ μ∥∑_{i=1}^{n} a_{i}^{2}∥^{\frac{1}{2}} I_{m}\}

z_{k+1} = \frac{1}{2} (b − a_{0} − ∑ R̃_{i}(z_{k}))

converges to the unique solution z̄ ∈ M_{μ} of the equation K̃(z) = b i.e. G(b) = \frac{z̄^{−1}}{2} .

Proof. This is an application of the well known Banach fix point theorem. We have to show that the function

F (z) = \frac{1}{2} (b − a_{0} − ∑ R̃_{i}(z))

maps the set M_{μ} contractively into itself. Then the unique fixed point of F is the claimed solution of the equation. From (3.10) we get that the image of any z ∈ M_{μ} satisfies

F (z) ≥ \frac{1}{2} (b − a_{0} − \frac{1}{2} ∑ a_{i}z^{−1}a_{i})

≥ \frac{1}{2} (c − \frac{1}{2μ}) ∥∑ a_{i}^{2}∥^{\frac{1}{2}}

= μ ∥∑ a_{i}^{2}∥^{\frac{1}{2}} .

This shows that F (M_{μ}) ⊆ M_{μ}. Assume now z′, z′′ ∈ M_{μ}. We have to estimate the norm of the difference F (z′) − F (z′′). From (3.8) we get that

∥F (z′) − F (z′′)∥ = \frac{1}{2} ∥∑ a_{i} (S_{i}^{+}(z′)^{−1} − S_{i}^{+}(z′′)^{−1}) a_{i}∥

≤ \frac{1}{2} ∥∑ a_{i}^{2}∥ max_{i} ∥S_{i}^{+}(z′)^{−1} − S_{i}^{+}(z′′)^{−1}∥

and it suffices to show that the derivative of S_{i}^{+}(z)^{−1} is bounded above uniformly by some constant strictly smaller than _{\frac{2}{∥∑ a_{i}^{2}∥}} for z ∈ M_{μ}. From (3.8) and (3.13) we get

D_{h}S_{i}^{+}(z)^{−1} = −S_{i}^{+}(z)^{−1} (D_{h}S_{i}^{+}(z)) S_{i}^{+}(z)^{−1}
= −2S_{i}^{+}(z)^{−1} (Id + L_{P̃_{i}(z)^{∗}}R_{P̃_{i}(z)})^{−1} [h]S_{i}^{+}(z)^{−1}.

Now for μ > \frac{1}{2} it follows from (3.11) that

∥P̃_{i}(z)∥ = ∥S_{i}^{+}(z)^{−1}a_{i}∥

≤ \frac{1}{2}∥z^{−1}∥∥a_{i}∥

≤ \frac{∥a_{i}∥}{2μ ∥∑ a_{i}^{2}∥^{\frac{1}{2}}}

< 1

and hence

∥(Id + L_{P̃_{i}(z)^{∗}}R_{P̃_{i}(z)})^{−1}∥ ≤ \frac{1}{1 − ∥P̃_{i}(z)∥^{2}} ≤ \frac{4}{4 − ∥a_{i}∥^{2}∥z^{−1}∥^{2}} .

For μ > \frac{1}{\sqrt{2}} this implies

∥DS_{i}^{+}(z)^{−1}∥ ≤ ∥z^{−1}∥^{2} \frac{2}{4 − ∥a_{i}∥^{2}∥z^{−1}∥^{2}}

≤ \frac{2}{∥∑ a_{i}^{2}∥} \frac{1}{4μ^{2} − \frac{∥a_{i}∥^{2}}{∥∑ a_{i}^{2}∥}}

< \frac{2}{∥∑ a_{i}^{2}∥}

and hence F is contractive.

Alternative proof of the lower inequality of theorem 1.1. This proof essentially is the same on the free group and on Fock space, and it relies on the following facts. We are given a differentiable function K̃(z) - the inverse of the Cauchy transform mapping the set of positive matrices into itself. The inverse of its derivative at y = G(z)^{−1} is given by

DK̃(y)^{−1} = −L_{y}R_{y}DG(z)

and maps the positive matrices into itself as well. Moreover there is a point z_{0} such that

K̃(z_{0}) = κ_{0}I_{m} (3.16)

and such that the derivative DK̃(z_{0}) is invertible and its inverse can be written as

DK̃(z_{0})^{−1} = ∑_{s∈S} L_{x_{s}^{∗}}R_{x_{s}} (3.17)

so that it also maps positive matrices to positive matrices. Let z(t) be the solution of the differential equation

z(0) = z_{0}
\frac{dz}{dt} (t) = −DK̃(z(t))^{−1}[I_{m}].

The curve z(t) satisfies K̃(z(t)) = κ(t)I_{m} with κ(t) = κ_{0} − t, since z(0) = κ_{0}I_{m} and

\frac{d}{dt}K̃(z(t)) = DK̃(z(t)) [\frac{dz}{dt}(t)] = −I_{m}.

Since K̃(z_{0}) is bounded, K̃(z(t)) cannot decrease ad infinitum, thus the solution exists in some maximal finite interval [0, t_{1}]. First observe that since ż(t) is negative definite, z(t) is bounded. There are two possible reasons which can make the solution stop at t_{1}: Either z(t) approaches the boundary of the cone of positive matrices which means that it tends to a non-invertible matrix, or the derivative DK̃(z) becomes singular. In the first case we shall exhibit a family of vectors ξ_{z} such that

\begin{aligned}∥ξ_{z}∥^{2} & ≥ tr(z^{−2}) \\ T ξ_{z} & = K̃(z)ξ_{z} + ξ′_{z} \end{aligned} (3.18)

where ∥ξ′_{z}∥ is uniformly bounded. From this it follows that ξ̄_{z} = \frac{ξ_{z}}{∥ξ_{z}∥} is an asymptotic set of eigenvectors, i.e.

lim_{t→t_{1}} T ξ̄_{z(t)} − κ(t)ξ̄_{z(t)} = 0

which shows that ∥T ∥ ≥ inf_{t→t_{1}} ∥K̃(z(t))∥.

In the second case by the positivity of the inverse of DK̃(z) it follows that actually DK̃^{−1}[I_{m}] becomes unbounded and it is enough to exhibit a family of vectors η_{z} satisfying

\begin{aligned}∥η_{z}∥^{2} & = tr(DK̃(z)^{−1}[I_{m}]) \\ T η_{z} & = K̃(z)η_{z} + η′_{z} \end{aligned} (3.19)

and such that η′_{z} isuniformly bounded. Then we can conclude as before that η̄_{z} = \frac{η_{z}}{∥η_{z}∥} contains a family of asymptotic eigenvectors. Concerning the details on the free group, the iteration procedure of proposition 3.5 provides us with a matrix z_{0} which satisfies the requirements (3.16) and (3.17). We only have to choose a starting point large enough such that series of the inverse of (3.12) converges. For the vectors ξ_{z} and η_{z} we can choose ω_{z,z^{−1}} and ω̃_{z} respectively from proposition 3.4.

4. Matrix valued operators on Fock space

In this section we will prove theorem 1.3 and compute the norm of the operator

T = a_{0} ⊗ I + ∑ a_{i} ⊗ l_{i} + a_{i}^{∗} ⊗ l_{i}^{∗}. (4.1)

First we have to compute the K-transform. Let x be a semicircular element in some noncommutative probability space (A, τ) and consider a multiple αx i.e. its moments are given by formula (1.3) multiplied by appropriate powers of α. The R-transform of αx takes the particularly simple form [VDN92]

R_{α}(z) = α^{2}z.

The matrix valued analogue (i.e. the matrix valued R-transform of a⊗x) can be obtained as in [Haa97]. Then we can use theorem 2.2 to get the R-transform of the free sum. But since we will include the projection p_{Ω} too, which makes the sum non-free, we derive the K-transform directly.

Proposition 4.1. (i) The K-transform of the operator T in (4.1) is given by

K(z) = z^{−1} + a_{0} + ∑ a_{i}^{∗}za_{i}. (4.2)

The modified K-transform

K̃(z) = K(z^{−1}) = z + a_{0} + ∑ a_{i}^{∗}z^{−1}a_{i}

has derivative

D_{h}K̃(z) = h − ∑ a_{i}^{∗}z^{−1}hz^{−1}a_{i}

and the vector ω̃_{z} = (I − ∑ z^{−1}a_{i} ⊗ l_{i})^{−1} I_{m} ⊗ Ω has norm

∥ω̃_{z}∥^{2} = tr (DK̃(z)^{−1}[I_{m}]) (4.3)

and satisfies

T ω̃_{z} = K̃(z) ⊗ I ω̃_{z} − z ⊗ Ω (4.4)

(ii) Adding the projection p_{Ω} = I − ∑ l_{i}l_{i}^{∗} to T has the following effect. The modified K-transform of

T_{0} = a_{0} ⊗ I + b ⊗ p_{Ω} + ∑ a_{i} ⊗ l_{i} + a_{i}^{∗} ⊗ l_{i}^{∗} (4.5) is given by

K̃_{0}(z) = K_{0}(z^{−1}) = K((b + z)^{−1}) = a_{0} + b + z + ∑ a_{i}^{∗}(b + z)^{−1}a_{i}. (4.6)

Proof. (i) In order to make the upcoming formulae more compact and readable, we will treat operators of the form a ⊗ I as “noncommutative scalars” and denote them by a instead. Define a vector ω_{z,y} in L_{2}(M_{m}) ⊗_{2} T (H) as follows

\begin{aligned}ω_{z,y} & = (I − ∑ za_{i} ⊗ l_{i})^{−1} y ⊗ Ω \\ & = ∑_{k=0}^{∞} ∑_{i_{1},...i_{k}} za_{i_{1}}za_{i_{2}} ⋯ za_{i_{k}}y ⊗ e_{i_{1},i_{2},... ,i_{k}}. \end{aligned} (4.7)

Then

T ω_{z,y} = a_{0}ω_{z,y} + ∑_{k=0}^{∞} ∑_{i_{1},... ,i_{k}} ∑_{i} a_{i}za_{i_{1}}za_{i_{2}} ⋯ za_{i_{k}}y ⊗ e_{ii_{1},i_{2},... ,i_{k}}

+ ∑_{k=1}^{∞} ∑_{i_{1},... ,i_{k}} a_{i_{1}}^{∗}za_{i_{1}}za_{i_{2}} ⋯ za_{i_{k}}y ⊗ e_{i_{2},... ,i_{k}}

= a_{0}ω_{z,y} + z^{−1} ∑_{k=1}^{∞} ∑_{i_{1},... ,i_{k}} za_{i_{1}}za_{i_{2}} ⋯ za_{i_{k}}y ⊗ e_{i_{1},i_{2},... ,i_{k}}

+ ∑_{i=1}^{∞} a_{i}^{∗}za_{i}(y ⊗ Ω + ∑_{k=0}^{∞} ∑_{i_{1},... ,i_{k}} za_{i_{1}}za_{i_{2}} ⋯ za_{i_{k}}y ⊗ e_{i_{2},... ,i_{k}})

= (z^{−1} + a_{0} + ∑ a_{i}^{∗}za_{i})ω_{z,y} − z^{−1}y ⊗ Ω.

It follows that

(z^{−1} + a_{0} + ∑ a_{i}^{∗}za_{i} − T ) ω_{z,z} = I_{m} ⊗ Ω

and hence

G (z^{−1} + a_{0} + ∑ a_{i}^{∗}za_{i}) = ⟨(z^{−1} + ∑ a_{i}^{∗}za_{i} − T )^{−1} (I_{m} ⊗ Ω), Ω⟩
= ⟨ω_{z,z}, Ω⟩
= z

i.e. the K-transform is given by (4.2). The norm of ω_{z,y} can be calculated as follows.

∥ω_{z,y}∥^{2} = ∑_{k=0}^{∞} ∑_{i_{1},... ,i_{k}} ∥za_{i_{1}}za_{i_{2}} ⋯ za_{i_{k}}y∥_{HS}^{2}

= tr(∑_{k=0}^{∞} ∑_{i_{1},... ,i_{k}} y^{∗}a_{i_{1}}^{∗}za_{i_{2}}^{∗}z ⋯ a_{i_{k}}^{∗}z^{2}a_{i_{k}}za_{i_{k−1}} ⋯ za_{i_{1}}y)

= tr(L_{y^{∗}}R_{y} ∑_{k=0}^{∞} ∑_{i_{1},... ,i_{k}} L_{a_{i_{1}}^{∗}za_{i_{2}}^{∗}z ··· a_{i_{k}}^{∗} z}R_{za_{i_{k}} za_{i_{k−1}} ··· za_{i_{1}}}[I_{m}])

= tr(L_{y^{∗}}R_{y} (Id − ∑ L_{a_{i}^{∗}z}R_{za_{i}})^{−1} [I_{m}])

and the norm of ω̃_{z} = ω_{z^{−1},I_{m}} can be identified as (4.3).

(ii) The operator (4.5) is not a sum of freely independent operators anymore. However, its K-transform can still be computed by the above method. Indeed, it is easy to see that we get

T_{0}ω_{z,y} = K(z)ω_{z,y} + (b − z^{−1})y ⊗ Ω

and hence, whenever b + z^{−1} is positive definite,

T_{0}ω_{(b+z^{−1})^{−1},z} = (a_{0} + b + z^{−1} + ∑ a_{i}^{∗}(b + z^{−1})^{−1}a_{i}) ω_{(b+z^{−1})^{−1},z} − I_{m} ⊗ Ω.

As before we deduce that the modified K-transform is given by (4.6).

The following iteration based on the Banach fix point theorem allows to compute G(b) for big enough values.

Proposition 4.2. Let a_{0}, a_{1}, …, a_{n} be m × m matrices with a_{0} selfadjoint and let s > 2∥ ∑_{i=1}^{n} a_{i}^{∗}a_{i}∥^{\frac{1}{2}} be a given real number. Assume further that b is an m × m matrix such that b − a_{0} ≥ sI_{m}. Then for any z_{0} ∈ M_{s} = \{z ∈ M_{m} : z ≥ \frac{s}{2}I_{m}\} the iteration defined recursively by

z_{k+1} = b − a_{0} − ∑ a_{i}^{∗}z_{k}^{−1}a_{i} (4.8)

converges to the unique solution z̄ ∈ M_{s} of the equation

z + a_{0} + ∑ a_{i}^{∗}z^{−1}a_{i} = b.

In other words, the Cauchy transform of T satisfies G(b) = z̄^{−1}.

Proof. In order to apply the Banach fix point theorem, we have to show that the function F : z ↦ b − a_{0} − ∑ a_{i}^{∗}z^{−1}a_{i} maps M_{s} into itself and that it is Lipschitz continuous there with Lipschitz constant strictly smaller than one. For the first part, assume that z ≥ \frac{s}{2}I_{m}. Then

∑ a_{i}^{∗}z^{−1}a_{i} ≤ \frac{2}{s} ∑ a_{i}^{∗}a_{i} ≤ ∥∑ a_{i}^{∗}a_{i}∥^{\frac{1}{2}} I_{m}

and hence

F (z) ≥ (s − ∥∑ a_{i}^{∗}a_{i}∥^{\frac{1}{2}} ) I_{m} ≥ \frac{s}{2}I_{m}.

To show Lipschitz continuity, assume that z′ and z′′ are matrices in M_{s}. Then we estimate the norm of the difference as follows.

∥F (z′) − F (z′′)∥ = ∥∑ a_{i}^{∗} (z′^{−1} − z′′^{−1}) a_{i}∥

≤ ∥∑ a_{i}^{∗}a_{i}∥ ∥z′^{−1}(z′′ − z′)z′′^{−1}∥

≤ \frac{4}{s^{2}} ∥∑ a_{i}^{∗}a_{i}∥ ∥z′′ − z′∥

and by assumption the Lipschitz constant appearing in the last line is strictly smaller than one.

Proof of the upper bound in Theorem 1.3. We shall use the full Fock space framework introduced in section 1. We will consider T = a_{0} ⊗ I + ∑_{i=1}^{n} a_{i} ⊗ l_{i} + a_{i}^{∗} ⊗ l_{i}^{∗} acting on the Hilbert space HS(M_{m}) ⊗_{2} T (H) by left multiplication. It suffices to estimate the action of T on finitely supported vectors of the form h = ∑_{k=0}^{∞} ∑ h_{i_{1},i_{2},... ,i_{k}} ⊗ e_{i_{1},i_{2},... ,i_{k}}. Here and later on, a sum over vectors e_{i_{1},... ,i_{k}} with k = 0 is understood as the vector Ω. We have

∥T h∥^{2} = ∑_{p=0}^{∞} ∑_{j_{1},j_{2},... ,j_{p}} ∥⟨T h, e_{j_{1},j_{2},... ,j_{p}}⟩∥_{HS}^{2} ;

18 FRANZ LEHNER

for fixed j_{1}, j_{2}, …, j_{p}, p ≥ 0 and z > 0 the estimate is as follows.

∥⟨T h, e_{j_{1},j_{2},... ,j_{p}}⟩∥_{HS}^{2}

= ∥a_{0}h_{j_{1},j_{2},... ,j_{p}} + ∑_{k=0}^{∞} ∑_{i_{1},i_{2},... ,i_{k}} ∑_{i} a_{i}h_{i_{1},i_{2},... ,i_{k}}⟨l_{i}e_{i_{1},i_{2},... ,i_{k}}, e_{j_{1},j_{2},... ,j_{p}}⟩

+ a_{i}^{∗}h_{i_{1},i_{2},... ,i_{k}}⟨l_{i}^{∗}e_{i_{1},i_{2},... ,i_{k}}, e_{j_{1},j_{2},... ,j_{p}}⟩∥_{HS}^{2}

= ∥a_{0}h_{j_{1},j_{2},... ,j_{p}} + a_{j_{1}}h_{j_{2},j_{3},... ,j_{p}} + ∑_{i} a_{i}^{∗}h_{i,j_{1},j_{2},... ,j_{p}} ∥_{HS}^{2}

= ∥a_{0}h_{j_{1},j_{2},... ,j_{p}} + z^{\frac{1}{2}} z^{− \frac{1}{2}} a_{j_{1}}h_{j_{2},j_{3},... ,j_{p}} + ∑_{i} a_{i}^{∗}z^{−1}a_{i}(z^{−1}a_{i})^{−1}h_{i,j_{1},j_{2},... ,j_{p}} ∥_{HS}^{2}.

At this point we can apply lemma 2.5 to the sequences

a0^{\frac{1}{2}} , z^{\frac{1}{2}} , (a_{1}^{∗}z^{−1}a_{1})^{\frac{1}{2}} , …, (a_{n}^{∗}z^{−1}a_{n})^{\frac{1}{2}}

and

a0^{\frac{1}{2}} h_{j_{1},j_{2},... ,j_{p}}, z^{− \frac{1}{2}} a_{j_{1}}h_{j_{2},j_{3},... ,j_{p}}, (a_{1}^{∗}z^{−1}a_{1})^{\frac{1}{2}} a_{1}^{−1}zh_{1,j_{1},j_{2},... ,j_{p}}, …

…, (a_{n}^{∗}z^{−1}a_{n})^{\frac{1}{2}} a_{n}^{−1}zh_{n,j_{1},j_{2},... ,j_{p}}

and we get the inequality

∥⟨T h, e_{j_{1},j_{2},... ,j_{p}}⟩∥_{HS}^{2} ≤ ∥a_{0} + z + ∑ a_{i}^{∗}z^{−1}a_{i}∥ tr(h_{j_{1},j_{2},... ,j_{p}}^{∗}a_{0}h_{j_{1},j_{2},... ,j_{p}} +

h_{j_{2},j_{3},... ,j_{p}}^{∗}a_{j_{1}}^{∗}z^{−1}a_{j_{1}}h_{j_{2},j_{3},... ,j_{p}} + ∑_{i} h_{i,j_{1},j_{2},... ,j_{p}}^{∗}zh_{i,j_{1},j_{2},... ,j_{p}}).

Again the modified K-transform K̃(z) = z + a_{0} + ∑ a_{i}^{∗}z^{−1}a_{i} appears and after reordering the following sums appropriately we get

∥T h∥_{HS}^{2} = ∑_{p=0}^{∞} ∑_{j_{1},j_{2},... ,j_{p}} ∥⟨T h, e_{j_{1},j_{2},... ,j_{p}}⟩∥_{HS}^{2}

≤ ∥K̃(z)∥ tr(∑_{p=0}^{∞} ∑_{j_{1},j_{2},... ,j_{p}} h_{j_{1},j_{2},... ,j_{p}}^{∗}a_{0}h_{j_{1},j_{2},... ,j_{p}}

+ ∑_{j_{1},j_{2},... ,j_{p}} h_{j_{2},j_{3},... ,j_{p}}^{∗}a_{j_{1}}^{∗}z^{−1}a_{j_{1}}h_{j_{2},j_{3},... ,j_{p}}

+ ∑_{i,j_{1},j_{2},... ,j_{p}} h_{i,j_{1},j_{2},... ,j_{p}}^{∗}zh_{i,j_{1},j_{2},... ,j_{p}})

= ∥K̃(z)∥ tr(∑_{p=0}^{∞} ∑_{j_{1},j_{2},... ,j_{p}} h_{j_{1},j_{2},... ,j_{p}}^{∗}(z + a_{0} + ∑ a_{i}^{∗}z^{−1}a_{i})h_{j_{1},j_{2},... ,j_{p}})

≤ ∥K̃(z)∥^{2}∥h∥^{2}.

Proof of the lower bound in Theorem 1.3. To show the reverse inequality we cannot make use of lemma 2.3, since the vacuum state is neither tracial nor faithful. However, the alternative argument at the end of section 3 translates exactly to the Fock space framework, using now the proposition 4.2 to find z_{0}. The vectors ξ_{z} and η_{z} can be taken to be ω_{z^{−1},z^{−1}} and ω_{z^{−1},I_{m}} respectively from proposition 4.1.

Proof of Theorem 1.8. In order to obtain the upper bound we only have to slightly modify the above proof. Since the vectors e_{j_{1},j_{2},... ,j_{p}} (p ≥ 1) do not feel the projection p_{Ω}, only the term ⟨T_{0}h, Ω⟩ is different from its counterpart ⟨T h, Ω⟩. Using the positivity of a + b and choosing z such that both z and b + z are positive and invertible, we can estimate similarly as before

∥⟨T_{0}h, Ω⟩∥_{2}^{2} = ∥(a_{0} + b)h_{Ω} + ∑ a_{i}^{∗}h_{i}∥_{2}^{2}
= ∥(a_{0} + b)^{\frac{1}{2}} (a_{0} + b)^{\frac{1}{2}} h_{Ω} + ∑ a_{i}^{∗}(b + z)^{−1}a_{i}a_{i}^{−1}(b + z)h_{i}∥_{2}^{2}
≤ ∥a_{0} + b + ∑ a_{i}^{∗}(b + z)^{−1}a_{i}∥ tr (h_{Ω}^{∗}(a + b)h_{Ω} + ∑ h_{i}^{∗}(b + z)h_{i})

We also replace z by z + b in the other terms and obtain for any index set j_{1}, …, j_{p}, p ≥ 1,

∥⟨T_{0}h, e_{j_{1},j_{2},... ,j_{p}}⟩∥_{HS}^{2} ≤

∥a_{0} + b + z + ∑ a_{i}^{∗}(b + z)^{−1}a_{i}∥ tr(h_{j_{1},j_{2},... ,j_{p}}^{∗}a_{0}h_{j_{1},j_{2},... ,j_{p}} +

h_{j_{2},j_{3},... ,j_{p}}^{∗}a_{j_{1}}^{∗}(b + z)^{−1}a_{j_{1}}h_{j_{2},j_{3},... ,j_{p}} + ∑_{i} h_{i,j_{1},j_{2},... ,j_{p}}^{∗}(b + z)h_{i,j_{1},j_{2},... ,j_{p}}).

Now note that 0 ≤ a_{0} + b + ∑ a_{i}^{∗}(b + z)^{−1}a_{i} ≤ a_{0} + b + z + ∑ a_{i}^{∗}(b + z)^{−1}a_{i} and we get in exactly the same manner as before that ∥T_{0}h∥ ≤ ∥K̃_{0}(z)∥ ∥h∥. In order to get the lower bound we can assume that one of the coefficients a_{i} (i ≥ 1), say a_{1}, is invertible. For if this is not the case, we can consider for any ε > 0 the perturbated operator T_{0}^{(ε)} = T_{0} + εI_{m} ⊗ (l_{n+1}^{∗} + l_{n+1}). Now εI_{m} is invertible and assuming that we have proved the claim when at least one coefficient is invertible, we can choose some z_{ε} with the required positivity properties such that

∥T_{0}^{(ε)}∥ ≥ ∥K_{0}^{(ε)}(z_{ε})∥ − ε.

Then the operator inequality K_{0}^{(ε)}(z_{ε}) = K_{0}(z_{ε}) + ε^{2}z_{ε}^{−1} ≥ K_{0}(z_{ε}) ≥ 0 implies that

∥T_{0}∥ ≥ ∥T_{0}^{(ε)}∥ − 2ε ≥ ∥K_{0}^{(ε)}(z_{ε})∥ − 3ε ≥ ∥K_{0}(z_{ε})∥ − 3ε

Since ε can be chosen arbitrary small, the claim follows. In order to show the claim when a_{1} is invertible, consider again a curve z(t) starting at some remote point z(0) = z_{0} where K̃(z_{0}) = κ_{0}I_{m} and with derivative \frac{dz}{dt}(t) = −DK̃_{0}(z(t))^{−1}[I_{m}]. As ∥K̃_{0}(z(t))∥ = κ_{0} − t decreases, it cannot happen that b + z comes close to a singular point, since in this case κ_{0}−t = τ_{m}(K̃_{0}(z(t))) ≥ τ_{m}(a_{1}a_{1}^{∗}(b+z)^{−1}) would explode. The two other possibilities which make z(t) stop at t_{1} are that DK̃_{0}(z(t))^{−1}[I_{m}] explodes or that z becomes singular. The former leads to the asymptotic eigenvector ω̃_{b+z} as before; in the latter case the family ω_{z^{−1},z^{−1}} satisfies T_{0}ω_{z^{−1},z^{−1}} = K̃_{0}(z)ω_{z^{−1},z^{−1}} −I_{m}⊗Ω and ∥ω_{z^{−1},z^{−1}}∥_{2}^{2} ≥ tr(z^{−2}) explodes, so that again we get a lower bound for the norm, namely lim_{t→t_{1}} ∥K̃_{0}(z(t))∥.

Remark 4.3. Although stated only for matrix coefficients, both formulae (1.1) and (1.4) hold with matrices replaced by operators. Indeed one can easily verify that the proof for the inequality “≤” carries over without modification, whereas lemma 2.3 does not seem to have an analogue in this setting. This can be overcome as follows. Let y_{i} denote either the free unitaries λ(g_{i}) or the free semicircular operators x_{i} and let T = a_{0} ⊗ I + ∑ a_{i} ⊗ y_{i} have coefficients in some matrix algebra M_{m}. First observe that embedding M_{m} into B(H) for some infinite dimensional Hilbert space and identifying the matrices a_{i} with the operators ã_{i} = [ \begin{matrix}a_{i} & 0 \\ 0 & 0 \end{matrix} ] we have that the norm of the operator T̃ = ∑ ã_{i} ⊗ x_{i} is equal to the norm of ∥T ∥. On the other hand, we have seen that

∥T̃∥ ≤ inf_{z̃∈B(H)_{+}^{−1}} ∥KT̃(z̃)∥ (4.9)

and the latter is smaller than

inf_{z∈M_{m}_{+}^{−1}} ∥K_{T} (z)∥ = ∥T ∥, (4.10)

since we may take z̃ of the form

z̃ = [\begin{matrix}z & 0 \\ 0 & z′ \end{matrix}]

with suitable positive invertible z′. This shows that equality holds in (4.9) if the coefficients are finite rank operators. This implies that the operator space structure defined the right hand side of (4.9) is completely isometric to the operator space structure defined by (4.10). Now Lemma 1.5 in [dCH85] implies that equality holds in (4.9) for arbitrary operator coefficients.

5. Applications to harmonic analysis on the free group and on Fock space

In this section we give applications of theorems 1.1 and 1.8 to compute the norms of arbitrary finitely supported operators in the C^{∗}-algebra generated by the free operators in consideration.

5.1. Free Group. It is known that the norm structure of a C^{∗}-algebra generated by unitaries u_{1}, u_{2}, …, u_{n} is determined by the operator space structure of the subspace spanned by I, u_{1}, u_{2}, …, u_{n}, see [Pis96]. The following gives a constructive proof of this fact. The claim is that one can compute the norm of arbitrary finite linear combinations of products (also called words) of the generating unitaries, provided that one knows how to compute matrix-valued linear combinations of the generating unitaries and the identity operator, i.e. the operator space structure of the spanned subspace. It is enough to give a procedure to compute matrix-valued linear combinations of products of length at most two of the generating unitaries. Then one can iterate this procedure and reduce the length of the involved words by a factor two. This is the object of the current section. We shall first consider the special case of scalar linear combinations of words of length smaller than or equal to two. To simplify notation we will assume that g_{0} is the identity element and hence λ(g_{0}) is the identity operator. Suppose that we are given a self-adjoint convolution operator T supported on the words of length 2, i.e. T has the form

T = ∑_{i,j=0}^{n} c_{ij} λ(g_{i}g_{j})

with c_{ij} ∈ C and c_{ij} = \overline{c_{ji}}. Assume moreover that the matrix C = [c_{ij}] ∈ M_{n+1} is positive definite, so that it has a self-adjoint square root A = [α_{ij}] (in what follows one can choose any matrix A such that A^{∗}A = C). Setting

G = [\begin{matrix}λ(g_{0}) & 0 & … & 0 \\ λ(g_{1}) & 0 & … & 0 \\ ⋮ & ⋮ & ... & ⋮ \\ λ(g_{n}) & 0 & … & 0 \end{matrix}]

it follows that

∥T ∥ = ∥∑ λ(g_{i})c_{ij}λ(g_{j})∥
= ∥G^{∗}CG∥
= ∥AG∥^{2}
= ∥∑ A_{i} ⊗ λ(g_{i})∥

where the coefficients are given by

A_{i} = [\begin{matrix}α_{i0} & 0 & … & 0 \\ α_{i1} & 0 & … & 0 \\ ⋮ & ⋮ & ... & ⋮ \\ α_{in} & 0 & … & 0 \end{matrix}] .

The norm can then be calculated by formula (1.1). Next assume that the coefficients c_{ij} are positive real numbers, C still being symmetric. It follows from [dlHRV93, Lemma 3.2] that the spectrum of T has nonvoid intersection with the positive real numbers and moreover max σ(T) ≥ | min σ(T)|. Hence for any positive real number λ we have ∥λI+T ∥ = λ+∥T ∥ and we can choose λ such that λI_{n+1} + C is positive definite. Then

∥T ∥ = ∥G^{∗}(λI_{n+1} + C)G∥ − (n + 1)λ

and we are back in the case of a positive definite coefficient matrix. A trick similar to the one used in section 1 helps with a non-selfadjoint coefficient matrix C. In this case we have that

∥T ∥ = ∥[\begin{matrix}0 & T^{∗} \\ T & 0 \end{matrix}]∥

The matrix T̃ = [ \begin{matrix}0 & T^{∗} \\ T & 0 \end{matrix} ] satisfies UT̃U^{∗} = −T̃, where U = [ \begin{matrix}I & 0 \\ 0 & −I \end{matrix} ] . Thus the spectrum of T̃ is symmetric around 0 and for any real number λ we have

∥[\begin{matrix}λ & T^{∗} \\ T & λ \end{matrix}]∥ = |λ| + ∥[\begin{matrix}0 & T^{∗} \\ T & 0 \end{matrix}]∥; (5.1)

choosing λ positive big enough so that the matrix

C̃_{λ} = [\begin{matrix}\frac{λ}{n+1}I_{n+1} & C^{∗} \\ C & \frac{λ}{n+1}I_{n+1} \end{matrix}]

is positive definite leads again to the case of a positive coefficient matrix, except that now we have to work with an extended matrix G̃ = [ \begin{matrix}G & 0 \\ 0 & G \end{matrix}] instead of G. Here instead of taking the positive square root of the extended matrix C̃_{λ}, it seems to be more efficient to take a Cholesky-type factor

R = [\begin{matrix}\begin{matrix}√ & \overline{λ} \\ √ & \overline{n+1} \end{matrix}I_{n+1} & \begin{matrix}√ & \overline{n+1} \\ √ & \overline{λ} \end{matrix}C^{∗} \\ 0 & D \end{matrix}]

where D is such that D^{∗}D + \frac{n+1}{λ}CC^{∗} = \frac{λ}{n+1}I_{n+1} and hence R^{∗}R = C̃_{λ}. This completes the special case of words of length two. In order to iterate this scheme, we have to extend it to matrix coefficients. Consider an operator of the form

T = ∑ C_{ij} ⊗ λ(g_{i}g_{j})

where C_{ij} ∈ M_{m}, i, j ∈ {0, …, n}. Again we have that

∥T ∥ = ∥[\begin{matrix}0 & T \\ T^{∗} & 0 \end{matrix}]∥

= ∥[\begin{matrix}(n + 1)λI_{m} ⊗ I & T \\ T^{∗} & (n + 1)λI_{m} ⊗ I \end{matrix}]∥ − (n + 1)λ

= ∥[\begin{matrix}G^{(n+1)}^{∗} & 0 \\ 0 & G^{(n+1)}^{∗} \end{matrix}] [\begin{matrix}λI_{n+1} ⊗ I_{m} ⊗ I & C^{(n+1)} ⊗ I \\ C^{(n+1)}^{∗} ⊗ I & λI_{n+1} ⊗ I_{m} ⊗ I \end{matrix}] [\begin{matrix}G^{(n+1)} & 0 \\ 0 & G^{(n+1)} \end{matrix}]∥ − (n + 1)λ

where G^{(n+1)} = ∑ e_{i1} ⊗ I_{m} ⊗ λ(g_{i}) and C^{(n+1)} = ∑ e_{ij} ⊗ C_{ij}. By choosing λ large enough the (n + 1)m × (n + 1)m matrix

[\begin{matrix}λI_{n+1} ⊗ I_{m} & C^{(n+1)} \\ C^{(n+1)}^{∗} & λI_{n+1} ⊗ I_{m} \end{matrix}]

can be made positive definite and we can reduce the norm computation to words of length one. Obviously this technique can be applied iteratively to convolution operators supported on arbitrary (finitely many) elements in the group Γ_{n}. Finally observe that the elements g_{0}g_{1}, g_{0}g_{2}, …, g_{0}g_{n} generate a copy of the free group F_{n} = Z ∗ Z ∗ ⋯ ∗ Z inside Γ_{n+1}, thus the above scheme can also be used to calculate the norm of finitely supported convolution operators on F_{n}.

5.2. Fock Space. The situation in Fock space is similar to the previous one. Any operator T supported on the words of length ≤ 2 can be represented as T = LCL^{∗} with some coefficient matrix C ∈ M_{2n+1} and where

L = [I, l_{1}^{∗}, l_{1}, …, l_{n}^{∗}, l_{n}] .

If the coefficient matrix C is positive definite, we can proceed as before. If it is not, we repeat the matrix trick with the following slight modification. Again the norm of T is the same as the norm of the amplified operator

T̃ = [\begin{matrix}0 & T \\ T^{∗} & 0 \end{matrix}] = [\begin{matrix}L & 0 \\ 0 & L \end{matrix}] [\begin{matrix}0 & C \\ C^{∗} & 0 \end{matrix}] [\begin{matrix}L^{∗} & 0 \\ 0 & L^{∗} \end{matrix}] .

Now we cannot immediately pass to the positive definite matrix [ \begin{matrix}λ & C \\ C^{∗} & λ \end{matrix}] since LL^{∗} = I + ∑ l_{i}^{∗}l_{i} + l_{i}l_{i}^{∗} = (n + 2)I − p_{Ω} is not quite a multiple of the identity. Thus we have to include p_{Ω} into L to get the identity matrix. Setting

L̃ = [p_{Ω}, I, l_{1}^{∗}, l_{1}, …, l_{n}^{∗}, l_{n}]

and

C̃ = [\begin{matrix}0 & 0 \\ 0 & C \end{matrix}] ∈ M_{2n+2}

we get that

[\begin{matrix}L̃ & 0 \\ 0 & L̃ \end{matrix}] [\begin{matrix}λI_{2n+2} & C̃ \\ C̃^{∗} & λI_{2n+2} \end{matrix}] [\begin{matrix}L̃^{∗} & 0 \\ 0 & L̃^{∗} \end{matrix}] = [\begin{matrix}(n + 2)λ & T \\ T^{∗} & (n + 2)λ \end{matrix}]

and from here one can proceed as before, using now theorem 1.8. Again this construction can be adapted to the matrix valued case which makes computation with words of arbitrary length possible.

6. Numerical computation

In view of the matrix manipulations in section 5 there might be some interest in evaluating the infima (1.1) and (1.4) for concrete matrices a_{i} – at least numerically – on a computer. This amounts to solving a constrained nonlinear optimization problem:

∥K̃(z)∥ → min!
z ∈ M_{m}, z > 0

Moreover, since the operator norm of a positive definite matrix is equal to its largest eigenvalue and since the eigenvalue map a ↦ λ_{i}(a) is differentiable at a if and only if the i-th eigenvalue of a is simple, the last assertions of theorem 1.1 and theorem 1.3 indicate that this optimization problem is essentially non-differentiable, so that standard Newton methods do not apply. However, the latter remarks also indicate where to look for the optimum, namely among the matrices z for which K̃(z) is a scalar multiple of the identity. Such a matrix is a zero of the matrix-valued function

z ↦ Φ(z) = K̃(z) − τ_{m}(K̃(z))I_{m},

which can be found using propositions 3.5 and 4.2 respectively. The function Φ is differentiable and thus can be put into a quasi-Newton method, which has the advantage of local quadratic convergence, while fix point iteration is only linearly convergent. Both methods together yield an efficient method to find a zero of Φ. Once such a zero is found, we can compute the tangent of the curve

γ = {z ∈ M_{m} : z > 0 and K̃(z) = λI_{m}, λ ∈ C} (6.1)

and minimize ||K̃(z)|| along the tangent. Iteration of these two steps then leads to an optimal z^{∗}. This is essentially equivalent to the solution of the differential equation considered at the end of the proof of theorem 1.1 in section 3. Recall that the Newton-Raphson method can be formulated as follows. Given a differentiable function Ψ : R^{n} → R^{n} and a point x_{k} ∈ R^{n}, one makes a linear approximation of Ψ around x_{k}

Ψ(x_{k} + h) = Ψ(x_{k}) + D_{h}Ψ(x_{k}) + o(||h||)

and takes the zero of this linearized version as the next candidate. It is given by

x_{k+1} = x_{k} − DΨ(x_{k})^{−1}[Ψ(x_{k})]

where DΨ(x_{k}) = [D_{i}Ψ_{j}(x_{k})] is the Jacobian of Ψ at the point x_{k}. Then it can be shown that provided x_{0} is close enough to a zero x^{∗} this sequence converges quadratically to x^{∗}, i.e. there is a constant C such that

||x_{k+1} − x^{∗}|| ≤ C||x_{k} − x^{∗}||^{2}.

This method requires the Jacobian to be invertible. In our case we know from lemma 2.3 that the derivative of K̃(z) is invertible on the curve (6.1) and hence in some neighbourhood of it. This implies that the derivative of Φ, which is given by D_{h}Φ(z) = D_{h}K̃(z) − τ_{m}(D_{h}K̃(z))I_{m}, has nontrivial null space (containing h_{0} = DK̃(z)^{−1}[I_{m}]) and Newton-Raphson cannot be applied. Hence we have to find another descent direction for the master function f(z) = ||Φ(z)||_{2}^{2} = tr(Φ(z)^{2}). A good candidate is h = −DK̃(z)^{−1}[Φ(z)], since

D_{h}Φ(z) = D_{h}K̃(z) − τ_{m}(D_{h}K̃(z))I_{m} = −Φ(z)

and hence

D_{h}f(z) = 2 tr (Φ(z)D_{h}Φ(z)) = −2f(z).

Indeed, this gives rise to the following quadratic convergent iteration.

Proposition 6.1. Let z_{0}^{∗} be a point on the curve γ given by (6.1) such that the Jacobian DK̃(z_{0}^{∗}) is regular. Then there exists a neighbourhood V of z_{0}^{∗} such that for any z_{0} ∈ V the iteration sequence z_{k} given by

z_{k+1} = z_{k} − DK̃(z_{k})^{−1}[Φ(z_{k})]

is well defined and the sequence δ_{k} = dist(z_{k}, γ) satisfies δ_{k+1} ≤ C(δ_{k})^{2}, i.e. z_{k} converges quadratically to a point on γ.

Proof. The proof is along the lines of the standard proof of quadratic convergence of the Newton method. We replace the euclidean metric around γ by the following one. Set

⟨h, h′⟩_{z} = τ_{m} (D_{h}K̃(z)D_{h′} K̃(z)) and ||h||_{z}^{2} = ⟨h, h⟩_{z}

Since DK̃(z) is invertible, this metric is locally uniformly equivalent to the standard metric. Hence it suffices to show that the iteration is quadratically convergent for this metric. We will abbreviate ⟨ . , . ⟩_{k} = ⟨ . , . ⟩_{z_{k}} and || . ||_{k} = || . ||_{z_{k}} respectively. It will be convenient to work with δ_{k} = dist(z_{k}, γ) according to this metric rather than the standard euclidean metric. Let z_{k} be close to γ and let z^{∗}(t) be a parametrization of γ. Since K has full rank, γ is differentiable and so is the function t ↦ ||z^{∗}(t) − z_{k}||_{k}^{2}. Thus if we choose t_{k} such that ∥z_{k} − z^{∗}(t_{k})∥_{k} = δ_{k}, the following necessary first order condition must be satisfied.

0 = \frac{1}{2} \frac{d}{dt} ∥z^{∗}(t) − z_{k}∥_{k}^{2}|_{t=t_{k}}

= τ_{m}(DK̃(z_{k}) [\frac{dz^{∗}}{dt} (t_{k})] DK̃(z_{k}) [z^{∗}(t_{k}) − z_{k}])

Now K̃(z^{∗}(t)) = κ(t)I_{m} for some scalar valued function κ and DK̃(z^{∗}(t)) [\frac{dz^{∗}}{dt} ] = κ′(t)I_{m}, whence it follows that

τ_{m} (DK̃(z_{k})[z^{∗}(t_{k}) − z_{k}]) = τ_{m}((DK̃(z^{∗}(t_{k})) − DK̃(z_{k})) [\frac{dz^{∗}}{dt} (t_{k})] DK̃(z_{k})[z^{∗}(t_{k}) − z_{k}])
= O (∥z^{∗}(t_{k}) − z_{k}∥_{k}^{2})

since DK̃ Lipschitz continuous and ||DK̃(z_{k}) − DK̃(z^{∗}(t_{k}))|| = O (∥z^{∗}(t_{k}) − z_{k}∥_{k}) . Next we use a first order Taylor approximation of Φ around z_{k} and we get

0 = Φ(z^{∗}(t_{k}))

= Φ(z_{k}) − DΦ(z_{k})[z_{k} − z^{∗}(t_{k})] + O(||z_{k} − z^{∗}(t_{k})||_{k}^{2})

= −DK̃(z_{k})[z_{k+1} − z_{k}] − DK̃(z_{k})[z_{k} − z^{∗}(t_{k})] + τ_{m}(DK̃(z_{k})[z_{k} − z^{∗}(t_{k})]) + O(||z_{k} − z^{∗}(t_{k})||_{k}^{2})

= −DK̃(z_{k})[z_{k+1} − z^{∗}(t_{k})] + O(||z_{k} − z^{∗}(t_{k})||_{k}^{2}) and hence

||z_{k+1} − z^{∗}(t_{k})||_{k} ≤ C||z_{k} − z^{∗}(t_{k})||_{k}^{2}.

This iteration can be combined with the globally convergent fix-point iteration from propositions 3.5 and 4.2 gives an efficient method to find a point z_{0} on γ. Now given a point z_{k} on γ, the next step involves a linesearch on the positive part of the tangent, i.e. {z_{k} + λh_{k} : λ > 0} to decrease K̃(z). Here h_{k} = −DK̃(z_{k})^{−1}[I_{m}] is a direction of descent for K̃(z), since K̃(z_{k}) is a multiple of the identity matrix I_{m} and h_{k} is chosen such that all the eigenvalues of K̃(z) decrease. Finding a parameter λ̃ which decreases ∥K̃(z)∥ on the tangent will yield a point z̃_{k+1} = z_{k} + λ̃h_{k} satisfying ∥K̃(z̃_{k+1})∥ < ∥K̃(z_{k})∥. In the hope that this is not too far off the curve γ we will then try to find a point z_{k+1} ∈ γ using the iteration from proposition 6.1 with starting point z̃_{k+1}. It is however not a good idea to take λ̃ to be the minimizer λ^{∗} of the function λ ↦ ∥K̃(z_{k} + λh_{k})∥ for the following reason. The critical value b = \frac{1}{∥T ∥}I_{m} is not the only singularity of the Cauchy transform G, and as soon as K̃(z) is close to another singularity of G one might expect that the Jacobian DK̃(z) becomes ill-conditioned if not singular, thus it will be difficult to get back to γ from such a point. This is what actually happens in numerical experiments, especially when z_{k} is close to the optimal z^{∗} and z_{k} + λ^{∗} gets very close to a singular point not on γ. It is therefore advisable to add a penalty function, for example a multiple of ∥K̃(z) − τ_{m}(K̃(z))∥_{2}^{2} in order to stay close enough to γ and away from other singular points. Numerically this works out fine and we have the following iteration scheme. After finding a point z_{0} ∈ γ repeat the following three steps

(i) Using the quasi-Newton iteration starting at z̃_{k}, find a point z_{k} ∈ γ. (ii) Compute the tangent vector h_{k} = −DK̃(z_{k})^{−1}[I_{m}]. (iii) Minimize ∥K̃(z)∥ + μ tr(Φ(z)^{2}) on the tangent {z_{k} + λh_{k} : λ ≥ 0} to get a point z̃_{k+1}

which decreases ∥K̃(z)∥. until some stopping criterion is fulfilled. A natural stopping criterion is given by the tangent vector h = DK̃(z)^{−1}[I_{m}]. Since its trace is equal to the norm of ω̃_{z} (see (4.3) and (3.15) above), one gets a good estimate from the identities (3.14) and (4.6), namely

∥T ∥ ≥ min spec(K̃(z)) − 2\frac{∥z∥_{HS}}{tr(h)}

in the case of the free group and

∥T ∥ ≥ min spec(K̃(z)) − \frac{∥z∥_{HS}}{tr(h)}

in the case of Fock space. The author has implemented this method in the Rlab environment [Sea] and can be contacted for further information.

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IMADA, Odense Universitet, Campusvej 55, DK-5230 Odense M, Denmark

E-mail address: lehner@imada.ou.dk