ON THE HAMILTONIAN STRUCTURE IN THE COMPUTATION OF SINGULAR VALUES FOR A CLASS OF HANKEL OPERATORS

P. A. Fuhrmann Department of Mathematics Ben-Gurion University of the Negev Beer Sheva, Israel

Abstract

The paper deals with the problem of computing the singular values of a Hankel operator with a symbol of the type m^{∗}w with m, w ∈ H_{+}^{∞}, m inner and w rational. In this case the problem reduces to the computation of zeroes of some functional determinant. This has been shown in Foias, Tannenbaum and Zames [1988], Lypchuk, Smith and Tannenbaum [1988], Smith [1989], Ozbay [1990]. The emphasis in this paper is to give a more illuminating and geometrical proof of the conjecture in Zhou and Khargonekar [1987]. This proof is based on the theory of polynomial models developed by the author, and it leans heavily on Smith [1989].

1 INTRODUCTION

The study of Hankel operators played always a central role in system theory, historically mainly due to its role in realization theory. This changed drastically with a shift of emphasis towards H^{∞} control. With the formulation of new problems, highlighted by Zames [1981], a well as the assimilation of the Adamjan, Arov and Krein [1968a,1968b,1971,1978] results into system theory, the use of Hankel operators widened and includes now the optimal control, sensitivity minimization, robust control, balancing and model reduction. In all of these problems the computation of Hankel singular values plays a significant role.

Since our interest is in the structured result conjectured by Zhou and Khargonekar [1987] we will work solely with the Hardy spaces of the left and right half planes. The simplification, due to the greater symmetry, is significant.

The approach presented here is based on the theory of polynomial models developed by the author. The particular problem studied in this paper, i.e. that of computing the singular values of a special class of Hankel operators, is in the intersection of algebra and analysis. This is an extremely rich area and algebra provides both powerful methods and illuminating insights. The results presented here are but one instance of how polynomial algebra can be used in the study of Hankel operators. Much more comprehensive studies in the same area are Fuhrmann [1990] and Fuhrmann and Ober [1991].

The properties of Hankel operators are completely determined by its symbol. If the symbol is, as we postulate, of the form ϕ = m^{∗}w with m, w ∈ H_{+}^{∞} with m inner and w rational then all such properties, including the computation of singular values and singular vectors, should in principle be directly computable from m and w. Now w determines a unique, up to isomorphism, minimal realization. Thus one should be able to compute all the required information from the inner function m and a minimal realization (A, B, C) of w. That this is possible and how to do it was conjectured and proved in a special case in Zhou and Khargonekar [1987]. Proofs of the conjecture followed by Foias, Tannenbaum and Zames [1988], Lypchuk, Smith and Tannenbaum [1988], Smith [1989], and Ozbay [1990]. The method presented here is of use mostly in the case that the rational function w is of low McMillan degree.

Of course one is interested in Hankel singular values and singular vectors mostly for reasons of Norm determination and optimal approximation by finite rank operators. In this connection the work of Adamjan, Arov and Krein [1971] is pioneering and still highly relevant.

2 HANKEL OPERATORS

We will study Hankel operators defined on half plane Hardy spaces rather than on those of the unit disc as was done by Adamjan, Arov and Krein [1971]. In this we follow the choice of Glover [1984]. This choice seems to be a very convenient ones as all results on duality simplify significantly, due to the greater symmetry between the two half planes in comparison to the unit disc and its exterior.

The setting is the Hardy spaces. Thus H_{+}^{2} is the Hilbert space of all analytic functions in the open right half plane with

||f||^{2} = sup_{x>0} \frac{1}{π} ∫_{−∞}^{∞} |f(x + iy)|^{2}dy.

The space H_{−}^{2} is similarly defined in the open left half plane. It is a theorem of Fatou that guarrantees the existence of boundary values of H_{±}^{2}-functions on the imaginary axis. Thus H_{±}^{2} can be considered as closed subspaces of L^{2}(iR), the space of Lebesgue square integrable functions on the imaginary axis. It follows from the Fourier-Plancherel and Paley-Wiener theorems that

L^{2}(iR) = H_{+}^{2} ⊕ H_{−}^{2},

with H_{+}^{2} and H_{−}^{2} the Fourier-Plancherel transforms of L^{2}(0, ∞) and L^{2}(−∞, 0) respectively. Also H_{+}^{∞} and H_{−}^{∞} will denote the spaces of bounded analytic functions on the open right and left half planes respectively. These spaces can be considered as subspaces of L^{∞}(iR), the space of Lebesgue measurable and essentially bounded functions on the imaginary axis. An extensive discussion of these spaces can be found in Hoffman [1962], Duren [1970] and Garnett [1981]. We will use the following conjugation f^{∗}(s) = f(−\overline{s})^{∗}.

We proceed to define Hankel operators and we do this directly in the frequency domain. Readers interested in the time domain definition and the details of the transformation into frequency domain are refered to Fuhrmann [1981], Glover [1984].

In the algebraic theory of Hankel operators the kernel and image of a Hankel operator are directly related to the coprime factorization of the symbol over the ring of polynomials. The details can be found for example in Fuhrmann [1983]. In the same way the kernel and image of a large class of Hankel operators are related to a coprime facorization over H^{∞}. This theme, originating in the work of Douglas, Shapiro and Shields [1971] and that of D. N. Clark, see Helton [1974], is developed extensively in Fuhrmann [1981]. Of course if the symbol of the Hankel operator is rational and in H_{−}^{∞} these two coprime factorizations are easily related.

We proceed to define Hankel operators.

Definition 2.1 Given a function ϕ ∈ L^{∞}(iR) the Hankel operator H_{ϕ} : H_{+}^{2} ⟶ H_{−}^{2} is defined by

H_{ϕ}f = P_{−}(ϕf), for f ∈ H_{+}^{2}.  (1)

The adjoint operator (H_{ϕ})^{∗} : H_{−}^{2} ⟶ H_{+}^{2} is given by

(H_{ϕ})^{∗}f = P_{+}(ϕ^{∗}f), for f ∈ H_{−}^{2}.  (2)

Here P_{−}, P_{+} are the orthogonal projections of L^{2}(iR) onto H_{−}^{2} and H_{+}^{2} respectively.

Of course the way we defined the Hankel operator was highly arbitrary. We can define also Ĥ_{ϕ} : H_{−}^{2} ⟶ H_{+}^{2} by

Ĥ_{ϕ}f = P_{+}ϕf, for f ∈ H_{−}^{2}.  (3)

The operator Ĥ_{ϕ} is sometimes referred to as the involuted Hankel operator. Notice that Ĥ_{ϕ} and H_{ϕ}^{∗} are different operators. Still, in the situation we study, the involuted Hankel operator provides useful information for the study of the Hankel operator itself.

The next theorem summarizes the basic facts concerning Hankel operators. Detailed information can be found in Fuhrmann [1981] and Nikolskii [1985].

Theorem 2.1 1. For every ψ ∈ H_{+}^{∞} the Hankel operator H_{ϕ} satisfies the functional

equation

P_{−}ψH_{ϕ}f = H_{ϕ}ψf, f ∈ H_{+}^{2}.  (4)

2. KerH_{ϕ} is an invariant subspace, i.e. for f ∈ KerH_{ϕ} and ψ ∈ H_{+}^{∞} we have ψf ∈ KerH_{ϕ} .

3. KerH_{ϕ} = mH_{+}^{2} with m inner if and only if ϕ = m^{∗}ψ for some ψ ∈ H_{+}^{∞} and m, ψ have no common inner factor.

Although we will not go into the details, it is of interest to point out the relation between certain Hankel operators and copressions of shifts.

Let m be inner. Then mH^{2} is a right invariant subspace of H_{+}^{2}, i.e. it is invariant under multiplication by all H_{+}^{∞} functions. We will put H(m) = H^{2} ⊖ mH^{2} = {mH^{2}}^{⊥}.

Theorem 2.2 Let m be inner in H_{+}^{∞} and ψ ∈ H_{+}^{∞}. Assume ψ and m have no common inner factor. Then, T_{ψ} : H(m) ⟶ H(m) is defined by

T_{ψ} = P_{H(m)}ψ|H(m).

1. For every ψ ∈ H_{+}^{∞} the Hankel operator H_{m^{∗}ψ} satisfies

T_{ψ}P_{H(m)} = mH_{m^{∗}ψ}.

||T_{ψ}|| = ||H_{m^{∗}ψ}|| = inf{||ψ − mq||_{∞} | q ∈ H^{∞}}.

3. The nontrivial singular values of T_{ψ} and H_{m^{∗}w} are equal.

3 SINGULAR VALUES OF HANKEL OPERATORS

We consider the class of Hankel operators with symbol given by ψ = m^{∗}w, where m ∈ H_{+}^{∞} is inner and w ∈ H_{+}^{∞} is rational. Since w is rational it has a unique representation of the

form w = \frac{b}{a}, with b, a coprime and a monic, stable of degree n. Clearly the coprimeness assumption is no restriction on the problem. We do however make the following assumption

Assumption 3.1 If ζ is a pole of w then −ζ is not a zero of w.

Assumtion 3.1 means that a ∧ b^{∗} = 1. The two coprimeness conditions are equivalent to b ∧ aa^{∗} = 1.

Given a bounded operatot T : H_{1} ⟶ H_{2} between two Hilbert spaces, then a vector f ∈ H_{1} is a singular vector associated with the singular value σ if

T^{∗}T f = σ^{2}f

Given a singular vector f then for g = \frac{1}{σ} T f we have T^{∗}g = \frac{1}{σ} T^{∗}T f = σf. Hence

T T^{∗}g = σT f = σ^{2}g.

Thus g so defined is a singular vector of T^{∗}. Moreover if

\{ \begin{matrix}T f & = & σg \\ T^{∗}g & = & σf \end{matrix}  (5)

then necessarily ||g|| = ||f||, for

||g||^{2} = (g, g) = (g, \frac{1}{σ} T f) == (\frac{1}{σ} T^{∗}g, f) = (f, f) = ||f||^{2}.

A pair (f, g) such that (5) holds is called a Schmidt pair of T, associated with the singular value σ. In much the same way as the singular vector determines a Schmidt pair so does a Schmidt pair deterrmine the singular vectors. Indeed if f, g satisfy (5) then

T^{∗}T f = T^{∗}σg = σT^{∗}g = σ^{2}f

Thus the determination of all Schmidt pairs is equivalent to the determination of all singular vectors.

The next lemma determines the kernel and image of the involuted Hankel operator. This will be of use in the sequel.

Lemma 3.1 Let ϕ = m^{∗}w with m, w ∈ H_{+}^{∞}, m inner and w rational. Let w = \frac{b}{a} with a, b coprime. Then

1. KerĤ_{ϕ} = \frac{a}{a^{∗}} H_{−}^{2}.

2. ImĤ_{ϕ} = X^{a} = \{\frac{r}{a} | deg r < deg a\}

\underline{Proof:}

1. Let g ∈ KerĤ_{ϕ} ⊂ H_{−}^{2}. Then P_{+}m^{∗} \frac{b}{a}g = 0, i.e. m^{∗} \frac{b}{a}g ∈ H_{−}^{2}, or \frac{b}{a}g = mh, for some

h ∈ H_{−}^{2}. This implies that g has zeroes coinciding with those of a. This means that

g = \frac{a}{a^{∗}} g′ for some g′ ∈ H_{−}^{2}.

Conversely if g ∈ \frac{a}{a^{∗}} H_{−}^{2} then g = \frac{a}{a^{∗}} g′ and

Ĥ_{ϕ}g = P_{+}m^{∗} \frac{b}{a} \frac{a}{a^{∗}} g′ = P_{+}m^{∗} \frac{b}{a^{∗}} g′ = 0

as m^{∗}, \frac{b}{a^{∗}} ∈ H_{+}^{∞}, and g′ ∈ H_{+}^{2},

2. From 1 it follows that

\{KerĤ_{ϕ}\}^{⊥} = H_{−}^{2} ⊖ \frac{a}{a^{∗}} H_{−}^{2} = X^{a^{∗}}

and hence ImĤ_{ϕ} = X^{a}.

In a way this simple lemma is the key to all that follows. The spaces ImĤ_{ϕ} and {KerĤ_{ϕ}}^{⊥}, although derived from analytic considerations, have been identified as well studied algebraic objects. Thus the road to using algebraic tools is opened.

We proceed now to the computation of singular values of H_{m^{∗}w}. Since in general this Hankel operator is infinite dimensional it has more than just pure point spectrum. We will denote by ρ_{∞} the essential spectral radius of H_{m^{∗}w}, but we will assume that ρ_{∞} < ||H_{m^{∗}w}||. Thus the norm is achieved at the largest singular value.

Lemma 3.2 (Smith) Let ρ > ρ_{∞}. Then ρ is a singular value of H_{m^{∗}w} if and only if there exist polynomials p, q, of degree ≤ n − 1 and not both zero, such that

f = \frac{mb^{∗}p + ρaq^{∗}}{ρ^{2}aa^{∗} − bb^{∗}} ∈ H_{+}^{2}  (6)

g = \frac{mb^{∗}q + ρap^{∗}}{ρ^{2}aa^{∗} − bb^{∗}} ∈ H_{+}^{2}  (7)

\underline{Proof:} Now ρ is a singular value of H_{m^{∗}w} if and only if there exist two H_{+}^{2} functions f and g for which:

\{ \begin{matrix}H_{m^{∗}w}f & = & ρg^{∗} \\ H_{m^{∗}w}^{∗}g^{∗} & = & ρf \end{matrix}  (8)

This in turn is equivalent to the existence of two H_{+}^{2} functions u and v for which:

m^{∗}wf + u = ρg^{∗}  (9)
mw^{∗}g^{∗} + v^{∗} = ρf  (10)

Solving for f we have

ρf = v^{∗} + mw^{∗}(\frac{m^{∗}wf + u}{ρ})

and hence

ρ^{2}f = ρv^{∗} + ww^{∗}f + mw^{∗}u

i.e.

f = \frac{mw^{∗}u + ρv^{∗}}{ρ^{2} − ww^{∗}}  (11)

Similarly for g we get

g^{∗} = \frac{m^{∗}wv^{∗} + ρu}{ρ^{2} − ww^{∗}}  (12)

Thus ρ is a singular value if and only if there exist u, v ∈ H_{+}^{2} for which

f = \frac{mw^{∗}u + ρv^{∗}}{ρ^{2} − ww^{∗}} ∈ H_{+}^{2}  (13)

and

g^{∗} = \frac{m^{∗}wv^{∗} + ρu}{ρ^{2} − ww^{∗}} ∈ H_{−}^{2}  (14)

Applying the projection P_{+} to equation (9) yields

u = −P_{+}m^{∗}wf

Now w = \frac{b}{a} and so u ∈ ImĤ_{m^{∗}w} = X^{a}. Similarly from equation (10)

v^{∗} = −P_{−}mw^{∗}g^{∗} ∈ ImĤ_{m^{∗}w}^{∗} = X^{a^{∗}}

Thus there exist polynomials p, q of degree < n = deg a such that u = \frac{p}{a} and v^{∗} = \frac{q^{∗}}{a^{∗}}. Substituting back into equations (13) and (14) respectively we get

f = \frac{m\frac{b^{∗}}{a^{∗}} \frac{p}{a} + ρ\frac{q^{∗}}{a^{∗}}}{ρ^{2} − \frac{bb^{∗}}{aa^{∗}}} ∈ H_{+}^{2}  (15)

g^{∗} = \frac{m^{∗} \frac{b}{a} \frac{q^{∗}}{a^{∗}} + ρ\frac{p}{a}}{ρ^{2} − \frac{bb^{∗}}{aa^{∗}}} ∈ H_{−}^{2}  (16)

or equivalently

f = \frac{mb^{∗}p + ρq^{∗}a}{ρ^{2}aa^{∗} − bb^{∗}} ∈ H_{+}^{2}  (17)

and

g^{∗} = \frac{m^{∗}bq^{∗} + ρpa^{∗}}{ρ^{2}aa^{∗} − bb^{∗}} ∈ H_{−}^{2}  (18)

Of course the last condition is equivalent to

g = \frac{mb^{∗}q + ρap^{∗}}{ρ^{2}aa^{∗} − bb^{∗}} ∈ H_{+}^{2}  (19)

We note that if there exist polynomials p and q of degree < n such that (17) and (18) hold then one easily checks that f and g so defined are singular vectors corresponding to the singular value ρ.

The conditions given by equations (6) and (7) can be characterized in terms of the zeroes of the polynomial Ω_{ρ} defined next. To clarify notation we will say, given an analytic function f and a polynomial Ω, that f is divisible by Ω, and write f = 0 mod(Ω), if f is analytic at all zeroes of Ω and the order of its zeroes is not less than the order of the zeroes of Ω.

Theorem 3.1 (Smith) Let ρ > ρ_{∞}. Let

Ω_{ρ}(z) = ρ^{2}a(z)a^{∗}(z) − b(z)b^{∗}(z).

Assume m has no zeroes coinciding with the zeroes of Ω_{ρ}. Then ρ is a singular value of H_{m^{∗}w} if and only if there exist polynomials p, q, of degree ≤ n − 1 and not both zero, such that

mb^{∗}p + ρaq^{∗} = 0 mod(Ω_{ρ})  (20)

and

mb^{∗}q + ρap^{∗} = 0 mod(Ω_{ρ})  (21)

If the latter condition is satisfied the corresponding singular vectors are given by (6) and (7).

\underline{Proof:} Put Ω_{ρ}(z) = ρ^{2}a(z)a^{∗}(z) − b(z)b^{∗}(z). Let ζ be an arbitrary right half plane zero of Ω_{ρ}. Then if u = \frac{p}{a} and v = \frac{q}{a} are related by the singular value/singular vector equation

(17) and (18) , then clearly f, g^{∗} ∈ H_{+}^{2} implies that the polynomial mb^{∗}p + ρaq^{∗} vanish at each zero ζ of Ω_{ρ} in the closed right half plane at least to the order of ζ as a zero of Ω_{ρ}.

Conversely, if mb^{∗}p + ρaq^{∗} vanish at each right half plane zero ζ at least to the order

of ζ as a zero of Ωρ then f = \frac{m\frac{b^{∗}}{a^{∗}} \frac{p}{a} + ρ\frac{q^{∗}}{a^{∗}}}{ρ^{2} − \frac{bb^{∗}}{aa^{∗}}} ∈ H_{+}^{2} and hence ρ is a singular value.

However the theorem claims more. Assume there exist p, q such that (20) and (21) hold. Then f and g defined by (17) and (19) respectively are in {KerH_{m^{∗}w}}^{⊥} = {mH_{+}^{2}}^{⊥} = H(m). Now m has a meromorphic continuation into the left half plane, unless the imaginary axis is its natural boundary in which case one needs to consider pseudo-meromorphic continuations, see Douglas, Shapiro and Shields [1971] or Fuhrmann [1981]. By our as-

sumtion m is analytic at all zeroes of Ω_{ρ}. Also g^{∗} = \frac{m^{∗}bq^{∗} + ρpa^{∗}}{Ω_{ρ}} is in ∈ H_{−}^{2}. Therefore

\frac{bq^{∗} + ρma^{∗}p}{Ω_{ρ}} is analytic at all zeroes of Ω_{ρ}. Now ρ\frac{a}{b} is, by the coprimeness of b and Ω_{ρ},

also analytic at all left half plane zeroes of Ω_{ρ}. Therefore ρ\frac{a}{b} \{\frac{bq^{∗} + ρma^{∗}p}{Ω_{ρ}}\} is analytic

at all left half plane zeroes of Ω_{ρ}. However this function can be rewritten as

\frac{ρaq^{∗} + ρ^{2}m\frac{aa^{∗}}{b} p}{Ω_{ρ}} = \frac{ρaq^{∗} + mb^{∗}p}{Ω_{ρ}} + \frac{mp}{b}

The right term is clearly analytic at zeroes of Ω_{ρ} so we have \frac{mb^{∗}p + ρaq^{∗}}{Ω_{ρ}} analytic at all

left half plane zeroes of Ω_{ρ}.

The sufficiency of the condition is obvious, in view of Lemma 3.2.

The interest of this result is in the fact that the computation of the singular values of the Hankel operator has been reduced to a problem in functional arithmetic modulo the polynomial Ω_{ρ}. Our aim is a direct interpretation of equations (20) and (21) in state space terms. To this end we make, in the next section, a digression into polynomial model theory.

4 POLYNOMIAL MODELS

In this section we summarize the main results concerning polynomial models. Although we will use the results solely in the scalar case, we will state them, for extra generality, in their multivariable form.

Our starting point is this basic result about free modules. A proof can be found in Van der Waerden [1949].

Theorem 4.1 Let R be a principal ideal domain and M a free left R−module with n basis elements. Then every R−submodule N of M is free and has at most n basis elements.

If V is a finite dimensional vector space over a field F then V [z], the space of vector polynomials is a free finitely generated module over the polynomial ring F [z]. Throughout we will assume a basis has been chosen and thus V will be identified with F^{n} and similarly V [z] with F^{n}[z]. Also we will identify F^{n}[z] and (F [z])^{n} and speak of its elements as polynomial vectors. Similarly elements of F^{m×n}[z] will be referred to as polynomial matrices. Because of the nature of the factorization results we are interested in, and for the consistency of notation, we will identify the field F with the real field R, noting that some of the results hold in greater generality.

The next theorem is the basic representation theorem for submnodules.

Theorem 4.2 A subset M of R^{n}[z] is a submodule of R^{n}[z] if and only if M = DR^{n}[z] for some D in R^{n×n}[z].

The following is the basic theorem that relates submodule inclusion to factorization.

Theorem 4.3 Let M = DR^{n}[z] and N = ER^{n}[z] then M ⊂ N if and only if D = EG for some G in R^{n×n}[z].

Let π_{+} and π_{−} denote the projections of R^{m}((z^{−1})) the space of truncated Laurent series on R^{m}[z] and z^{−1}R[[z^{−1}]], the space of formal power series vanishing at infinity, respectively. Since

R^{m}((z^{−1})) = R^{m}[z] ⊕ z^{−1}R^{m}[[z^{−1}]]  (22)

π_{+} and π_{−} are complementary projections.

Given a nonsingular polynomial matrix D in R^{m×m}[z] we define two projections π_{D} in R^{m}[z] and π^{D} in z^{−1}R^{m}[[z^{−1}]] by

π_{D}f = Dπ_{−}D^{−1}f for f ∈ R^{m}[z]  (23)
π^{D}f = π_{−}D^{−1}π_{+}Dh for h ∈ z^{−1}R^{m}[[z^{−1}]]  (24)

and define two linear subspaces of R^{m}[z] and z^{−1}R^{m}[[z^{−1}]] by

X_{D} = Imπ_{D}.  (25)

and

X^{D} = Imπ^{D}.  (26)

An element f of R^{m}[z] belongs to X_{D} if and only if π_{+}D^{−1}f = 0, i.e. if and only if D^{−1}f is a strictly proper rational vector function.

We turn X_{D} into an R[z]−module by defining

p · f = π_{D}pf for p ∈ R[z], f ∈ X_{D}.  (27)

Since Kerπ_{D} = DR^{m}[z] the following isomorphism result holds.

Theorem 4.4 With the previously defined module structure X_{D} is isomorphic to R^{m}[z]/DR^{m}[z].

Similarly, we turn X^{D} into an R[z]−module by defining

p · h = π_{−}ph for p ∈ R[z], h ∈ X^{D}.  (28)

In X_{D} we will focus on a special map S_{D}, a generalization of the classical companion matrix, which corresponds to the action of the identity polynomial z, i.e.,

S_{D}f = π_{D}zf for f ∈ D.

Thus the module structure in X_{D} is identical to the module structure induced by S_{D} through p · f = p(S_{D})f. With this definition the study of S_{D} is identical to the study of the module structure of X_{D}. In particular the invariant subspaces of S_{D} are just the submodules of X_{D} which we proceed to investigate.

The interpretation of factorization of polynomial matrices on the level of polynomial models is described next.

Theorem 4.5 A subset M of X_{D} is a submodule, or equivalently an S_{D} invariant subspace, if and only if M = D_{1}X_{D_{2}} for some factorization D = D_{1}D_{2} with D_{i} ∈ R^{n×n}[z].

We summarize now the connection between the geometry of invariant subspaces and the arithmetic of polynomial matrices.

Theorem 4.6 Let M_{i}, i = 1, …, s be submodules of X_{D}, having the representations M_{i} = E_{i}X_{F_{i}}, that correspond to the factorizations

D = E_{i}F_{i}.

Then the following statements are true.

(i) M_{1} ⊂ M_{2} if and only if E_{1} = E_{2}R, i.e. if and only if E_{2} is a left factor of E_{1}.

(ii) ∩_{i=1}^{s}M_{i} has the representation E_{ν}X_{F_{ν}} with E_{ν} the l.c.r.m. of the E_{i} and F_{ν} the g.c.r.d. of the F_{i}.

(iii) M_{1} + ⋯ + M_{s} has the representation E_{μ}X_{F_{μ}} with E_{μ} the g.c.l.d. of the E_{i} and F_{μ} the l.c.l.m. of all the F_{i}.

Corollary 4.1 Let D = E_{i}F_{i}, for i = 1, …, s. Then

(i) We have

X_{D} = E_{1}X_{F_{1}} + ⋯ + E_{s}X_{F_{s}}

if and only if the E_{i} are left coprime.

(ii) We have ∩_{i=1}^{s}E_{i}X_{F_{i}} = 0 if and only if the F_{i} are right coprime.

(iii) The decomposition

X_{D} = E_{1}X_{F_{1}} ⊕ ⋯ ⊕ E_{s}X_{F_{s}}

is a direct sum if and only if D = E_{i}F_{i} for all i, the E_{i} are left coprime and the F_{i} are right coprime.

The next result summarizes the relation between factorization and the spectral decomposition of linear maps.

Theorem 4.7 Let D(z) ∈ R^{n×n}[z] be nonsingular and let d(z) = detD(z) be its characteristic polynomial. Suppose d has a factorization d = e_{1} ⋯ e_{s} with the e_{i} pairwise coprime. Then D admits factorizations

D = D_{i}E_{i}

with detD_{i} = d_{i}, detE_{i} = e_{i} and such that

X_{D} = D_{1}X_{E_{1}} ⊕ ⋯ ⊕ D_{s}X_{E_{s}}  (29)

Moreover

det(S_{D}|D_{i}X_{E_{i}}) = e_{i}.  (30)

Denoting by T̃ the transpose of the matrix T we define, for an element A(z) = ∑_{j=−∞}^{∞} A_{j}z^{j} of R^{m×m}((z^{−1})), Ã by

Ã(z) = ∑_{j=−∞}^{∞} Ã_{j}z^{j}.  (31)

In R^{m}((z^{−1})) × R^{m}((z^{−1})) we define a symmetric bilinear form [f, g] by

[f, g] = ∑_{j=−∞}^{∞} f̃_{j}g_{−j−1}  (32)

where f(z) = ∑_{j=−∞}^{∞} f_{j}z^{j} and g(z) = ∑_{j=−∞}^{∞} g_{j}z^{j}. It is clear that, as both f and g are truncated Laurent series, that the sum in (32) is well defined, containing only a finite number of nonzero terms. We denote by T^{∗} the adjoint of a map T relative to the bilinear form of (32), i.e. T^{∗} is the unique map that satisfies

[T f, g] = [f, T^{∗}g]  (33)

for all f, g ∈ R^{m}((z^{−1})). We use this global bilinear form to obtain a concrete representation of X_{D}^{∗} the dual space of X_{D}.

Theorem 4.8 X_{D}^{∗} The dual space of X_{D} can be identified with XD̃ under the pairing

< f, g >= [D^{−1}f, g]  (34)

for f ∈ X_{D} and g ∈ XD̃. Moreover the module structures of X_{D} and XD̃ are related through

S_{D}^{∗} = SD̃  (35)

The following result shows how the characterization of submodules of X_{D} and their relation to factorizations is reflected by duality.

Theorem 4.9 Let M ⊂ X_{D} be a submodule, represented as M = EX_{G} for some factorization D = EG into nonsingular factors. Then M^{⊥} is a submodule of XD̃ and is given by M^{⊥} = G̃XẼ.

The next two theorems are the central tools in polynomial model theory. They provide the characterization of all module homomorphisms and the condition for their invertibility. For the relation to the commutant lifting theorem and the spectral mapping theorem see Fuhrmann [1981] and Nikolskii [1985].

Theorem 4.10 Let Z : X_{D} ⟶ X_{D_{1}} be the module homomorphism defined by

Zf = π_{D_{1}}Nf

with

ND = D_{1}M

holding. Then

(i) KerZ = EX_{G}, where D = EG and G is a g.c.r.d. of D and N.

(ii) ImZ = E_{1}X_{G_{1}}, where D_{1} = E_{1}G_{1} and E_{1} is a g.c.l.d. of D_{1} and M.

(iii) Z is surjective if and only if N and D_{1} are left coprime.

(iv) Z is injective if and only if D and M are right coprime.

Recall that a vector space V is symplectic if it is equipped with a nondegenerate, alternating bilinear form. A linear map H in a symplectic space V is Hamiltonian if H^{∗} = −H relative to this form. A map R in V is a symplectic if it is invertible and leaves the form invariant.

The canonical example of a symplectic space is R^{2n} with the bilinear form induced by

J = ( \begin{matrix}0 & I \\ −I & 0 \end{matrix} ) .  (36)

A Hamiltonian map H in this case is given by

H = ( \begin{matrix}A & P \\ Q & −Ã \end{matrix} )  (37)

with P, Q symmetric. R is symplectic if and only if R̃JR = J.

Contrary to inner product spaces, there are self orthogonal elements and subspaces in symplectic spaces. A subspace L of a symplectic space V is called Lagrangian if it is a maximal self orthogonal subspace.

We modify now the previous approach to duality to accomodate the study of symplectic spaces. To this end we introduce now a global alternating form on R^{m}((z^{−1}))×R^{m}((z^{−1})) in the following way. We define now a new bilinear form on R^{m}((z^{−1})) × R^{m}((z^{−1})) by

{f, g} = [τf, g] for f, g ∈ R^{m}((z^{−1}))  (38)

where τ : R^{m}((z^{−1})) → R^{m}((z^{−1})) is defined by

(τf)(z) = f(−z).  (39)

Lemma 4.1 The bilinear form defined by (38) on R^{m}((z^{−1})) × R^{m}((z^{−1})) is alternating, i.e.

{f, g} = −{g, f}.  (40)

Given a rational matrix function Φ we define its Hamiltonian or parahermitian conjugate, Φ^{∗} by

Φ^{∗}(z) = Φ(−\overline{z})^{∗}.  (41)

This implications of this symmetry were originally studied by Brockett and Rahimi [1972]. An extensive study of Hamiltonian symmetry and its associated realization theory can be found in Fuhrmann [1984].

Given a map Z : R^{m}((z^{−1})) → R^{m}((z^{−1})) we will denote by Z^{∇} the map, assuming it exists, which satisfies

{Zf, g} = {f, Z^{∇}g}  (42)

for all f, g ∈ R^{m}((z^{−1})). Since the forms [ , ], and hence also { , }, are nondegenerate the map Z^{∇}, if it exists, is unique. The map Z^{∇} will be called the Hamiltonian adjoint of Z.

Assume now that we have two nondegenerate bilinear forms on V × V^{∗} and V^{∗} × V which satisfy << x, y >>= − << y, x >> for all x ∈ V and y ∈ V^{∗}. Note that while the two forms are distinct we do not distinguish between them, as it is always clear from the context which form we use. We say x is orthogonal to y if << x, y >>= 0. Given a subset M ⊂ V we define M^{⊥} as usual by

M^{⊥} = {y ∈ V^{∗} | << x, y >>= 0 for all x ∈ M}.  (43)

Since full submodules of R^{m}[z] are of the form DR^{m}[z] with D a nonsingular polynomial matrix it is of interest to characterize its orthogonal complement. The result should be compared with Theorem 4.9.

We can use these isomorphisms to define a pairing between X_{D} and X_{(D^{∗})} by

Theorem 4.11 X_{D}^{∇} The dual space of X_{D} can be identified with X_{(D^{∗})} under the pairing

<< f, g >>= {D^{−1}f, g} = [JD^{−1}f, g]  (44)

for all f ∈ X_{D} and g ∈ X_{(D^{∗})}. Moreover the module structures of X_{D} and X_{(D^{∗})} are related through

(S_{D})^{∗} = −S_{(D^{∗})}.  (45)

We note at this point that if D is a nonsingular Hamiltonian symmetric polynomial matrix, i.e. if D^{∗} = D, then X_{D} with the metric of (44) is a symplectic space. Much use of this will be made later.

The pairing of elements of X_{D} and X_{D^{∗}} given by (44) allows us to compute, for a subset V of X_{D}, the annihilator V^{⊥}, i.e., the set of all g ∈ X_{D^{∗}} such that << f, g >>= 0 for all f ∈ V. Since (S_{D})^{∗} = −S_{D^{∗}}, it follows that if V ⊂ X_{D} is a submodule, then so is V^{⊥} in X_{D^{∗}}. However, we know that submodules are related to factorizations of D into nonsingular factors. The annihilator of V can be concretely identified.

Theorem 4.12 Let V ⊂ X_{D} be a submodule. Then V^{⊥} is a submodule of X_{(D^{∗})}. Moreover if V = EX_{F} where D = EF is a factorization of D into nonsingular factors, then

V^{⊥} = F^{∗}X_{E^{∗}}.  (46)

If we assume D̃(−z) = D(z), i.e. D^{∗} = D, then X_{D} is a symplectic space and S_{D} a Hamiltonian map. For details see Fuhrmann [1984]. One way to obtain Hamiltonian symmetric polynomial matrices is to study those which have a factorization of the form D(z) = Ẽ(−z)E(z) or D = E^{∗}E.

Theorem 4.13 Let D = E^{∗}E. Then E^{∗}X_{E} is a Lagrangian S_{D}-invariant subspace of X_{D}.

5 REALIZATION THEORY

As usual, given a proper rational matrix G we will say a system (A, B, C, D) is a realization of G if

G = D + C(zI − A)^{−1}B.

We will use the notation G = [A, B, C, D]. We will be interested in realizations associated with rational functions having the following representations

G = V T^{−1}U + W.  (47)

Our approach to the analysis of these systems is to associate with each representation of the form (47), a state space realization in the following way. We choose X_{T} as the state space and define the triple (A, B, C), with A : X_{T} ⟶ X_{T}, B : R^{m} ⟶ X_{T}, and C : X_{T} ⟶ R^{p} by

\{ \begin{matrix}A = S_{T} \\ Bξ = π_{T} Uξ, \\ Cf = (V T^{−1}f)_{−1} \\ D = G(∞). \end{matrix}  (48)

We call this the associated realization to the polynomial matrix P,

P = ( \begin{matrix}T & U \\ −V & W \end{matrix} ) .

Theorem 5.1 The system given by (48) is a realization of G = V T^{−1}U + W. This realization is reachable if and only if T and U are left coprime and observable if and only if T and V are right coprime.

Theorem 5.2 Let G be a normalized bicausal isomorphismi, and let G = T^{−1}D be a left coprime factorization. Then

1. The polynomial models X_{D} and X_{T} contain the same elements.

2. Let B be a common basis of the two polynomial models. Let (A_{T} , B_{T} , C_{T} , I) and (A_{D}, B_{D}, C_{D}, I) be the matrix representation of the shift realizations of G and G^{−1} respectively in the given basis. Then

\begin{aligned}A_{D} & = & A_{T} − B_{T} C_{T} \\ B_{D} & = & −B_{T} \\ C_{D} & = & C_{T} \end{aligned}  (49)

Let G_{1} = [A_{1}, B_{1}, C_{1}, D_{1}] and G_{2} = [A_{2}, B_{2}, C_{2}, D_{2}] be two transfer functions realized in the state spaces X_{1} and X_{2} respectively. If the number of inputs of the second system equals the number of outputs of the first we can feed those outputs to the second system. This gives rise to the series coupling and the corresponding transfer function is

G_{2}G_{1} = [( \begin{matrix}A_{1} & 0 \\ B_{2}C_{1} & A_{2} \end{matrix} ) , ( \begin{matrix}B_{1} \\ B_{2}D_{1} \end{matrix} ) , ( D_{2}C_{1} C_{2} ) , D_{2}D_{1}] .

We will use also the notation G_{2}G_{1} = [A_{2}, B_{2}, C_{2}, D_{2}] × [A_{1}, B_{1}, C_{1}, D_{1}].

Definition 5.1 We say that (A, B, C, D) acting in R^{2n} is a Hamiltonian system if, with J defined by (36), we have

\begin{aligned}ÃJ & = & −JA \\ JB & = & C̃ \\ D & = & D̃ \end{aligned}  (50)

The main result concerning Hamiltonian realizations is the following.

Theorem 5.3 (i) A transfer function G has a Hamiltonian realization if and only if it is Hamiltonian symmetric, i.e. if G(−\overline{z})^{∗} = G(z).

(ii) Two minimal Hamiltonian realizations of G are symplectically equivalent.

For an extensive study of Hamiltonian symmetry see Fuhrmann [1984].

6 BASES AND THE BEZOUTIAN

In the previous section all development has been basis free. Once a choice of basis is made for a polynomial model then all operators, and so realizations, have associated matrix representations. Different choices of basis highlight different properties of operators and systems. We will not go into all such possibilities and will restrict ourselves to the few instances needed in the sequel. Some more complex results in this direction, and relevant to the topic of this paper can be found in Fuhrmann [1990] and Fuhrmann and Ober [1991].

X finite dimensional vector space over F and let X^{∗} be its dual space. {e_{1}, …, e_{n}} a basis for X then {f_{1}, …, f_{n}} in X^{∗} is the dual basis if

< e_{i}, f_{j} >= δ_{ij} 1 ≤ i, j ≤ n.

We will investigate next the implications of duality in the context of polynomial models. In particular we will focus on an important pair of dual bases. Let X_{q} be the polynomial model associated with the polynomial q(z) = z^{n} + q_{n−1}z^{n−1} + ⋯ + q_{0}.

We consider the following very natural pair of bases in X_{q}. The first one we will refer to as the standard basis B_{st} = {1, z, …, z^{n−1}} whereas the second, the control basis is given by B_{co} = {e_{1}, …, e_{n}} where

e_{i}(z) = π_{+}z^{−i}q = q_{i} + q_{i+1}z + ⋯ + z^{n−i}, i = 1, …, n.

These bases are important in the study of Bezoutians, introduced next.

The Bezoutian is a well studied quadratic form associated with two polynomials. It has many applications in root location, coprimeness characterization, the Cauchy index, characterization of output feedback invariants, etc. A comprehensive exposition and further references can be found in Helmke and Fuhrmann [1989].

Given p(z), q(z) of polynomials with real coefficients, q(z) monic of degree n and p of degree ≤ n, the Bezoutian B(q, p) of q and p is defined

\begin{aligned}B(q, p)(z, w) & = & \frac{q(z)p(w) − p(z)q(w)}{z − w} \\ & = & ∑_{i=1}^{n} ∑_{j=1}^{n} b_{ij}z^{i−1}w^{j−1} \end{aligned}

B = (b_{ij}) is a symmetric matrix.

The following theorem is central for the understanding of Bezoutians inasmuch as it reduces its study to a basis free object, namely a module homomorphism of which the Bezoutian is just a matrix representation. Proofs can be found in Fuhrmann [1981] and Fuhrmann and Helmke [1989].

Theorem 6.1 1. The standard basis and the control basis are a dual pair of bases for

X_{q}, i.e.

B_{co}^{∗} = B_{st}

2. p(S_{q}) is a self adjoint operator in the indefinite metric < , >, i.e.

p(S_{q})^{∗} = p(S_{q})

3. The Bezoutian is the matrix representation of the self adjoint operator p(S_{q}) with respect to the dual pair of bases, the standard and the control, i.e.

B(q, p) = [p(S_{q})]_{co}^{st}.

Since p(S_{q}) is invertible whenever p and q are coprime, it follows that, coprimeness assumed, the columns of B(q, p) are linearly independent, Thus a basis of R^{n}. Of course this basis has a functional representation in the polynomial model X_{q}. This is considered next.

Lemma 6.1 Let p, q ∈ R[z] be coprime, with deg p < deg q. Then the polynomials

p(z)q_{k}(z) − q(z)p_{k}(z) = ∑_{i=1}^{n} b_{ik}z^{i−1}  (51)

constitute a basis for X_{q}. Here

p_{k} = π_{+}z^{−k}p, q_{k} = π_{+}z^{−k}q.

\underline{Proof:} Rewrite the definition of the Bezoutian as

q(z)p(w) − p(z)q(w) = ∑_{i=1}^{n} ∑_{j=1}^{n} b_{ij}z^{i−1}(z − w)w^{j−1}.

Applying the operation π_{+}w^{−k} to both sides and equating w = z we get

\begin{matrix}q(z)p_{k}(w) − p(z)q_{k}(w) \\ = π_{+} ∑_{i=1}^{n} ∑_{j=1}^{n} b_{ij}z^{i−1}w^{−k}(z − w)w^{j−1}|_{w=z} \\ = ∑_{i=1}^{n} z^{i−1} ∑_{j=1}^{n} b_{ij}π_{+}w^{−k}(z − w)w^{j−1}|_{w=z} \end{matrix}  (52)

Now

π_{+}w^{−k}(z − w)w^{j−1}|_{w=z} = \{ \begin{matrix}0 & j ≠ k \\ −1 & j = k \end{matrix}  (53)

and hence the result follows.

Note that if we choose p(z) = 1 then this basis reduces to the control basis of X_{q}.

7 HAMILTONIAN STRUCTURE

In this section we will give our proof of the Zhou-Khargonekar result. Before proceeding to that we will present in the next theorem a summary of relevant results on various Hamiltonian realizations.

Theorem 7.1 Let w = a^{−1}b be a strictly proper, stable rational function, with a and b coprime polynomials. Thus a is stable and we assume it to be monic of degree n.

1. X_{aa^{∗}} is a symplectic space, and the following are direct sum representations for it.

X_{aa^{∗}} = X_{a} ⊕ aX_{a^{∗}}  (54)

and

X_{aa^{∗}} = a^{∗}X_{a} ⊕ aX_{a^{∗}}.  (55)

Whereas the first representation is a direct sum of two subspaces only one of which is invariant, the second representation is as a direct sum of two Lagrangian invariant subspaces of S_{aa^{∗}}.

2. Assume the polynomial b is coprime with both a and a^{∗}. Then the following is a direct sum representation of the polynomial model

X_{aa^{∗}} = b^{∗}X_{a} ⊕ aX_{a^{∗}}.

Since X_{aa^{∗}} and X_{Ω_{ρ}} contain the same elements, this is also a direct sum representation of X_{Ω_{ρ}}, i.e.

X_{Ω_{ρ}} = b^{∗}X_{a} ⊕ aX_{a^{∗}}.

3. Let {e_{1}, …, e_{n}} be the control basis associated with the monic polynomial a. Then the set B = {1, s, …, s^{n−1}, ae_{1}^{∗}, …, ae_{n}^{∗}} is a symplectic basis for X_{aa^{∗}}, i.e.

[I]_{B}^{B^{∗}} = ( \begin{matrix}0 & I \\ −I & 0 \end{matrix} ) = J.

Let f_{i} be defined, as in Lemma 6.1, by f_{i}(z) = b(z)a_{k}(z) − a(z)b_{k}(z). Then B_{1} = {b^{∗}, …, b^{∗}z^{n−1}, af_{1}^{∗}, …, af_{n}^{∗}} is also a basis for X_{aa^{∗}}.

4. Let w = a^{−1}b. Then the shift realization of w has the following matrix representation with respect to the standard basis of X_{a}

A = ( \begin{matrix}0 & . & . & . & −a_{0} \\ 1 & & & & . \\ & . & & & . \\ & & . & & . \\ & & & 1 & −a_{n−1} \end{matrix} ) , B = ( \begin{matrix}b_{1} \\ . \\ . \\ . \\ b_{n} \end{matrix} ) , C = ( 0 . . 0 1 ) ,  (56)

5. w(−z) has the following realization

A = ( \begin{matrix}0 & −1 \\ & & . \\ & & & . \\ & & & & −1 \\ a_{0} & . & . & . & a_{n−1} \end{matrix} ) , B = ( \begin{matrix}0 \\ . \\ . \\ . \\ 1 \end{matrix} ) , C = ( b_{1} … b_{n} ) .  (57)

6. γ(z) = 1 − \frac{w(−z)w(z)}{ρ^{2}} has the following Hamiltonian realization

\begin{matrix}A = ( \begin{matrix}0 & . & . & . & −a_{0} \\ 1 & & & & . \\ & . & & & . \\ & & . & & . \\ & & & 1 & −a_{n−1} \\ & & & & & 0 & −1 \\ & & & & & & & . \\ & & 0 & & & & & & . \\ & & & & & & & & & −1 \\ & & & & 1 & a_{0} & . & . & . & a_{n−1} \end{matrix} ) \\ B = \frac{1}{ρ} ( \begin{matrix}b_{1} \\ . \\ . \\ . \\ b_{n} \\ 0 \\ . \\ . \\ . \\ 0 \end{matrix} ) , C = \frac{1}{ρ} ( 0 … 0 b_{1} … b_{n} ) , D = 1. \end{matrix}  (58)

This is the matrix representation of the shift realization of γ with respect to the basis B = {1, s, …, s^{n−1}, ae_{1}^{∗}, …, ae_{n}^{∗}}. It has the same representation with respect to the basis B_{1} = {b^{∗}, …, b^{∗}z^{n−1}, af_{1}^{∗}, …, af_{n}^{∗}} 7. γ(z)^{−1} = (1 − \frac{w(−z)w(z)}{ρ^{2}})^{−1} has the following minimal realization given by

\begin{matrix}( & \begin{aligned}0 & . & . & . & −a_{0} \\ 1 & & & & . \\ & . & & & . & & & −\frac{1}{ρ^{2}} (b_{i}b_{j}) \\ & & . & & . \\ & & & 1 & −a_{n−1} \\ & & & & & 0 & −1 \\ & & & & & & & . \\ & & 0 & & & & & & . \\ & & & & & & & & & −1 \\ & & & & 1 & a_{0} & . & . & . & a_{n−1} \end{aligned} & ) \\ B = \frac{1}{ρ} ( \begin{matrix}b_{1} \\ . \\ . \\ . \\ b_{n} \\ 0 \\ . \\ . \\ . \\ 0 \end{matrix} ) , C = \frac{1}{ρ} ( 0 … 0 b_{1} … b_{n} ) , D = 1. \end{matrix}  (59)

This is the matrix representation of the shift realization of γ(z)^{−1} in the polynomial model X_{Ω_{ρ}} with respect to the basis {b^{∗}, …, b^{∗}z^{n−1}, af_{1}^{∗}, …, af_{n}^{∗}}.

8. Let (A, B, C) be an arbitrary minimal realization of the rational function w = \frac{b}{a}.

Then the transfer function γ(z) = 1 − \frac{w(−z)w(z)}{ρ^{2}} has a minimal realization given by

[( \begin{matrix}A & 0 \\ \frac{1}{ρ^{2}} C̃C & −Ã \end{matrix} ) , ( \begin{matrix}B \\ 0 \end{matrix} ) , ( 0 −B̃ ) , 1] .  (60)

This realization and the shift realization, in the polynomial model X_{aa^{∗}}, based on the representation γ(z) = 1 − \frac{1}{ρ^{2}} \frac{b^{∗}b}{a^{∗}a} are symplectically equivalent.

9. If π_{1} and π_{2} are the parallel projections on the two direct summands of

X_{aa^{∗}} = X_{a} ⊕ aX_{a^{∗}}

then

π_{1}S_{aa^{∗}} | X_{a} = S_{a}  (61)
π_{2}S_{aa^{∗}} | X_{a} = S_{a^{∗}}  (62)

The same holds true for the direct sum representation

X_{aa^{∗}} = b^{∗}X_{a} ⊕ aX_{a^{∗}}.  (63)

10. γ(z)^{−1} = (1 − \frac{w(−z)w(z)}{ρ^{2}})^{−1} has a minimal realization given by

[( \begin{matrix}A & −\frac{1}{ρ}BB̃ \\ \frac{1}{ρ}C̃C & −Ã \end{matrix} ) , ( B̃ ; 0 ) , ( 0 −B̃ ) , 1] .

It also has a minimal shift realization, in the polynomial model X_{Ω_{ρ}}, based on the

representation γ(z)^{−1} = 1+\frac{ρ^{2}b^{∗}b}{Ω_{ρ}}. The two realizations are symplectically equivalent.

In particular S_{Ω_{ρ}} and F_{ρ} defined by

F_{ρ} = ( \begin{matrix}A & −\frac{1}{ρ}BB̃ \\ \frac{1}{ρ}C̃C & −Ã \end{matrix} )

are similar.

11. The polynomial Ω_{ρ} is, up to a constant factor, the characteritic polynomial of F_{ρ}.

\underline{Proof:}

1. That X_{aa^{∗}} is a symplectic space follows from Theorem 4.13. The direct sum representation (55) follows from Corollary 4.1. (54) can be checked directly.

2. We consider the module homomorphism Z in X_{aa^{∗}} induced by b^{∗}, i.e. b^{∗}(S_{aa^{∗}})

Zf = b^{∗}(S_{aa^{∗}})f = π_{aa^{∗}}b^{∗}f

This is an isomorphism, i.e. b^{∗}(S_{aa^{∗}}) is invertible if and only if b^{∗} ∧ aa^{∗} = 1, and this was assumed.

Next we study how X_{a} and aX_{a^{∗}} transform under this map. Clearly X_{a} ⟶ b^{∗}X_{a} by f ↦ b^{∗}f, whereas, since aX_{a^{∗}} is a submodule,

b^{∗}(S_{aa^{∗}})aX_{a^{∗}} = ab^{∗}(S_{a^{∗}})X_{a^{∗}}.

Also note that a^{∗} ∧ b^{∗} = 1 implies the invertibility of b^{∗}(S_{a^{∗}}). Thus the direct sum representation follows.

X_{aa^{∗}} = b^{∗}X_{a} ⊕ aX_{a^{∗}}

Consider the basis B_{1} which is the image under the map b^{∗}(S_{aa^{∗}}) . As b^{∗}(S_{aa^{∗}}) commutes with S_{aa^{∗}} it follows that S_{aa^{∗}} has the same matrix representation as before.

3. We use the following computational rules

{f^{∗}, g^{∗}} = −{f, g}
<< f, g >>= − << g, f >>

and

<< af, g >>=<< f, a^{∗}g >>

Then

\begin{aligned}<< s^{i}, ae_{j}^{∗} >> & = & \{\frac{s^{i−1}}{aa^{∗}} , ae_{j}^{∗}\} = \{s^{i−1}, \frac{e_{j}^{∗}}{a^{∗}} \} = −\{\frac{e_{j}^{∗}}{a^{∗}} , s^{i−1}\} \\ & = & \{\frac{e_{j}}{a} , (−s)^{i−1}\} = [\frac{e_{j}}{a} , s^{i−1}] =< e_{j}, s^{i−1} >= δ_{ij} \end{aligned}

The basis B_{1} = {b^{∗}, …, b^{∗}z^{n−1}, af_{1}^{∗}, …, af_{n}^{∗}} is the image, under the invertible map b^{∗}(S_{aa^{∗}}), of the basis B.

4. This is a simple computation.

5. Follows from the previous realization by Hamiltonian duality. This is also the matrix representation of the shift realization of w(−z) = b(−z)a(−z) in the basis {e_{1}^{∗}, …, e_{n}^{∗}} of X_{a^{∗}}.

6. This can be seen in several ways. This realization is the series connection of the realizations (56) and (57), with the factor ρ taken into consideration. This is also

the matrix representation of the shift realization of γ(z) = 1 − \frac{b^{∗}}{ρ} (aa^{∗})^{−1} \frac{b}{ρ} in the

basis B = {1, z, …, z^{n−1}, ae_{1}^{∗}, …, ae_{n}^{∗}}

7. Follows from the realization (59) and Theorem 5.2.

8. From w(z) = C(zI − A)^{−1}B follows w(−z) = B̃(zI + Ã)^{−1}C̃. The series connection of the two realizations implies the result.

9. Given two pairs of systems [A_{1}, B_{1}, C_{1}, D_{1}], [A_{2}, B_{2}, C_{2}, D_{2}] as well as [A′_{1}, B′_{1}, C′_{1}, D′_{1}], [A′_{2}, B′_{2}, C′_{2}, D′_{2}] which are isomorphic in pairs, then the series connection of the first pair is isomorphic to the series connection of the later pair. We also use the fact that the series connection of the product of two shift realizations, under coprimeness assumptions, is isomorphic to the shift realization of their product. The proof given in Fuhrmann [1976] can be adapted to cover this case.

10. Recall that, for f ∈ X_{a} we have S_{a}f = zf − aξ, where ξ = (a^{−1}f)_{−1}. So

S_{aa^{∗}}f = zf = zf − aξ + aξ = S_{a}f + aξ,

therefore

π_{1}S_{aa^{∗}}f = S_{a}f

Equation (62) follows from the invariance of aX_{a^{∗}}.

Similarly, if f ∈ X_{a} we have

\begin{aligned}S_{aa^{∗}}b^{∗}f & = & π_{aa^{∗}}zb^{∗}f = zb^{∗}f \\ & = & b^{∗}\{zf − aξ + aξ\} = b^{∗}S_{a}f + ab^{∗}ξ. \end{aligned}

11. Follows by a simple result on inverse systems.

12. Follows from the similarity of F_{ρ} and S_{Ω_{ρ}} and the fact that Ω_{ρ} is the characteristic polynomial of S_{Ω_{ρ}}.

Before we state the next theorem we clarify how analytic functions act on matrices. Here we follow the Riesz-Dunford functional calculus, see Dunford and Schwartz [1957]. Thus assume we are given a matrix A and a function m analytic in a neighbourhood of the spectrum of A. Let μ be a polynomial such that m − μ has at each eigenvalue of A a zero of order not less than the order of the zero of the minimal polynomial of A at that point. Then we define m(A) = μ(A). Clearly this definition is independent of the polynomial μ.

We can pass on now to the proof of the main result.

Theorem 7.2 (Zhou-Khargonekar) Let (A, B, C) be a minimal realization of w ∈ H_{+}^{∞}. Let m(z) be an inner function in the right half plane Π_{+} and assume that m is analytic at all zeroes of Ω_{ρ}. Let F_{ρ} be the Hamiltonian matrix

F_{ρ} = ( \begin{matrix}A & \frac{1}{ρ}BB̃ \\ −\frac{1}{ρ}C̃C & −Ã \end{matrix} ) .

Then ρ > ρ_{∞} is a singular value of H_{m^{∗}w} if and only if

det[m(F_{ρ})]_{11} = 0.

In particular

inf{||w − mq||_{∞} | q ∈ H^{∞}}_{+} = sup{ρ | det[m(F_{ρ})]_{11} = 0}

\underline{Proof:} It suffices to show that the statement is equivalent to that of Theorem 3.1.

The statement of the theorem is independent of the particular realization of w taken. Indeed if R is a similarity in the state space then

( \begin{matrix}R & 0 \\ 0 & R̃^{−1} \end{matrix} )

is a symplectic map. In fact

( \begin{matrix}R̃ & 0 \\ 0 & R^{−1} \end{matrix} ) ( \begin{matrix}0 & I \\ −I & 0 \end{matrix} ) ( \begin{matrix}R & 0 \\ 0 & R̃^{−1} \end{matrix} ) = ( \begin{matrix}0 & I \\ −I & 0 \end{matrix} )

and

( \begin{matrix}R^{−1}AR & \frac{1}{ρ}R^{−1}BB̃R̃^{−1} \\ \frac{1}{ρ}R̃C̃CR & −R̃ÃR̃^{−1} \end{matrix} ) = ( \begin{matrix}R^{−1} & 0 \\ 0 & R̃ \end{matrix} ) ( \begin{matrix}A & \frac{1}{ρ}BB̃ \\ \frac{1}{ρ}C̃C & −Ã \end{matrix} ) ( \begin{matrix}R & 0 \\ 0 & R̃^{−1} \end{matrix} )

As by our assumption m is analytic at the zeroes of Ω_{ρ} then there exists a unique polynomial μ of degree < deg(Ω_{ρ}) such that m − μ = 0mod(Ω_{ρ}). In particular equation (20) transforms into the polynomial equation

μ^{∗}p + ρaq∗ = 0mod(Ω_{ρ})

or equivalently

π_{Ω_{ρ}}μ^{∗}p = μ(S_{Ω_{ρ}})b^{∗}p = −ρaq∗.

Let π_{1} be the projection on the first component relative to the direct sum X_{Ω_{ρ}} = b^{∗}X_{a} ⊕ aX_{a^{∗}}. Thus π_{1}p(S_{Ω_{ρ}})|b^{∗}X_{a} is a singular map. But S_{Ω_{ρ}} is isomorphic to F_{ρ} and so π_{1}f(S_{Ω_{ρ}})|b^{∗}X_{a} is isomorphic to [F_{ρ}]_{11}. Hence the result follows.

EXAMPLE:

We conclude by computing the special case:

w(s) = \frac{1}{αs + 1} ϕ(s) = e^{−hs}

w is realized by (−\frac{1}{α}, \frac{1}{√α}, \frac{1}{√α}, 0) and hence

F_{ρ} = ( \begin{matrix}−\frac{1}{α} & \frac{1}{αρ} \\ −\frac{1}{αρ} & \frac{1}{α} \end{matrix} ) .

So

e^{−hF_{ρ}} = e^{−h( \begin{matrix}−\frac{1}{α} & \frac{1}{αρ} \\ −\frac{1}{αρ} & \frac{1}{α} \end{matrix} )} = e^{− \frac{h}{α}( \begin{matrix}−1 & \frac{1}{ρ} \\ −\frac{1}{ρ} & 1 \end{matrix} )}.

The eigenvalues of F_{ρ} are

λ = \frac{±i√1 − ρ^{2}}{ρ}

The corresponding eigenvectors are

( \begin{matrix}1 \\ ρ ± i√1 − ρ^{2} \end{matrix} ) .

Since

\begin{matrix}( \begin{matrix}1 & 1 \\ ρ + i√1 − ρ^{2} & ρ − i√1 − ρ^{2} \end{matrix} )^{−1} = \\ \frac{i}{2√1 − ρ^{2}} & ( \begin{matrix}ρ − i√1 − ρ^{2} & −1 \\ −(ρ + i√1 − ρ^{2}) & 1 \end{matrix} ) \end{matrix}

we have

\begin{matrix}( \begin{matrix}−1 & \frac{1}{ρ} \\ −\frac{1}{ρ} & 1 \end{matrix} ) = \frac{i}{2√1 − ρ^{2}} ( \begin{matrix}1 & 1 \\ ρ + i√1 − ρ^{2} & ρ − i√1 − ρ^{2} \end{matrix} ) \\ × ( \begin{matrix}\frac{i√1−ρ^{2}}{ρ} & 0 \\ 0 & \frac{−i√1−ρ^{2}}{ρ} \end{matrix} ) ( \begin{matrix}ρ − i√1 − ρ^{2} & −1 \\ −(ρ + i√1 − ρ^{2}) & 1 \end{matrix} ) \end{matrix}

This allows us to compute e^{−hF_{ρ}}. In particular

\begin{matrix}[e−hF^{ρ}]_{11} = \\ \frac{i}{2√1−ρ^{2}} & [(ρ − i√1 − ρ^{2})e^{− \frac{ih^{√}1−ρ^{2}}{αρ}} − (ρ + i√1 − ρ^{2})e^{\frac{ih^{√}1−ρ^{2}}{αρ}} ] \end{matrix}

Thus

det[e^{−hF_{ρ}}]_{11} = 0 ⟺

e\frac{2ih√1 − ρ^{2}}{αρ} = \frac{ρ − i√1 − ρ^{2}}{ρ + i√1 − ρ^{2}} = (ρ − i√1 − ρ^{2})^{2}

Or

e\frac{ih√1 − ρ^{2}}{αρ} = ±(ρ − i√1 − ρ^{2}).  (64)

Putting y = \frac{h√1 − ρ^{2}}{αρ}, then equation (64) reduces to

cos y = ρ, sin y = −√1 − ρ^{2}

and hence to

tan y = −\frac{−√1 − ρ^{2}}{ρ} .

Finally we get

tan y + \frac{αy}{h} = 0.

So if y is the smallest zero of this equation, namely the zero that satisfies π/2 < y < π, then

ρ = \frac{1}{√1 + \frac{α^{2}y^{2}}{h^{2}}} .

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