On Von Neumann’s Inequality for Matrices of Complex Polynomials

We prove that every matrix ( ) k n F M ∈  is associated with the smallest positive integer ( ) 1 d F ≠ such that ( ) d F F ∞ 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. In application, we show that von Neumann’s inequality holds up to the constant 2 n for matrices of the algebra ( ) k n M  .


Introduction
In 1951, von Neumann [1] showed that for any contraction linear operator T on a Hilbert space the inequality ( ) , p T p ∞ ≤ holds for all complex polynomials ( ) p z over the unit disk, where p ∞ 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 { } 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 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 3 3 × 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 ques-

Proof of the Main Result
In this section, we give the proof of Theorem , , can be written as ( ) can be regarded as a polynomial over n  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 , x y of G such that 0 x > and 0 y > , there corresponds a positive integer n such that nx y > .
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 a nb ≤ .
The Archimedean property of the total order on  allows us to show that every element ( ) , .
For matrices of ( ) , k r n Ω  , we have the following: , .  1 2  3  1 2  3  1 2 3  , ,,   , , , be a matrix of complex polynomials of three variables of ( ) is a positive matrix of complex polynomials and ( ) , , , , 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 be a matrix of complex polynomials over n  . Then ( ) It is easy to say that ( ) In other words, It is straightforward to observe that if we add (2.1) and (2.2), one has ( ) ( ) ( ) . It follows that Therefore, there exists a positive constant n K such that


We also notice that the fraction ( ) depends on the expression of F not on the number of variables of F. However, the constant 2 n depends on the number of variables of F. This means that for the sufficiently large number of variables, the inequality (2.6) is trivial.