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

Lemma 5.1. Let
R = a unital full subring of A(D), S := C(T),
G := Z,
ι := w.
Then (A1)–(A4) are satisfied.

Proof. (A1) and (A2) are clear. (A3) is evident from the definition of w. Finally, we will show below that (A4) holds.
Suppose that f ∈ R ∩ (inv C(T)) is invertible as an element of R. Then obviously f is also invertible as an element of
A(D). Hence it has no zeros or poles in D. For r ∈ (0,1), define f_r ∈ A(D) by f_r(z) = f (rz) (z ∈ D). Then f_r also has no zeros or poles in D, and has a holomorphic extension across T. From the Argument Principle (applied to f_r), it follows that w( f_r) = 0. But I f_r − f I_∞ → 0 as r ր 1. Hence w( f ) = lim_r→1 w( f_r) = lim_r→1 0 = 0.
Suppose, conversely, that f ∈ R ∩ (inv C(T)) is such that w( f ) = 0. For all r ∈ (0,1) sufficiently close to 1, we have that f_r ∈ inv C(T). Also, by the local constancy of w, for r sufficiently close to 1, w( f_r) = w( f ) = 0. By the Argument principle, it then follows that f_r has no zeros in D. Equivalently, f has no zeros in rD. But letting r ր 1, we see that f has no zeros in D. Moreover, f has no zeros on T either, since f ∈ inv C(T). Thus f has no zeros in D. Consequently, we conclude that f is invertible as an element of A(D). (Indeed, f is invertible as an element of C(D), and it is also then clear that this inverse is holomorphic in D.) Finally, since R is a full subring of A(D), we can conclude that f is invertible also as an element of R. ✷

Besides A(D) itself, some other examples of such R are:

(1) RH^∞(D), the set of all rational functions without poles in D.
(2) The Wiener algebra W ⁺(D) of all functions f ∈ A(D) that have an absolutely convergent Taylor series about the origin:

n=0
^∞

| f_n| < +∞, where f (z) =

n=0
^∞

f_nzⁿ (z ∈ D).

(3) ∂⁻ⁿ H^∞(D), the set of f : D → C such that f , f ⁽¹⁾, f ⁽²⁾,..., f ⁽ⁿ⁾ belong to H^∞(D). Here H^∞(D) denotes the Hardy
algebra of all bounded and holomorphic functions on D.
An application of our main result (Theorem 4.1) yields the following Nyquist criterion. We note that invertibility of f in C(T) just means that f belongs to C(T) and it has no zeros on T.
Corollary 5.2. Let R be a unital full subring of A(D). 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.
(2) (a) det(I − C P) belongs to C(T),
coprime factorization of P, and C = D_C ¹N_C be a left coprime factorization of C. Then the following are equivalent:

(c) w(det(I − C P)) + w(det D_P ) + w(det D_C ) = 0.

(b) det(I − C P), det D_P , det D_C have no zeros on T, and
It can be shown that Y = T satisfies the generalized argument principle for A(D); see [17, Corollary 1.25]. Moreover, we know that if a function in A(D) is invertible, then by considering the map r → f_r|_T : [0,1] → inv C(T), we see that f belongs to the connected component of inv C(T) that contains 1. So it is of the form f |_T = e^g for some g ∈ C(T). Hence f |_T has a continuous logarithm on T. So we can take S = C(T). Moreover, if exp C(T) denotes the connected component in
inv C(T) which contains the constant function 1 on T, then G = (inv C(T)/exp C(T)) is isomorphic to Z (see for example [7, Corollary 2.20]), and ι can be taken as the natural homomorphism from inv C(T) to Z given by the winding number.

708 A. Sasane / J. Math. Anal. Appl. 370 (2010) 703–715

Remarks 5.3.
(1) RH^∞(D) is a projective free ring since it is a Bezout domain. Also A(D), W ⁺(D), or ∂⁻ⁿ H^∞(D) are projective free rings, since their maximal ideal space is D, which is contractible; see [1]. Thus if R is one of RH^∞(D), A(D), W ⁺(D) or ∂⁻ⁿ H^∞(D), then the set S(R, 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].
(2) The result in Corollary 5.2 was known in the special cases when R is RH^∞(D) or A(D); see [25].
(3) We remark that in the case of single input single output systems, namely when p = m = 1, the result in Theorem 5.2
can be interpreted graphically on the basis of what is called the Nyquist plot; see for example [9, §5.5]. In the case of multi-input multi-output systems (namely when p and/or m are larger than 1), the possibility of having an analogous graphical Nyquist diagram similar to the single input single output case was investigated in [16].

5.2. Almost periodic functions
The algebra AP of complex valued (uniformly) almost periodic functions is the smallest closed subalgebra of L^∞(R) that contains all the functions e_λ := e^iλy. Here the parameter λ belongs to R. For any f ∈ AP, its Bohr–Fourier series is defined by the formal sum

λ
where


f_λe^iλy, y ∈ R, (1)

f_λ := lim
N→∞



1

[−N,N]
2N □


e^−iλy f (y)dy, λ ∈ R,

and the sum in (1) is taken over the set σ( f ) := {λ ∈ R | f_λ = 0}, called the Bohr–Fourier spectrum of f . The Bohr–Fourier spectrum of every f ∈ AP is at most a countable set.
The almost periodic Wiener algebra AP W is defined as the set of all AP such that the Bohr–Fourier series (1) of f converges absolutely. The almost periodic Wiener algebra is a Banach algebra with pointwise operations and the norm

AP W ⁺ = ^□ f ∈ AP W ^□ σ( f ) ⊂ [0,∞)^□.
I f I := ^□_λ∈R | f_λ|. Set
AP⁺ = ^□ f ∈ AP ^□ σ( f ) ⊂ [0,∞)^□,

1
Then AP⁺ (respectively AP W ⁺) is a Banach subalgebra of AP (respectively AP W ). For each f ∈ inv AP, we can define the average winding number w( f ) ∈ R of f as follows:

w( f ) = lim
T →∞

See [14, Theorem 1, p. 167].

_2T ^□arg^□ f (T )^□ − arg^□ f (−T )^□□.

Lemma 5.4. Let
R := a unital full subring of AP⁺,
S := AP,
G := R,
ι := w.
Then (A1)–(A4) are satisfied.

Proof. (A1) and (A2) are clear. (A3) follows from the definition of w. Finally, (A4) follows from [3, Theorem 1, p. 776] which
says that f ∈ AP⁺ satisfies
inf

Im(s)□0□

f (s)^□ > 0 (2)
if and only if inf_y∈R | f (y)| > 0 and w( f ) = 0. But

y∈R□

f (y)^□ > 0
inf

A. Sasane / J. Math. Anal. Appl. 370 (2010) 703–715 709

is equivalent to f being an invertible element of AP by the corona theorem for AP (see for example [10, Exercise 18, p. 24]). Also the equivalence of (2) with that of the invertibility of f as an element of AP⁺ follows from the Arens–Singer corona theorem for AP⁺ (see for example [2, Theorems 3.1, 4.3]). Finally, the invertibility of f ∈ R in R is equivalent to the invertibility of f as an element of AP⁺ since R is a full subring of AP⁺. ✷

Remark 5.5. Specific examples of such R are AP⁺ and AP W ⁺. More generally, let Σ ⊂ [0,+∞) be an additive semigroup (if λ,μ ∈ Σ, then λ + μ ∈ Σ) and suppose 0 ∈ Σ. Denote

AP W_Σ = ^□ f ∈ AP W ^□ σ( f ) ⊂ Σ^□.
AP_Σ = ^□ f ∈ AP ^□ σ( f ) ⊂ Σ^□,
Then AP_Σ (respectively AP W_Σ ) is a unital Banach subalgebra of AP⁺ (respectively AP W ⁺). Let Y_Σ denote the set of all maps θ : Σ → [0,+∞] such that θ(0) = 0 and θ(λ + μ) = θ(λ) + θ(μ) for all λ,μ ∈ Σ. Examples of such maps θ are the following. If y ∈ [0,+∞), then θ_y, defined by θ_y(λ) = λy, λ ∈ Σ, belongs to Y_Σ . Another example is θ_∞, defined as follows:

_{θ∞(λ) =} 0 if λ = 0,

+∞ if λ = 0.

So in this way we can consider [0,+∞] as a subset of Y_Σ .
The results [2, Proposition 4.2, Theorem 4.3] say that if Y_Σ ⊂ [0,+∞], and f ∈ AP_Σ (respectively AP W_Σ ), then f ∈
inv AP_Σ (respectively ∈ inv AP W_Σ ) if and only if (2) holds. So in this case AP_Σ and AP W_Σ are unital full subalgebras of AP⁺.
An application of our main result (Theorem 4.1) yields the following Nyquist criterion. We note that invertibility of f in AP just means that f belongs to AP and is bounded away from zero on R again by the corona theorem for AP.