On Von Neumann’s Inequality for Matrices of Complex Polynomials ()
1. Introduction
In 1951, von Neumann [1] showed that for any contraction linear operator T on a Hilbert space the inequality
holds for all complex polynomials
over the unit disk, where
denotes the supremum norm of p over the unit disk. This result was generalised by many people. In particular, Brehmer [2] proved in 1961 that von Neumann’s inequality also holds for families
of commuting operators on a complex Hilbert space with
In 1963, Ando [3] established the natural generalisation of von Neumann’s inequality for polynomials in two commuting contractions. In 1974, Varopoulos [4] proved that the analogue of von Neumann’s inequality fails for 3 or more commuting contractions. There are several such counterexamples in the literature [5] . In 1978, Lubin [6] proved that if
are commuting contractions on a Hilbert space, then
(1.1)
for any polynomial
over
. von Neumann’s inequality holds for commutative families of isometries and doubly commuting sets of contractions [7] . Recent work of Kosiński on the three point Pick interpolation problem on polydisc shows that von Neumann’s inequality holds for
commuting contractive matrices [8] . In 2020, Mouanda proved that von Neumann’s inequality holds for n-tuples of upper (or lower) complex triangular Toeplitz (or circulant) contractions [9] . This result was first extended to matrices of complex polynomials in 2021 by Mouanda [10] .
This result, which has many engineering applications, is a fundamental tool in operator theory [5] [11] .
In this paper, we are mainly concerned with the following long-standing question: Does von Neumann’s inequality hold up to some constant for n-tuples of commuting contractions? First of all, we show that every
is associated with the smallest positive integer
such that
is always bigger than the sum of the operator norms of the Fourier coefficients of F. We establish some inequalities for matrices of complex polynomials.
Theorem 1.1. Let
be the algebra of complex polynomials over
, let
be a matrix of complex polynomials and let
be a set of commuting contractions on the Hilbert space
. Then there exists a sequence
of positive numbers such that
with
where
is a positive integer such that
and
In application, we show that the von Neumann inequality holds up to the constant
for matrices of the algebra
.
2. Proof of the Main Result
In this section, we give the proof of Theorem 1.1. Given
a complex polynomial over
. Then
Suppose that
are complex polynomials over
. Then the
-matrix
can be written as
is a polynomial over
with matrix coefficients. Let
denote the algebra of complex polynomials over
. Given
complex polynomials over
. Then
Denote by
Then the
-matrix
can be written as
In other words,
The
-matrix
can be regarded as a polynomial over
with matrix coefficients. Each element of the algebra
has this representation. In our case, the set
is the set of Fourier coefficients of
. Let us set
where the supremum is taken over the family of all n-tuples of commuting contractions on all Hilbert spaces. It is easy to see that
is finite, since it is bounded by the sum of the operator norms of the Fourier coefficients of F, and that this quantity defines a norm on the algebra
of matrices of polynomials over
. For each polynomial P in
, there is always an n-tuple of contractions where this supremum is achieved. Therefore,
is a normed algebra.
Definition 2.1. Let G be a group. An order >, on G, is called archimedean if it has the following property: to every pair of elements
of G such that
and
, there corresponds a positive integer n such that
.
The order axiom for the real line states that every real number is less than some natural number. This is equivalent to the assertion that for any two positive real numbers a and b there is a positive integer n such that
.
The Archimedean property of the total order on
allows us to show that every element
of
is associated to the smallest positive integer
such that
is always bigger than the sum of the operator norms of the Fourier coefficients of
.
Theorem 2.2. Let
be an element of
. Then there exists a smallest positive interger
such that
Proof. Let
be a
-matrix of complex polynomials over
. In general, we have
and
The Archimedean property of the total order on
allows us to claim that there exists a smallest positive integer
such that
口
Remark 2.3. Each element F of the algebra
has a finite number of Fourier coefficients and each Fourier coefficient of an element F of
is bounded by its supremum norm
over the polydisc
.
Let
be an element of
and let
be the number of the Fourier coefficients of F. Denote by
Theorem 2.4. Let
be the set of complex polynomials over
and let
be a subspace of
. Then
Proof. Let
be an element of
. Therefore,
. Remark 2.3 allows us to claim that
with
It follows that
For matrices of
, we have the following:
Therefore,
Finally,
口
Theorem 2.4 allows us to state that, for every n-tuple
of commuting contractions on a Hilbert space
, one has
We can factorize matrices of complex polynomials. Assume that
and let
be a matrix of complex polynomials of three variables of
. Then
there exists a constant
,
, such that
Let us notice that
is a positive matrix of complex polynomials and
is not unique. Denote by
and
It is straightforward to see that
This implies that there exists a matrix
of complex polynomials over
such that
What we need to notice is that the sets
are not bounded. However, the factorization of matrices of complex polynomials, in terms of the product of matrices of complex polynomials of two variables, allows us to claim that the set
is bounded by
.
Proof of Theorem 1.1
Let
be a matrix of complex polynomials over
. Then
there exists a constant
,
, such that
Denote by
the matrix of complex polynomials with positive matrices as Fourier coefficients. The matrix of complex polynomials
satisfies the von Neumann’s inequality. Denote by
and
. As we can see
, are matrices of complex polynomials of two variables with positive matrices as Fourier coefficients and
, are matrices of complex polynomials of one variable with positive matrices as Fourier coefficients. These polynomials have exactly the same number of Fourier coefficients than
and they all satisfy the von Neumann inequality. It is straightforward to see that
and
Denote by
That is,
. It is not difficult to see that there exists a matrix
of complex polynomials over
such that
In other words,
It is straightforward to see that
In general, the matrix
is not unique. The structure of the matrix
depends on the structure of
. Let
be an n-tuple of commuting contractions on a Hilbert space
. The von Neumann inequality for matrices of complex polynomials of two (or one) variable(s) allows us to say that
This implies that
Suppose that
It follows that
This implies that,
Thus,
Also, we have
It is easy to say that
Therefore,
We can claim that
since, in general,
It is clear that
This implies that
Due to the fact that
and
we can say that for every
, there exist two positive real numbers
and
such that
and
Thus,
and
Therefore,
It is not difficult to see that
Now, we can claim that
There exists a positive real number
such that
This allows us to say that
In other words,
(2.1)
with
. Let us notice that
and
(2.2)
It is straightforward to observe that if we add (2.1) and (2.2), one has
(2.3)
. It follows that
(2.4)
Therefore, there exists a positive constant
such that
Define the sequence
of matrices of complex
polynomials over
by setting
with
,
, such that
The elements of the sequence
satisfy the von
Neumann inequality. Let
be two new sequences of matrices of complex polynomials over
defined by
and
. For every
, one has
A straightforward calculation shows that there exists a new sequence
of matrices of complex polynomials over
such that
We have exactly the same result as in (2.4). In other words,
(2.5)
Denote by
. Finally, for every
, there exists a sequence
of positive numbers such that
Let us notice that the constant
depends only on the Fourier coefficients of the matrix of complex polynomials
. Let us estimate the growth of
depending on the values of m. Let us examine the number of terms of the product
This matrix of complex polynomials has exactly
terms and there exists a matrix of complex polynomials
.
A simple calculation shows that
In other words,
The expression of
can be deduced from the matrix of complex polynomials
This means that the expression of the matrix
can be computed from the
remaining terms of the matrix of complex polynomials
It follows that
with
Recall that
Therefore, we can claim that
口
3. Application
In this section, we show that von Neumann’s inequality holds up to the constant
.
Remark 3.1. Let
be a matrix of complex polynomials over
. There exists a positive constant
such that
and
where
is a positive integer such that
We can now show that von Neumann’s inequality hold up to the constant
.
Theorem 3.2. Let
be the algebra of complex polynomials over
, let
be a matrix of complex polynomials over
and let
be an n-tuple of commuting contractions on a Hilbert space
. Then
Proof. Let
be a matrix of complex polynomials over
. Theorem 1.1 allows us to claim that there exists a sequence
of matrices of complex polynomials over
such that
where
is a positive integer such that
Recall that
Remark 3.1 allows us to claim that there exists a positive constant
such that
and
It follows that
This implies that
Therefore,
Due to the fact that
, implies that
Therefore,
Finally, if
is an n-tuple of commuting contractions on a Hilbert space
, we have
口
We also notice that the fraction
,
,
,
depends on the expression of F not on the number of variables of F. However, the constant
depends on the number of variables of F. This means that for the sufficiently large number of variables, the inequality (2.6) is trivial.