On the Numerical Range of Square Matrices with Coefficients in a Degree 2 Galois Field Extension

Let L be a degree 2 Galois extension of the field K and M an n×n matrix with coefficients in L. Let 〈 , 〉 : Ln × Ln −→ L be the sesquilinear form associated to the involution L −→ L fixing K. We use 〈 , 〉 to define the numerical range Num(M) of M (a subset of L), extending the classical case K = R, L = C and the case of a finite field introduced by Coons, Jenkins, Knowles, Luke and Rault. There are big differences with respect to both cases for number fields and for all fields in which the image of the norm map L −→ K is not closed by addition, e.g., c ∈ L may be an eigenvalue of M , but c / ∈ Num(M). We compute Num(M) in some case, mostly with n = 2. For any integer n > 0 and any field L let Mn,n(L) be the L-vector space of all n × n matrices with coefficients in L. Let K be a field and L a degree 2 Galois extension of K. Call σ the generator of the Galois group of the extension K →֒ L. Thus σ : L −→ L is a field isomorphism, σ : L −→ L is the identity map and K = {t ∈ L | σ(t) = t}. For any u = (u1, . . . , un) ∈ L, v = (v1, . . . , vn) ∈ L set 〈u, v〉 := ni=1 σ(ui)vi. The map 〈 , 〉 : L × L −→ L is sesquilinear, i.e. for all u, v, w ∈ L and all c ∈ L we have 〈u + v, w〉 = 〈u,w〉 + 〈v, w〉, 〈u, v + w〉 = 〈u, v〉+ 〈u,w〉, 〈cu, w〉 = σ(c)〈u,w〉 and 〈u, cw〉 = c〈u,w〉. Set Cn(1) := {u ∈ L | 〈u, u〉 = 1}. For any M ∈ Mn,n(M) set Num(M) := {〈u,Mu〉 | u ∈ Cn(1)}. Since Cn(1) 6= ∅, we have Num(M) 6= ∅. As in the classical case when K = R, L = C and σ is the complex conjugation the subset Num(M) of L is called the numerical range of M ([4], [5], [6], [11]). When K is a finite field the numerical range was introduced in [3] and [1]. In particular [3] built a bridge between the classical case and the finite field case and at certain points we will duly quote the parts of [3], which we adapt to our set-up. Assume for the moment L = K(i) with K ⊂ R and σ the complex conjugation. In this case, calling Num(M)C ⊂ C the usual numerical range of M , we have Num(M) ⊆ Num(M)C and hence Num(M) is a bounded subset of C. But even in this case there are many differences, in particular as for number fields not every element of K is a square. The main differences come from the structures of the sets ∆ and ∆n defined below. Let ∆ ⊆ K be the image of the norm map NormL/K : L −→ K, i.e. set ∆ := {aσ(a) | a ∈ L} ⊆ K. If a ∈ K, then σ(a) = a and hence NormL/K(a) = a. Thus ∆ contains all squares of elements of K. In particular 0 ∈ ∆ and 1 ∈ ∆. Since the norm map NormL/K is multiplicative, ∆ is closed under multiplication. If c ∈ ∆̂ := ∆ \ {0}, say c = σ(a)a for some a ∈ L \ {0}, then 1/c = σ(a)a and hence ∆̂ is a multiplicative group. For any integer n > 0 let ∆n be the set of all 2010 Mathematics Subject Classification. 15A33; 15A60; 12D99; 12F99.