Math. Anal. Appl. 370 (2010) 703-715 Contents lists available at

Corollary 5.20. Let P ∈ S(H^∞(D), p,m) and C ∈ S(H^∞(D),m, p). Moreover, let P = N_P D_P ¹ be a right coprime factorization of P,

(1) C stabilizes P.
(2) (a) det(I − C P) ∈ H^∞(D) + C(T).
(b) Let F₁, F₂, F₃ be the harmonic extensions to D, of
and C = D_C ¹N_C be a left coprime factorization of C. Then the following are equivalent:


respectively. There exist δ,ǫ > 0 such that

(c) ι(det(I^□F_iCre^it□□ □ ǫ, 1 − δ < r < 1, i = 1,2,3.
f₁ := det(I − C P), f₂ := det D_P , f₃ := det D_C ,

P)) + ι(det D_P ) + ι(det D_C ) = 0.



Remark 5.21. It was proved by Inouye [13] that the set S(H^∞(D), p,m) of plants possessing a left and a right coprime factorization coincides with the class of plants that are stabilizable.

5.6. The polydisk algebra
Let

Dⁿ := ^□(z₁,..., z_n) ∈ Cⁿ: |z_i| < 1 for i = 1,...,n^□,




Tⁿ := ^□(z₁,..., z_n) ∈ Cⁿ: |z_i| = 1 for i = 1,...,n^□.
Dⁿ := ^□(z₁,..., z_n) ∈ Cⁿ: |z_i| □ 1 for i = 1,...,n^□,
The polydisk algebra A(Dⁿ) is the set of all functions f : Dⁿ → C such that f is holomorphic in Dⁿ and continuous on Dⁿ.
If f ∈ A(Dⁿ), then the function f_d defined by z → f (z,..., z) : D → C belongs to the disk algebra A(D), and in particular
also to C(T). The map

f → ( f |_Tⁿ , f_d) : A^□D^n□ → C^□T^n□ × C(T)

is a ring homomorphism. This map is also injective, and this is an immediate consequence of Cauchy’s formula; see [22, p. 4–5]. We recall the following result; see [22, Theorem 4.7.2, p. 87].

Proposition 5.22. Suppose that Ψ = (ψ₁,...,ψ_n) is a continuous map from D into Dⁿ, which carries T into Tⁿ and the winding number of each ψ_i is positive. Then for every f ∈ A(Dⁿ), f (Ψ (D) ∪ Tⁿ) = f (Dⁿ).
Lemma 5.23. Let

S := C^□T^n□ × C(T),
R = a unital full subring of A^□D^n□,
G := Z,

Then (A1)–(A4) are satisfied.
ι := ^□(g,h) → w(h)^□.
Proof. (A1) and (A2) are clear. (A3) was proved earlier in Subsection 5.1. Finally, we will show below that (A4) holds, following [6].
Suppose that f ∈ A(Dⁿ) is such that f |_Tⁿ ∈ inv C(Tⁿ), f_d ∈ inv C(T) and that w( f_d) = 0. We use Proposition 5.22, with Ψ (z) := (z,..., z) (z ∈ D). Then we know that f will have no zeros in Dⁿ if f (Ψ (D)) does not contain 0. But since f_d ∈ inv C(T) and w( f_d) = 0, it follows that f_d is invertible as an element of A(D) by the result in Subsection 5.1. But this implies that f (Ψ (D)) does not contain 0.
Now suppose that f ∈ A(Dⁿ) with f |_Tⁿ ∈ inv C(Tⁿ), f_d ∈ inv C(T), and that it is invertible as an element of A(Dⁿ). But then in particular, f_d is an invertible element of A(D), and so again by the result in Subsection 5.1, it follows that w( f_d) = 0. ✷
Besides A(Dⁿ) itself, another example of such R is RH^∞(Dⁿ), the set of all rational functions without poles in Dⁿ.
An application of our main result (Theorem 4.1) yields the following Nyquist criterion.

A. Sasane / J. Math. Anal. Appl. 370 (2010) 703–715 715
Corollary 5.24. Let R be a unital full subring of A(Dⁿ). Let P ∈ S(R, p,m) and C ∈ S(R,m, p). Moreover, let P = N_P D_P ¹ be a right

(1) C stabilizes P.
coprime factorization of P, and C = D_C ¹N_C be a left coprime factorization of C. Then the following are equivalent:

(b) ι(det(I − C P)) + ι(det D_P ) + ι(det D_C ) = 0.
(2) (a) det(I − C P), det D_P , det D_C belong to inv(C(Tⁿ) × C(T)), and
Remark 5.25. By [1], it follows that A(Dⁿ) is a projective free ring, since its maximal ideal space the polydisk Dⁿ is con- tractible. Thus the set S(A(Dⁿ), p,m) of plants possessing a left and a right coprime factorization coincides with the class of plants that are stabilizable by [20, Theorem 6.3].
Corollary 5.24 was known in the special case when R = P; see [6].

Acknowledgments
The author thanks Alban Quadrat for mentioning the problem of obtaining a Nyquist criterion for infinite dimensional control systems, for Refs. [3] and [5], and for a discussion on this area. Thanks are also due to the anonymous referee for useful suggestions.
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