POLYNOMIAL INEQUALITIES FOR NON-COMMUTING OPERATORS^{∗}

JOHN E. MCCARTHY ^{†} AND RICHARD M. TIMONEY ^{‡}

Abstract. We prove an inequality for polynomials applied in a symmetric way to non-commuting operators.

Key words. Ando inequality, non-commuting, symmetric functional calculus

AMS subject classifications. 15A60

1. Introduction. J. von Neumann [9] proved an inequality about the norm of a polynomial applied to a contraction on a Hilbert space H. Let 𝔻 be the unit disk and 𝕋 the unit circle in ℂ, and for any polynomial p let ∥p∥_{X} be the supremum of the modulus of p on the set X. The result is that

T ∈ B(H), ∥T ∥ ≤ 1 ⇒ ∥p(T)∥ ≤ ∥p∥\overline{𝔻}.  (1.1)

For polynomials p(z) = p(z_{1}, z_{2}, …, z_{n}) = ∑_{|α|≤N} c_{α}z^{α} in n variables we use the standard multi-index notation (where α = (α_{1}, α_{2}, …, α_{n}) has 0 ≤ α_{j} ∈ ℤ for 1 ≤ j ≤ n, |α| = ∑_{j=1}^{n} α_{j}, z^{α} = ∏_{j=1}^{n} z_{j}^{α_{j}}). There is an obvious way of applying p to an n-tuple T = (T_{1}, T_{2}, …, T_{n}) of commuting operators T_{j} ∈ B(H) (1 ≤ j ≤ n), namely

p(T) = p(T_{1}, T_{2}, …, T_{n}) = ∑_{|α|≤N} c_{α}T^{α}

(with T^{α} = ∏_{j=1}^{n} T_{j}^{α_{j}} and T_{j}^{0} = I).

T. Andô [2] proved an extension of von Neumann’s inequality to pairs of commuting contractions.

Theorem 1.1 (Andô). If T_{1}, T_{2} ∈ B(H), max(∥T_{1}∥, ∥T_{2}∥) ≤ 1, T_{1}T_{2} = T_{2}T_{1} and p(z) = p(z_{1}, z_{2}) is a polynomial, then

∥p(T_{1}, T_{2})∥ ≤ ∥p∥\overline{𝔻}^{2}.

∗Received by the editors on Month x, 200x. Accepted for publication on Month y, 200y Handling Editor: .

† Washington University, St. Louis and Trinity College, Dublin (mccarthy@wustl.edu). Partially supported by National Science Foundation Grants DMS 0501079 and DMS 0966845.

‡Trinity College, Dublin (richardt@maths.tcd.ie).

The purpose of this note is to look for analogues of Andô’s inequality that are satisfied by non-commuting operators. For a polynomial p in n variables and an ntuple of operators T = (T_{1}, …, T_{n}) we define p_{sym}(T) to be a symmetrized version of p applied to T (we make this precise in Section 2). We are looking for results of the form:

For all n-tuples T of operators in a certain set, there is a set K_{1} in ℂ^{n} such that

∥p_{sym}(T)∥ ≤ ∥p∥_{K_{1}}.  (1.2)

and

For all n-tuples T of operators in a certain set, there is a set K_{2} in ℂ^{n} and a constant M such that

∥p_{sym}(T)∥ ≤ M ∥p∥_{K_{2}}.  (1.3)

Our main result is:

Theorem 4.6 There are positive constants M_{n} and R_{n} such that, whenever T = (T_{1}, T_{2}, …, T_{n}) ∈ B(H)^{n} satisfies

∥ ∑_{i=1}^{n} ζ_{i}T_{i}∥ ≤ 1 ∀ζ_{i} ∈ \overline{𝔻},

and p is a polynomial in n variables, then

∥p_{sym}(T)∥ ≤ ∥p∥_{R_{n}\overline{𝔻}^{n}}  (1.4)
∥p_{sym}(T)∥ ≤ M_{n}∥p∥\overline{𝔻}^{n}.  (1.5)

Moreover, one can choose R_{2} = 1.85, R_{3} = 2.6, M_{2} = 4.1 and M_{3} = 16.6.

2. Tuples of noncommuting contractions. There are several natural ways one might apply a polynomial p(z_{1}, z_{2}) in two variables to pairs T = (T_{1}, T_{2}) ∈ B(H)^{2} of operators. A simple case is for polynomials of the form p(z_{1}, z_{2}) = p_{1}(z_{1}) + p_{2}(z_{2}) where we could naturally consider p(T_{1}, T_{2}) to mean p_{1}(T_{1}) + p_{2}(T_{2}).

A recent result of Drury [4] is that if p(z_{1}, z_{2}) = p_{1}(z_{1}) + p_{2}(z_{2}), T_{1}, T_{2} ∈ B(H) (no longer necessarily commuting), max(∥T_{1}∥, ∥T_{2}∥) ≤ 1, then

∥p(T_{1}, T_{2})∥ ≤ \sqrt{2}∥p∥\overline{𝔻}^{2}.  (2.1)

Moreover, Drury [4] shows that the constant \sqrt{2} is best possible.

One way to apply a polynomial p(z_{1}, z_{2}) = ∑_{j,k=0}^{n} a_{j,k}z_{1}^{j}z_{2}^{k} to two noncommuting operators T_{1} and T_{2} is by mapping each monomial z_{1}^{j}z_{2}^{k} to the average over all possible products of j number of T_{1} and k number of T_{2}, and then extend this map by linearity to all polynomials. We use the notation p_{sym}(T_{1}, T_{2}) and the formula

p_{sym}(T_{1}, T_{2}) = ∑_{j,k=0}^{n} \frac{a_{j,k}}{\binom{j+k}{j}} ∑_{S∈P(j+k,j)} ∏_{i=1}^{j+k} T_{2−χ_{S}(i)}

where P(j + k, j) denotes the subsets of {1, 2, …, j + k} of cardinality j. The empty product, which arises for j = k = 0, should be taken as the identity operator. The notation ∏_{i=1}^{j+k} T_{2−χ_{S}(i)} is intended to mean the ordered product

T_{2−χ_{S}(1)}T_{2−χ_{S}(2)} ⋯ T_{2−χ_{S}(j+k)},

and χ_{S}(·) denotes the indicator function of S.

Remarks 2.1. The operation p ↦ p_{sym}(T_{1}, T_{2}) is not an algebra homomorphism (from polynomials to operators). It is a linear operation and does not respect squares in general.

For example, if p(z_{1}, z_{2}) = z_{1}^{2} + z_{2}^{2}, then
p_{sym}(T_{1}, T_{2}) = T_{1}^{2} + T_{2}^{2}

but for q(z_{1}, z_{2}) = (p(z_{1}, z_{2}))^{2} = z_{1}^{4} + z_{2}^{4} + 2z_{1}^{2}z_{2}^{2} we have

(p_{sym}(T_{1}, T_{2}))^{2} = T_{1}^{4} + T_{2}^{4} + T_{1}^{2}T_{2}^{2} + T_{2}^{2}T_{1}^{2} ≠ q_{sym}(T_{1}, T_{2})

in general.

Similarly for p(z_{1}, z_{2}) = 2z_{1}z_{2} and
q(z_{1}, z_{2}) = (p(z_{1}, z_{2}))^{2} = 4z_{1}^{2}z_{2}^{2},

p_{sym}(T_{1}, T_{2}) = T_{1}T_{2} + T_{2}T_{1},

(p_{sym}(T_{1}, T_{2}))^{2} = T_{1}T_{2}T_{1}T_{2} + T_{1}T_{2}^{2}T_{1} + T_{2}T_{1}^{2}T_{2} + T_{2}T_{1}T_{2}T_{1} ≠ q_{sym}(T_{1}, T_{2})

in general.

However in the very restricted situation that p(z_{1}, z_{2}) = α+βz_{1}+γz_{2} and q = p^{m}, then we do have q_{sym}(T_{1}, T_{2}) = (p_{sym}(T_{1}, T_{2}))^{m}.

The symmetrizing idea generalizes in the obvious way to n > 2 variables. We will use the notation p_{sym}(T) for n-tuples T ∈ B(H)^{n} for n ≥ 2.

3. Example. The analogue of Andô’s inequality for n ≥ 3 commuting Hilbert space contractions and polynomials norms on 𝔻^{n} is known to fail (see Varopoulos [8], Crabb & Davie [3], Lotto & Steger [6], Holbrook [5]).

The explicit counterexamples of Kaijser & Varopoulos [8], and Crabb & Davie [3]) have p(T) nilpotent (and so of spectral radius 0). While the examples of Lotto & Steger [6] and Holbrook [5]) do not have this property, they are obtained by perturbing examples where p(T) is nilpotent (and so p(T) has relatively small spectral radius).

It is not known whether there is a constant C_{n} so that the multi-variable inequality

∥p(T)∥ = ∥p(T_{1}, T_{2}, …, T_{n})∥ ≤ C_{n}∥p∥\overline{𝔻}^{n}  (3.1)

holds for all polynomials p(z) in n variables and for all n-tuples T of commuting Hilbert space contractions. However, it is well-known that a spectral radius version of Andô’s inequality is true — indeed, it holds in any Banach algebra.

Proposition 3.1. If p is a polynomial in n variables and T = (T_{1}, T_{2}, …, T_{n}) is an n-tuple of commuting elements in a Banach algebra, each with norm at most one, then

ρ(p(T)) = lim_{m→∞} ∥(p(T))^{m}∥^{1/m} ≤ ∥p∥\overline{𝔻}^{n}  (3.2)

(for ρ(·) the spectral radius).

Proof. We consider a fixed n. It follows from the Cauchy integral formula, that if max_{1≤j≤n} ∥T_{j}∥ ≤ r < 1, then

∥p(T)∥ = ∥p(T_{1}, T_{2}, …, T_{n})∥ ≤ C_{r}∥p∥\overline{𝔻}^{n}  (3.3)

for a constant C_{r} depending on r (and n).

To see this write

p(T) = \frac{1}{(2πi)^{n}} ∫_{ζ∈𝕋^{n}} ∏_{j=1}^{n} p(ζ) ∏_{j=1}^{n}(ζ_{j} − T_{j})^{−1} dζ_{1} dζ_{2} … dζ_{n}

and estimate with the triangle inequality. This shows that C_{r} = (1 − r)^{−n} will work.

Applying (3.3) to powers of p and using the spectral radius formula, we get

ρ(p(T)) ≤ ∥p∥\overline{𝔻}^{n},

(provided max_{1≤j≤n} ∥T_{j}∥ ≤ r < 1). However, for the general case max_{1≤j≤n} ∥T_{j}∥ = 1, we can apply this to rT to get

ρ(p(T)) = lim_{r→1^{−}} ρ(p(rT)) ≤ ∥p∥_{∞}.

Example 3.2.

Let p(z, w) = (z − w)^{2} + 2(z + w) + 1 = z^{2} + w^{2} − 2zw + 2(z + w) + 1,

T_{1} = (\begin{matrix}cos(π/3) & sin(π/3) \\ sin(π/3) & − cos(π/3) \end{matrix}) = (\begin{matrix}1/2 & \sqrt{3}/2 \\ \sqrt{3}/2 & −1/2 \end{matrix}) ,

T_{2} = (\begin{matrix}cos(π/3) & − sin(π/3) \\ − sin(π/3) & − cos(π/3) \end{matrix}) .

Note that ∥p∥_{𝔻^{2}} ≥ p(1, −1) = 5. To show that ∥p∥_{𝔻^{2}} ≤ 5, consider the homogeneous polynomial

q(z_{1}, z_{2}, z_{3}) = z_{1}^{2} + z_{2}^{2} + z_{3}^{2} − 2z_{1}z_{2} − 2z_{1}z_{3} − 2z_{2}z_{3}

and observe first that p(z, w) = q(z, w, −1). Moreover

∥p∥_{𝔻^{2}} = ∥p∥_{𝕋^{2}} = ∥q∥_{𝕋^{3}} = ∥q∥_{𝔻^{3}},

by homogeneity of q and the maximum principle. Holbrook [5, Proposition 2] gives a proof that ∥q∥_{𝔻^{3}} = 5.

We have

p_{sym}(T_{1}, T_{2}) = (T_{1} − T_{2})^{2} + 2(T_{1} + T_{2}) + I
= (\begin{matrix}0 & \sqrt{3} \\ \sqrt{3} & 0 \end{matrix})^{2} + 2 (\begin{matrix}1 & 0 \\ 0 & −1 \end{matrix}) + I
= (\begin{matrix}3 & 0 \\ 0 & 3 \end{matrix}) + (\begin{matrix}2 & 0 \\ 0 & −2 \end{matrix}) + (\begin{matrix}1 & 0 \\ 0 & 1 \end{matrix})
= (\begin{matrix}6 & 0 \\ 0 & 2 \end{matrix})

So ∥p_{sym}(T_{1}, T_{2})∥ = 6 > 5 = ∥p∥_{𝔻^{2}}.

Remark 3.3. The example has hermitian T_{1} and T_{2} and a polynomial with real coefficients and yet ρ(p_{sym}(T_{1}, T_{2})) > ∥p∥_{𝔻^{2}}. Thus even Proposition 3.1 does not hold for non-commuting pairs.

The referee has provided an argument to show that for the polynomial p of Example 3.2, one has the inequality ∥p_{sym}(T_{1}, T_{2})∥ ≤ 6 for all contractions T_{1} and T_{2} (and thus the example is optimal for that p). This inequality is a substantial improvement over using the sum of the absolute values of the coefficients of p, so one is led to ask how well can one bound ∥p_{sym}(T)∥ for general p?

6 J. E. McCarthy and R. M. Timoney

4. ∥∑ ζ_{i}T_{i}∥ ≤ 1. In this section, we shall consider n-tuples T = (T_{1}, …, T_{n}) of operators, not assumed to be commuting, and we shall make the standing assumption:

∥ ∑_{i=1}^{n} ζ_{i}T_{i}∥ ≤ 1  ∀ ζ_{i} ∈ \overline{𝔻}.  (4.1)

This will hold, for example, if the condition

∑_{i=1}^{n} ∥T_{i}∥ ≤ 1  (4.2)

holds. We wish to derive bounds on ∥p_{sym}(T)∥. We start with the following lemma:

Lemma 4.1. If S ∈ B(H) and ∥S∥ < 1 then ℜ((I + S)(I − S)^{−1}) ≥ 0.

Proof.

2ℜ((I + S)(I − S)^{−1})
= (I − S^{∗})^{−1}(I + S^{∗}) + (I + S)(I − S)^{−1}
= (I − S^{∗})^{−1}[(I + S^{∗})(I − S) + (I − S^{∗})(I + S)](I − S)^{−1}
= 2(I − S^{∗})^{−1}[I − S^{∗}S](I − S)^{−1}
≥ 0.

If p(z) = ∑ c_{α}z^{α}, define

Γp(z) = ∑ c_{α} \frac{α!}{|α|!}z^{α}  (4.3)

(as usual, α! means α_{1}! ⋯ α_{n}!). We let Λ denote the inverse of Γ:

Λ ∑ d_{α}z^{α} = ∑ d_{α} \frac{|α|!}{α!}z^{α}.

Proposition 4.2. Let T = (T_{1}, T_{2}, …, T_{n}) ∈ B(H)^{n} satisfy (4.1) and p(z) be a polynomial in n variables. Then

∥p_{sym}(T)∥ ≤ ∥Γp∥\overline{𝔻}^{n}.  (4.4)

Proof. We first restrict to the case

ζ = (ζ_{1}, ζ_{2}, …, ζ_{n}) ∈ 𝕋^{n} ⇒ ∥ζ · T ∥ = ∥∑_{j=1}^{n} ζ_{j}T_{j}∥ < 1

and hence by Lemma 4.1 the operator

(I + ζ · T)(I − ζ · T)^{−1} = (I + ζ · T) ∑_{j=0}^{∞}(ζ · T)^{j} = I + 2 ∑_{j=1}^{∞}(ζ · T)^{j}

has positive real part

K(ζ, T) = ℜ ((I + ζ · T)(I − ζ · T)^{−1})
= I + ∑_{j=1}^{∞}(ζ · T)^{j} + ∑_{j=1}^{∞}(ζ̄ · T^{∗})^{j}
= 2ℜ [ ∑_{α_{1},...,α_{n}=0}^{∞} \frac{|α|!}{α!}ζ^{α}(z^{α})_{sym}(T)] − I.

We can compute that for polynomials p(z) = p(z_{1}, z_{2}, …, z_{n}),

p_{sym}(T) = ∫_{𝕋^{n}} Γp(ζ)K(ζ̄, T) dσ(ζ)

with dσ indicating normalised Haar measure on the torus 𝕋^{n} (and ζ̄ = (ζ̄_{1}, ζ̄_{2}, …, ζ̄_{n})).

As

K(ζ̄, T) dσ(ζ)

is a positive operator valued measure on 𝕋^{n}, we then have a positive unital linear map C(𝕋^{n}) → B(H) given by f ↦ ∫_{𝕋^{n}} f(ζ)K(ζ̄, T) dσ(ζ). As this map is then of norm 1, we can conclude

∥p_{sym}(T)∥ ≤ ∥Γp∥\overline{𝔻}^{n}.

For the remaining case sup_{ζ∈𝕋^{n}} ∥ζ · T ∥ = 1, we have

∥p_{sym}(T)∥ = lim_{r→1^{−}} ∥p_{sym}(rT)∥ ≤ ∥Γp∥\overline{𝔻}^{n}.

Remark 4.3. The technique of the above proof is derived from methods of [7]. Now we want to estimate ∥Γp∥\overline{𝔻}^{N}.

8 J. E. McCarthy and R. M. Timoney

Proposition 4.4. For each n ≥ 2 there is a constant M_{n} so that

∥Γp∥\overline{𝔻}^{n} ≤ M_{n} ∥p∥\overline{𝔻}^{n}.

Moreover,

M_{2} ≤ 4.07
M_{3} ≤ 16.6

Proof. Define

J(η) = ∑_{α_{1}=0,...,α_{n}=0}^{∞} \frac{α!}{|α|!}η^{α}. (4.5)

Then

Γp(z) = ∫_{𝕋^{n}} p(ζ)[J(z_{1}ζ̄_{1}, …, z_{n}ζ̄_{n})]dσ(ζ).  (4.6)

To use (4.6), we break J into two parts — the sum J_{0} where the minimum of the α_{i} is 0, and the remaining terms J_{1}.

J_{1}(η) = ∑_{α_{1}=1,...,α_{n}=1}^{∞} \frac{α!}{|α|!}η^{α}.

Case: n = 2. Here,

∫_{𝕋^{2}} p(ζ)J_{0}(z_{1}ζ̄_{1}, z_{2}ζ̄_{2})dσ(ζ) = p(z_{1}, 0) + p(0, z_{2}) − p(0, 0).  (4.7)

So the norm of the left-hand side of (4.7) is dominated by 3∥p∥\overline{𝔻}^{2}.

For J_{1}, we will use the estimate

|∫_{𝕋^{2}} p(ζ)J_{1}(z_{1}ζ̄_{1}, z_{2}ζ̄_{2})dσ(ζ)| ≤ ∥p∥_{∞}∥J_{1}∥_{L^{1}} ≤ ∥p∥_{∞}∥J_{1}∥_{L^{2}}.

We have

∥J_{1}∥_{L^{2}}^{2} = ∑_{α_{1},α_{2}=1}^{∞} (\frac{α_{1}!α_{2}!}{(α_{1} + α_{2})!})^{2}
= ∑_{α_{1}=1}^{∞} \frac{1}{(α_{1} + 1)^{2}} + ∑_{α_{2}=2}^{∞} \frac{1}{(α_{2} + 1)^{2}} + ∑_{α_{1},α_{2}=2}^{∞} (\frac{α_{1}!α_{2}!}{(α_{1} + α_{2})!})^{2}
≤ (\frac{π^{2}}{3} − \frac{9}{4}) + ∑_{k=4}^{∞}(k − 3) (\frac{2}{k(k − 1)})^{2}
≤ (1.069)^{2}.

(In the penultimate line, we let k = α_{1} + α_{2}; there are k − 3 terms with this sum, and the largest they can be is when either α_{1} or α_{2} is 2.) Adding the two estimates, we get M_{2} ≤ 4.07.

Case: n = 3. Again, we estimate the contributions of J_{0} and J_{1} separately. We have

∫ p(ζ)J_{0}(z_{1}ζ̄_{1}, z_{2}ζ̄_{2}, z_{3}ζ̄_{3})dσ(ζ)
= Γp(0, z_{2}, z_{3}) + [Γp(z_{1}, 0, z_{3}) − p(0, 0, z_{3})]
+ [Γp(z_{1}, z_{2}, 0) − p(z_{1}, 0, 0) − p(0, z_{2}, 0) + p(0, 0, 0)]

where we have had to subtract some terms to avoid double-counting. Thus the contribution of J_{0} is at most 3M_{2} + 4.

To calculate the contribution of J_{1}, we make the following estimate on ∥J_{1}∥_{L^{2}}, which is valid for all n ≥ 3:

We want to bound

∑_{α_{1}=1,...,α_{n}=1}^{∞} (\frac{α!}{|α|!})^{2}  (4.8)

Let k = |α| in (4.8). Note first that the number of terms for each k is the number of ways of writing k as a sum of n distinct positive integers (order matters), and this is exactly \binom{k−1}{n−1}. Moreover, as each α_{i} is at least 1, we have

\frac{α!}{|α|!} ≤ \frac{1}{k(k − 1) ⋯ (k − n + 2)}.

Therefore (4.8) is bounded by

∑_{k=n}^{∞} \binom{k − 1}{n − 1} (\frac{1}{k(k − 1) ⋯ (k − n + 2)})^{2}

= ∑_{k=n}^{∞} \frac{k − n + 1}{(n − 1)!k} \frac{1}{k(k − 1) ⋯ (k − n + 2)}.

The terms on the right-hand side of (4.9) decay like 1/k^{n−1}, so the series converges for all n ≥ 3. When n = 3, the series is

∑_{k=3}^{∞} \frac{k − 2}{2k^{2}(k − 1)} ≤ (0.381)^{2}.

Therefore M_{3} ≤ 3M_{2} + 4.381 < 16.59.

We now proceed by induction on n. The contribution from J_{0} is dominated by applying Γ to the restriction of p to the slices with one or more coordinates equal to 0, and these are bounded by the inductive hypothesis. The contribution from J_{1} is bounded by (4.8).

We have proved that the polydisk is an M-spectral set for T; we can make the constant one by enlarging the domain.

Proposition 4.5. There is a constant R_{n} so that

∥Γp∥\overline{𝔻}^{n} ≤ ∥p∥_{R_{n}\overline{𝔻}^{n}}.  (4.9)

Moreover,

R_{2} ≤ 1.85
R_{3} ≤ 2.6

Proof. Let L(η) = 2ℜJ(η) − 1. Adding terms that are not conjugate analytic powers of ζ inside the bracket in (4.6) will not change the value of the integral, so, writing zζ̄ for the n-tuple (z_{1}ζ̄_{1}, …, z_{n}ζ̄_{n}), we get

Γp(z) = ∫_{𝕋^{n}} p(ζ)[L(zζ̄)]dσ(ζ).  (4.10)

As L is real and has integral 1, if we can choose r_{n} so that if |z_{i}| ≤ r_{n} for each i then L(zζ̄) is non-negative for all ζ, then its L^{1} norm would equal its integral, and so we would get from (4.10) that

|Γp(z)| ≤ ∥p∥\overline{𝔻}^{n}.

Letting R_{n} = 1/r_{n} gives (4.9). As the series (4.5) converges absolutely for all η ∈ 𝔻^{n}, and L(0) = 1, the existence of some r_{n} now follows by continuity.

Let us turn now to obtaining quantitative estimates.

Case: n = 2. Adding terms to J that are not analytic will not affect the integral (4.10), so let us consider

L′(η) = ℜ [\frac{1 + η_{1}}{1 − η_{1}} ] · ℜ [\frac{1 + η_{2}}{1 − η_{2}} ] − ∑_{α_{1}=1,α_{2}=1}^{∞}(1 − \frac{α!}{|α|!})(η_{1}^{α_{1}} − η̄_{1}^{α_{1}})(η_{2}^{α_{2}} − η̄_{2}^{α_{2}}).

Then L′ has integral 1 and (4.10) is unchanged if L is replaced by L′. So we wish to find the largest r so that L′ is positive on r𝔻^{2}.

It can be checked numerically that r = 0.5406 works, so the best R_{2} is smaller than the reciprocal of 0.5406, which is less than 1.85.

Case: n = 3. As in the case n = 2, we consider the kernel

L′(η) = ℜ [\frac{1 + η_{1}}{1 − η_{1}} ] · ℜ [\frac{1 + η_{2}}{1 − η_{2}} ] · ℜ [\frac{1 + η_{3}}{1 − η_{3}} ]

− ∑_{α_{1}=1,α_{2}=1,α_{3}=0}^{∞}(1 − \frac{α!}{|α|!})(η_{1}^{α_{1}} − η̄_{1}^{α_{1}})(η_{2}^{α_{2}} − η̄_{2}^{α_{2}})(η_{3}^{α_{3}} + η̄_{3}^{α_{3}}).

(Note that there is a plus in the last factor to keep L′ real.) Again, a computer search can find r so that L′ is positive on r𝔻^{3}, and r = .39 works, so R_{3} < 2.6.

Combining Propositions 4.2, 4.4 and 4.5, we get the main result of this section.

Theorem 4.6. There are positive constants M_{n} and R_{n} such that whenever T = (T_{1}, T_{2}, …, T_{n}) ∈ B(H)^{n} satisfies (4.1) and p(z) is a polynomial in n variables, then

∥p_{sym}(T)∥ ≤ ∥p∥_{R_{n}\overline{𝔻}^{n}}  (4.11)
∥p_{sym}(T)∥ ≤ M_{n}∥p∥\overline{𝔻}^{n}.  (4.12)

Moreover, one can choose R_{2} = 1.85, R_{3} = 2.6, M_{2} = 4.1 and M_{3} = 16.6.

Remark 4.7. Another way to estimate ∥p_{sym}(T)∥, under the assumption (4.2), would be to crash through with absolute values. Let Δ_{n} = {z ∈ ℂ^{n} : ∑_{j=1}^{n} |z_{j}| ≤ 1} and let r_{n} denote the Bohr radius of Δ_{n}, i.e. the largest r such that whenever p(z) = ∑ c_{α}z^{α} has modulus less than or equal to one on Δ_{n}, then q(z) = ∑ |c_{α}|z^{α} has modulus bounded by one on rΔ_{n}. One then has the estimate that, under the hypothesis (4.2), and writing C_{n} = 1/r_{n},

∥p_{sym}(T)∥ ≤ ∥q∥_{Δ_{n}} ≤ ∥p∥_{C_{n} Δ_{n}}.  (4.13)

It was shown by L. Aizenberg [1, Thm. 9] that

\frac{1}{3e^{1/3}} < r_{n} ≤ \frac{1}{3}.

So the estimate in (4.11) for pairs satisfying (4.2) does not follow from (4.13).

5. n-tuples of contractions. In an attempt to use the above technique for tuples T ∈ B(H)^{n} such that max_{1≤j≤n} ∥T_{j}∥ ≤ 1, we consider restricting ζ to belong to Δ_{n}, and we replace σ by some probability measure μ supported on Δ_{n}.

Suppose we can find some function q such that

Λ_{μ}(q)(z) := ∫_{Δ_{n}} q(ζ)ℜ\frac{1 + ζ̄ · z}{1 − ζ̄ · z} dμ(ζ)  (5.1)

equals p(z). We do not actually need q to be a polynomial; having an absolutely convergent power series on Δ_{n} (in ζ and ζ̄) is enough.

Lemma 5.1. With notation as above, assume Λ_{μ}(q) = p and that T ∈ B(H)^{n} is an n-tuple of contractions. Then

∥(p)_{sym}(T)∥ ≤ ∥q∥_{suppt(μ)} ≤ sup{|q(z)| : z ∈ Δ_{n}}.

Proof. We assume first that max_{1≤j≤n} ∥T_{j}∥ < 1 and use the notation K(ζ, T) from the proof of Proposition 4.2 (which is permissible as ∥ζ · T ∥ < 1 for ζ ∈ Δ_{n}). We have

(Λ_{μ}q)_{sym}(T) = ∫_{Δ_{n}} q(ζ)K(ζ̄, T) dσ(ζ)

and hence the inequality ∥(p)_{sym}(T)∥ ≤ ∥q∥_{suppt(μ)} follows as in the previous proof.

If max_{1≤j≤n} ∥T_{j}∥ = 1, we deduce the result from ∥(p)_{sym}(rT)∥ ≤ ∥q∥_{Δ_{n}} for 0 < r < 1.

Remark 5.2. For an arbitrary measure μ, there might be no q such that Λ_{μ}(q) = p. If μ is chosen to be circularly symmetric, though, one gets

Λ_{μ}(z^{α}) = [\frac{|α|!}{α_{1}! … α_{n}!} ∫ |ζ^{α}|^{2}dμ(ζ)] z^{α}.  (5.2)

As long as none of the moments on the right of (5.2) vanish, inverting Λ_{μ} is now straightforward.

To make use of the lemma to bound p_{sym}(T) we need to find a way to choose another polynomial q and a μ on Δ_{n} so that p = Λ_{μ}q and ∥q∥_{Δ_{n}} is small. We do not know a good way to do this.

Question 1. What is the smallest constant R_{n} such that, for every n-tuple T of contractions and every polynomial p, one has

∥p_{sym}(T)∥ ≤ ∥p∥_{R_{n}\overline{𝔻}^{n}}?  (5.3)

We do not know if one can choose R_{n} smaller than the reciprocal of the Bohr radius of the polydisk, even when n = 2.

Question 2. Is there a constant M_{n} such that, for every n-tuple T of contractions and every polynomial p, one has

∥p_{sym}(T)∥ ≤ M_{n} ∥p∥\overline{𝔻}^{n}?  (5.4)

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