Page 104 - Textos de Matemática Vol. 44

94 N. KRUPNIK AND Y. SPIGEL
Theorem 1.2. Let F(A,B) be a non-trivial polynomial of second degree. The following statements are equivalent:
(i) (ii)
(1.2)
(iii) (1.3)
The polynomial F(A,B) is 3-admissible.
The polynomial F(A,B) admits a representation
F(A,B)=a(A2 +B2)+bAB+cBA+d(A+B)+qE, where a,b,c,d,q ∈ C and a2 = bc.
The polynomial F(A,B) admits a special factorization
F (A, B) = (pA + rB + sE)(rA + pB + sE) + γE,
(i) (ii)
(1.4)
(iii) (1.5)
The polynomial F(A,B) is 4-admissible.
The polynomial F(A,B) can be represented (up to a constant factor) in the following form.
F(A,B)=A3 +B3 +ABA+BAB−AB2 −A2B−BA2 −B2A +a(A2 +B2)+bAB+cBA+d(A+B)+qE,
wherea=−b+c andd=−(b−c)2. 2 16
The polynomial F(A,B) admits a special factorization
F (A, B) = (A − B − αE) (A + B + βE) (A − B + αE) + γE,
Correspondingly
where p,q,r,s,q ∈ C.
The polynomial F(A,B) is admissible.
(iv)
The proof of this theorem is given in Section 4.
Theorem 1.3. Let F(A,B) be a non-trivial polynomial of third degree. As- sume that F(A,B) is not of second degree. Then the following statements are equivalent:
F (B, A) = (A − B + αE) (A + B + βE) (A − B − αE) + γE, whereα, β, γ∈C.
(1.6)
(iv) F(A,B) is admissible.
The proof of this theorem is given in Section 5.
In Sections 2 and 3 respectively we give some sufficient and some necessary conditions for the polynomials of any degree to be admissible.
In Section 6 we discuss some results related to polynomials of degrees n ≥ 4. In particular, we prove the following challenging statement:
Theorem 1.4. For any number n ∈ N, n ≥ 2 there exists a non-trivial ad- missible polynomial F(A,B) of degree n.