October 02, 2025
We characterize the inverse of an analytic Fredholm operator-valued function \(A(z)\) near an isolated singularity within a general Banach space framework. Our approach relies on the sequential factorization of \(A(z)\) via Fredholm quotient operators. By analyzing the properties of these quotient operators near an isolated singularity, we fully characterize the Laurent series expansion of the inverse of \(A(z)\) in terms of its Taylor coefficients around the singularity. These theoretical results are subsequently applied to characterize the solution of a general autoregressive law of motion in a Banach space.
Let \(\mathcal{B}\) be a complex Banach space and \(\mathcal{L}_{\mathcal{B}}\) be the space of bounded linear operators acting on \(\mathcal{B}\) with operator norm \(\|\cdot\|_{\mathcal{L}_{\mathcal{B}}}\). Let \(A(z)\) be an analytic \(\mathcal{L}_{\mathcal{B}}\)-valued function defined on an open and connected subset \(U\) of \(\mathbb{C}\). We then consider the Taylor series of \(A(z)\) around \(z_0 \in U\) as follows: \[\label{eq01} A(z) = \sum_{j= 0}^\infty A_{j,z_0}(z-z_0)^j, \quad A_{j,z_0} \in \mathcal{L}_{\mathcal{B}}, \quad z\in U,\tag{1}\] This paper studies the inversion of the above operator-valued function when \(z_0\) is an isolated singularity and the derivation of the expression of \(A(z)^{-1}\) around \(z=z_0\). Instances of this problem arise in various applications, such as asymptotic linear programming ([1], [2]), Markov chains ([3]), and linear control theory ([4]). Additional examples can be found in [5] and [6], and we also discuss a specific example in Section 1.2.
Specifically, this paper aims to provide a closed-form expression for \(A(z)^{-1}\) around an isolated singularity \(z_0\) when \(A(z)\) is a Fredholm-valued function (i.e.,\(\dim(\ker A(z))<\infty\) and \(\dim(\mathop{\mathrm{coker}}A(z))<\infty\) for \(z\in U\)). We provide a recursive formula for determining the coefficients in the Laurent series of \(A(z)^{-1}\) around \(z=z_0\), in terms of the Taylor coefficients \(\{A_{j,z_0}\}_{j\geq 1}\) appearing in and the associated projections, for any order of the pole at \(z=z_0\); moreover, we also demonstrate that the order of the pole can be characterized by those operators. These results are achieved by the factorization of a Fredholm operator-valued function into a perturbed identity operator and a Fredholm quotient, along with a closed-form expression for the latter in terms of \(\{A_{j,z_0}\}_{j\geq 1}\). As detailed in Section 1.1, this approach not only distinguishes our inversion results from existing ones but also provides a useful recasting of previous findings in a Hilbert/Euclidean space setting.
Inversion of an operator-valued function has been studied in various contexts. Particularly, when \(A(z)\) is a linear operator pencil (i.e., \(A_j=0\) for \(j\geq 2\)), much is already known in a Hilbert space setting ([7]) and in a more general Banach space setting ([8]–[11]). In a more general case with a polynomial or analytic operator-valued function was mostly discussed in a finite dimensional setting ([5], [6], [12], [13]). More recently, [14] and [15] consider the inversion of an analytic Fredholm-valued function in a Hilbert space setting in order to characterize solutions to an autoregressive (AR) law of motion applied to function-valued random elements. There are also some results on the inversion of \(A(z)^{-1}\) in a Banach space setting for Fredholm-valued functions ([16], [17]) and for possibly non-Fredholm-valued functions ([18]); however, a closed-form expression of \(A(z)^{-1}\), which is desired in the present paper, has only been explored in a limited case when the order of the pole at \(z=z_0\) is one or two ([17]).
Suppose that a random sequence \(X_t\), taking values in a Banach space, satisfies an AR law of motion given by \[\label{eq001} \sum_{j=0}^\infty A_{0,j} X_{t-j} = \varepsilon_t, \quad t\geq 1,\tag{2}\] where \(\varepsilon_t\) is another sequence of random elements that typically exhibits simpler dynamics than \(X_t\), such as an independently and identically distributed sequence. Then the operator-valued function \[A(z) = \sum_{j=0}^\infty A_{0,j} z^j\] is called the characteristic polynomial of the AR law of motion. The case where \(A(z)\) is invertible for every \(z\) satisfying \(|z|\leq 1 +\eta\) for some \(\eta>0\) but not for \(z=1\) has received considerable attention in the literature on time series analysis. In this case, it turns out that the behavior of solutions to the AR law of motion crucially depends on the local behavior of \(A(z)^{-1}\) around \(z=1\) (see e.g., [19]). A complete characterization of such solutions (except for the initial condition at time zero) reduces to answering the following two questions: (i) what is the order of the pole at \(z=1\), and (ii) what is an explicit expression for the Laurent series of \(A(z)^{-1}\) around \(z=1\)?
When \(\varepsilon_t\) is understood as a stationary sequence, (i) and (ii) are the key questions of the so-called Granger-Johansen representation theory in the literature on time series analysis, which has been studied either in a (finite-dimensional) Euclidean space ([13], [20]–[23]) or a Hilbert space ([14], [15], [24]) or a Banach space ([17], [18], [25]) setting. A recent work by [17] appears to be successful to some extent in this direction, as it provided answers to (i) and (ii) in a Banach space setting when \(A(z)\) is Fredholm and the pole order is one or two. As shown in that paper, answering these questions requires solving a certain system of operator equations using properly defined generalized inverses. However, since those results are derived using operator algebra after specifying the pole order as one or two, they do not, in general, characterize \(A(z)^{-1}\) near a pole of any arbitrary order in a unified framework. The present paper overcomes this limitation by providing a unified approach to characterizing \(A(z)^{-1}\).
We have only considered the case where \(A(z)\) has a real unit root (i.e., \(A(z)\) is not invertible at \(z=1\)). Additionally, there are other papers concerning different cases, such as a complex unit root (see, e.g., [26], [27]), with potential applications in statistical time series analysis. In this literature, the local behavior of \(A(z)^{-1}\) is crucial for understanding the behavior of the random sequence determined by .
We hereafter always assume that \(A(z)\) is an analytic (i.e., complex-differentiable) operator-valued function given in and, for each \(z \in U\), \(A(z)\) is a Fredholm operator acting on a separable Banach space \(\mathcal{B}\), where \(U\) is an open and connected set. The Fredholm property of \(A(z)\) aligns with assumptions made in some preceding articles (see, e.g., [14], [15]) that study the inversion of analytic operator-valued functions. It is noteworthy that any linear operator acting on a finite-dimensional vector space is Fredholm. In practice, it is of particular interest to characterize \(A(z)^{-1}\) using the coefficients \(\{A_{j,z_0}\}_{j \geq 0}\) in , and this is precisely the focus of the present paper. The subsequent theoretical results primarily rely on the factorization of the Fredholm-valued function, as described in the following proposition:
Proposition 1. Suppose that \(A(z)\) is invertible for any \(z\) in a punctured neighborhood of \(z_0\), but not at \(z=z_0\). Let \(P_1\) denote any projection on \(\mathop{\mathrm{ran}}A(z_0)\) and let \(Q_1 = I - P_1\). Then there exists an analytic index-zero Fredholm operator-valued function \(A^{[1]}(z)\) satisfying \[A(z) = [P_1 + (z-z_0)Q_1]A^{[1]}(z), \label{fredfac}\qquad{(1)}\] and \(A^{[1]}(z)^{-1}\) has a pole of order \(d-1\) at \(z=z_0\).
The above proposition may be understood as a natural extension of some existing results developed in a Hilbert space setting to a Banach space setting; see e.g., Theorem 2.4 of [28] and Theorem 3.4 of [29]. [30] obtains a similar result when \(A(z)\) satisfies an additional condition. Given that \(I=P_1 + Q_1\), the operator \(P_1 + (z-z_0)Q_1\) is understood as a perturbed identity near \(z=z_0\). Its inverse is simply given by \(P_1 + (z-z_0)^{-1}Q_1\), and thus \(A^{[1]}(z)\) may be regarded as the quotient operator (see, e.g., [31], [32]) obtained by dividing by the perturbed identity. It is also important to note that the resulting operator \(A^{[1]}\) is similar to \(A(z)\) in the sense that both are analytic Fredholm-valued functions, but differentiated in that \(A^{[1]}\) has a pole of order \(d-1\) at \(z=z_0\). This makes it possible to apply the same factorization repeatedly, for \(k=1,\ldots, d\) with \(d > 0\), \[\label{eqfreadholmq} A^{[k-1]}(z) =(P_{k}+ (z-z_0)Q_{k}) A^{[k]}(z), \quad \text{with} \quad A^{[0]}(z)=A(z),\tag{3}\] until we obtain an analytic operator-valued function \(A^{[d]}(z)\), which is invertible at \(z=z_0\). Noting that, for each \(z\), \(A^{[k]}(z)\) can be understood as a quotient operator and that the order of the pole of \(A^{[k]}(z)^{-1}\) at \(z = z_0\) is reduced by \(k\) compared to that of \(A^{[0]}(z)^{-1}=A(z)^{-1}\), we will refer to \(A^{[k]}\) as the Fredholm quotient of order \(k\) hereafter. The repetition described by leads to the following generalization of Proposition 1:
Proposition 2 (Fredholm factorization). Let the assumptions of Proposition 1 hold and let \(A^{[0]} = A(z)\). Then for some \(d>0\) \[\begin{align} A(z) = (P_1+ (z-z_0)Q_1) \cdots(P_{d} + (z-z_0)Q_{d})A^{[d]}(z), \label{fredfac2} \end{align}\qquad{(2)}\] where \(A^{[d]}(z)\) is analytic, \(A^{[d]}(z_0)\) is invertible, \(\mathop{\mathrm{ran}}P_j = \mathop{\mathrm{ran}}A^{[j-1]}\) and \(P_j = I - Q_j\) for \(j=1,\ldots,d\). Moreover, \[\begin{align} \label{eqinclusion} \mathop{\mathrm{ran}}P_{1} \subset \mathop{\mathrm{ran}}P_{2} \subset \cdots \subset \mathop{\mathrm{ran}}P_{d}. \end{align}\qquad{(3)}\]
Similar results to the above proposition, considering the case where \(\mathcal{B}\) is a Hilbert space, can be found in [28] and [29]. Proposition 2 will be a key input to the subsequent discussion. Note that if \(A^{[d]}(z_0)\) is invertible, then it is surjective. This implies that any additional factorization given by when applied to , changes nothing and yields \(A^{[d]}(z)=A^{[d+1]}(z_0)=\cdots\) since \(P_{d+1}=P_{d+2}=\cdots=I\) (and thus \(Q_{d+1}=Q_{d+2}=\cdots =0\)). By requiring \(P_d \neq I\) in Proposition 2, we may understand that \(d\) in is the (uniquely determined) order of the pole at \(z=z_0\).
Remark 1. Note that \(P_{k}\) is given by the projection onto \(\mathop{\mathrm{ran}}A^{[k-1]}(z_0)\) in Proposition 2 for \(k=1,\ldots,d\), and there are no other requirements; that is, \(\ker P_k\) can be any arbitrarily convenient choice. This means that, in a Hilbert space setting, \(P_k\) can be set as an orthogonal projection whose kernel is given by \([\mathop{\mathrm{ran}}A^{[k-1]}]^\perp\).
The Fredholm quotient of order \(k\), \(A^{[k]}\), and the projection \(P_{k+1}\) onto its range play a crucial role in the subsequent discussion. These can further be characterized in terms of the Taylor coefficients \(\{A_{j,z_0}\}_{j\geq 0}\) of \(A(z)\) via the recursive formula to be given in Proposition below. Hereafter, for the convenience in presenting the subsequent theoretical results, we assume that \[\label{eqpcondi} P_0 = 0, \quad P_{d} \neq I, \quad \text{and} \quad P_{d+j} = I \quad \text{for j \geq 0},\tag{4}\] where, as discussed above, the latter two conditions are not restrictions but should be understood as normalizations that allow us to interpret the integer \(d\) appearing in Proposition 2 as the pole order of \(A(z)^{-1}\) at \(z=z_0\).
Proposition 3 (Fredholm quotient of order \(k\)). Let everything be as in Proposition 2 with \(d>0\), and suppose holds. Then, for \(k=1,\ldots,d\), the Fredholm quotient of order \(k\) is given by \[A^{[k]}(z_0) = \sum_{j=0}^{k-1}(P_{j+1}-P_j)A_{j,z_0} + (I-P_k)A_{k,z_0} \label{fredfactor}\qquad{(4)}\] and for all \(0\leq j,\ell \leq k-1\) with \(j\neq \ell\) \[\begin{align} \label{eqprojec} (P_{j+1}-P_j)(P_{\ell+1}-P_\ell) = 0 \quad \text{and} \quad (P_{j+1}-P_j)(I-P_k) = 0. \end{align}\qquad{(5)}\]
From , and the facts that \(\mathop{\mathrm{ran}}P_k = \mathop{\mathrm{ran}}A^{[k-1]}(z_0)\) and \(\sum_{j=0}^{k-2}(P_{j+1}-P_j) + (I-P_{k-1}) = I\) (with \(P_0=0\)), we find that, for \(k=1,\ldots,d\), \[\begin{align} \mathop{\mathrm{ran}}P_{k} &= \sum_{j=0}^{k-2}(P_{j+1}-P_j) \mathop{\mathrm{ran}}A^{[k-1]}(z_0) + (I-P_{k-1})\mathop{\mathrm{ran}}A^{[k-1]}(z_0) \notag\\ &= \mathop{\mathrm{ran}}A_{0,z_0} + \mathop{\mathrm{ran}}(P_2-P_1)A_{1,z_0} + \cdots + \mathop{\mathrm{ran}}(I-P_{k-1}) A_{k-1,z_0} \label{eqadg}. \end{align}\tag{5}\] Moreover, noting that \(\mathop{\mathrm{ran}}P_{d+1} = \mathop{\mathrm{ran}}A^{[d]}(z_0) = \mathcal{B}\), we also find that \[\begin{align} \mathop{\mathrm{ran}}P_{d+1} &= \sum_{j=0}^{d-1}(P_{j+1}-P_j) \mathop{\mathrm{ran}}A^{[d]}(z_0) + (I-P_{d})\mathop{\mathrm{ran}}A^{[d]}(z_0) \notag \\ &= \mathop{\mathrm{ran}}A_{0,z_0} + \mathop{\mathrm{ran}}(P_2-P_1)A_{1,z_0} + \cdots + \mathop{\mathrm{ran}}(I-P_{d}) A_{d,z_0} = \mathcal{B}. \label{eqadg2} \end{align}\tag{6}\]
The results given by Propositions 1-3 and - give us a few necessary and sufficient conditions for \(A(z)^{-1}\) to have a pole of order \(d\) at \(z=z_0\).
Proposition 4. Let everything be as in Proposition 2 with \(d>0\), and suppose holds. The following are equivalent conditions:
(i) \(A(z)^{-1}\) has a pole of order \(d\) at \(z=z_0\)
(ii) \(A^{[d]}(z_0)\) is invertible while \(A^{[d-1]}(z_0)\) is not.
(iii) \(\mathcal{B} = \sum_{j=0}^d \mathop{\mathrm{ran}}(P_{j+1}-P_{j}) A_{j,z_0}\) with \(P_{d}\neq I\).
(iv) \(\{0\} = \cap_{j=0}^d \ker (P_{j+1}-P_{j}) A_{j,z_0}\) with \(P_{d}\neq I\).
We first note that, in the case where \(\mathcal{B}\) is a Hilbert space, the condition ([cor1d]) in Proposition 4 can be equivalently written as the following direct sum condition: \[\label{eq001a} \mathcal{B} = \sum_{j=0}^d [\ker (P_{j+1}-P_j)A_{j,z_0}]^{\perp}.\tag{7}\] In the literature on time series analysis, when \(z_0 = 1\), the conditions given in Proposition 4 are often referred to as the I(\(d\)) condition (see, e.g., [33]), which guarantees that \(X_t\), satisfying with the characteristic polynomial equivalent to \(A(z)\) in (meaning that the Taylor series of the characteristic polynomial around \(z=z_0\) is given by ), contains a \(d\)-th order integrated component \(\widetilde{\varepsilon}_{d,t} = \sum_{s=1}^{t} \widetilde{\varepsilon}_{d-1,s}\) with \(\widetilde{\varepsilon}_{0,t}=\varepsilon_t\). The conditions ([cor1c]) and ([cor1d]) with \(z_0 = 1\) are particularly interesting when compared to the existing characterizations of the I(\(d\)) condition in a potentially infinite-dimensional setting. [14] provide a certain direct sum condition for \(X_t\) to contain a \(d\)-th order integrated component in a Hilbert space setting. [15] provide similar direct sum conditions for \(d = 1\) and \(2\). These existing conditions, in fact, involve Moore-Penrose inverses of various operators depending on \(A_{j,z_0}\) and certain orthogonal projections, and thus are expressed in quite complicated forms unless \(d = 1\). However, our direct sum conditions given in ([cor1c]) and are written in terms of the Fredholm quotients, depending only on \(\{A_{j,z_0}\}_{j=0}^d\) and \(\{P_j\}_{j=0}^{d+1}\) (furthermore, from Proposition 3 and , \(P_j\) can be characterized in terms of \(A_{j,z_0}\)). Importantly, this formulation applies in a straightforward manner to any arbitrary order \(d\), thereby highlighting both its simplicity and generality. It should also be noted that our conditions in Proposition 4 are developed under a more general Banach space setting. To compare the differences between ours and the recent results, see [15] [14], and [17].
In this section, we assume that \(A(z)^{-1}\) has a pole of order \(d\) at \(z=z_0\), and then derive a systematic and complete way to express \(A(z)^{-1}\) in terms of \(\{A_{j,z_0}\}_{j\geq 0}\) and \(\{P_j\}_{j=0}^{d+1}\) (with \(P_{d+1}=I\) and \(P_0=0\)) using the Fredholm quotient \(A^{[d]}(z)\). We let \[\label{eqexpanG} A^{[d]}(z) = \sum_{j=0}^\infty G_{j,z_0} (z-z_0)^{j}\tag{8}\] and \[\label{eqexpanH} (A^{[d]}(z))^{-1} = \sum_{j=0}^\infty H_{j,z_0} (z-z_0)^{j}.\tag{9}\] The operators \(G_{j,z_0}\) in and \(H_{j,z_0}\) in can be explicitly expressed as a function of \(\{P_j\}_{j=0}^{d+1}\) and \(\{A_{j,z_0}\}_{j\geq 0}\) as follows:
Proposition 5. Let everything be as in Proposition 2 with \(d>0\), and suppose holds. Then for all \(k \geq 0\), \[\begin{align} G_{k,z_0}&= \sum_{j=0}^{d}(P_{j+1}-P_{j})A_{k+j,z_0}, \quad j \geq 0, \notag \\ H_{j,z_0} &= \begin{cases} G_{0,z_0}^{-1} \quad &\text{if j=0}, \\ - G_{0,z_0}^{-1} \sum_{k=1}^j G_{k,z_0}H_{j-k,z_0}, \quad &\text{if j\geq1}. \label{eqpropo} \end{cases} \end{align}\qquad{(6)}\]
Combined with the results given in Proposition 5, the following result provides a closed-form representation of \(A(z)^{-1}\):
Proposition 6. Let everything be as in Proposition 2 with \(d>0\), and suppose holds. Then the inverse \(A(z)^{-1}\) admits the Laurent series \[A(z)^{-1}= \sum_{j=-d}^{\infty} \Psi_j (z-z_0)^{j},\] where \[\Psi_j = \begin{cases} \sum_{\ell=0}^{j+d}H_{j+d-\ell,z_0} (P_{d+1-\ell}-P_{d-\ell}), & \text{if j <0},\\ \sum_{\ell=0}^{d} H_{j+\ell,z_0} (P_{\ell+1}-P_{\ell}), & \text{if j \geq0}. \end{cases} \notag\]
In this section, we characterize solutions to the AR law of motion with the characteristic polynomial equivalent to \(A(z)\) in , using our theoretical results. The cases \(d=1\) or \(d=2\) are regarded as empirically relevant in the time series literature, and we focus on these as applications of our theoretical framework. However, since our characterization of the solutions follows directly from the results in Sections 2 and 3, the representation of \(X_t\) for \(d\geq 3\) requires only a minor modification. We hereafter let \(\pi_j(k)\) be defined as follows: \(\pi_0(k)=1,\) \(\pi_1(k)=k\), and \(\pi_j(k)=k(k-1)\cdots (k-j+1)/j!\) for \(j \geq 2\). The following results show the desired characterization.
Proposition 7. Assume that the AR law of motion holds with the characteristic polynomial equivalent to \(A(z)\) in , which is further assumed to be invertible for any \(z\) satisfying \(|z|\leq 1 + \eta\) for some \(\eta>0\), except at \(z=1\). Then the following hold:
(i) If \(A^{[1]}(1) = A_{0,1} - (I-P_1)A_{1,1}\) is invertible, then \(d=1\) and \(X_t\) can be represented as follows: \[\label{eqi1repre} X_t = \tau_0 + \Psi_{-1} \sum_{s=1}^t \varepsilon_s + \nu_t, \quad t\geq 1,\qquad{(7)}\] where \(\nu_t = \sum_{j=0}^\infty \Phi_j \varepsilon_{t-j}\), \(\Phi_j = \sum_{k=j}^\infty (-1)^{k-j} \pi_j(k)\Psi_k\), and \(\Psi_j\) is determined as in Proposition 6 with \(d=1\).
(ii) If \(A^{[2]}(1) = A_{0,1}+(P_2-P_1)A_{1,1}+(I-P_2)A_{2,1}\) is invertible while \(A^{[1]}(1)\) is not, then \(d=2\) and \(X_t\) can be represented as follows: for some \(\tau_0, \tau_1 \in \mathcal{B}\), \[X_t = \tau_0 + \tau_1 t + \Psi_{-2}\sum_{r=1}^s \sum_{s=1}^t \varepsilon_s + \Psi_{-1}\sum_{s=1}^t \varepsilon_s + \nu_t,\] where \(\nu_t = \sum_{j=0}^\infty \Phi_j \varepsilon_{t-j}\), \(\Phi_j = \sum_{k=j}^\infty (-1)^{k-j} \pi_j(k)\Psi_k\), and \(\Psi_j\) is determined as in Proposition 6 with \(d=2\).
Even though it is not the main focus of this paper, it can be shown that \(\|\Phi_j\|_{\mathcal{L}{\mathcal{B}}}\) decreases exponentially as \(j\) increases, under the assumptions of Proposition 7; see, e.g., the proofs of Propositions 3.1 and 4.1 in [17]. The characterization of solutions to the AR law of motion , as in Proposition 7, is a central problem in the time series analysis literature. As discussed in Section 1.2, the most well-known results in this context are the Granger-Johansen representation theorems. In this regard, Proposition 7 may be viewed as a version of the Granger-Johansen representation theorem, derived from our previous Fredholm factorization and the properties of the associated Fredholm quotients, in a more general setup.
Proof of Proposition 1. Since \(P_1\) is an orthogonal projection and \(Q_1=I-P_1\), the perturbed identity \(P_1 + (z-z_0) Q_1\) is invertible for \(z \in U \setminus \{z_0\}\), where \(U\) is an open and connected set as introduced in Section 1, and its inverse is given by \[\begin{align} \label{eqpf00} [P_1 + (z-z_0) Q_1]^{-1} = P_1 + (z-z_0)^{-1} Q_1. \end{align}\tag{10}\] Let \(A^{[1]}(z)\) be an operator-valued function that is defined as \(A^{[1]}(z) = [P_1 + (z-z_0) Q_1]^{-1}A(z) = P_1A(z) + (z-z_0)^{-1}Q_1A(z)\) for \(z\in U\setminus\{z_0\}\) and \(A^{[1]}(z_0) = P_1A(z_0) + Q_1A_{1,z_0}(z_0)\). Then \(A^{[1]}(z)\) is holomorphic at \(z \in U \setminus \{z_0\}\) and also at \(z=z_0\) since \[\begin{align} \frac{A^{[1]}(z)-A^{[1]}(z_0)}{z-z_0}&= P_1\left(\frac{A(z)-A(z_0)}{z-z_0}\right) + Q_1\left(\frac{A(z)-A(z_0)}{(z-z_0)^2} - \frac{A_{1,z_0}(z_0)}{z-z_0} \right), \label{eqpf01} \end{align}\tag{11}\] where we used the fact that \(Q_1A(z)=Q_1(A(z)-A(z_0))\) since \(Q_1A(z_0)=0\). Note that the right hand side of converges to \(P_1 A_{1,z_0} + Q_1 A_{2,z_0}\) as \(z\to z_0\). We next show that \(A^{[1]}(z)\) is an index-zero Fredholm operator for all \(z\in U\). Note that \[A^{[1]}(z) = A(z) + K(z),\] where \[K(z) = \begin{cases} - Q_1 A(z) + (z-z_0)^{-1} Q_1 A(z)& \quad \text{if z\neq z_0},\\ Q_1A_{1,z_0}(z_0) & \quad \text{if z= z_0}. \end{cases}\] Since \(Q_1\) is a finite dimensional projection due to the Fredholm property of \(A(z)\), \(K(z)\) is obviously compact for all \(z \in U\). This implies that \(A^{[1]}(z)\) and \(A(z)\) are Fredholm operators of the same index (see e.g., Theorem 3.11 of [34]).
We next show that \(A^{[1]}(z)^{-1}\) has a pole of order \(d-1\) at \(z=z_0\). Note that \[A(z)^{-1} = A^{[1]}(z)^{-1}[P_1 + (z-z_0)^{-1} Q_1],\] which implies that \(A^{[1]}(z)^{-1}\) has a pole of order at most \(d-1\) at \(z=z_0\). Moreover, from and the fact that \(A^{[1]}(z) = [P_1 + (z-z_0) Q_1]^{-1}A(z)\) for any \(z \in U\, \setminus\, \{z_0\}\), we have \[\begin{align} A^{[1]}(z)^{-1} = A(z)^{-1}P_1 + (z-z_0) A(z)^{-1} Q_1. \label{lemeq001} \end{align}\tag{12}\] Note that the second term in has a pole of order \(d-1\) at \(z=z_0\). Moreover, observe that, for any \(x \in \mathcal{B}\), there exists \(y\) such that \(P_1x=A(z_0)y\). We then note that \[\begin{align} |z-&z_0|^{d-1}\|A^{-1}(z)P_1x\| = |z-z_0|^{d-1}\|A^{-1}(z)A(z_0)y\| \notag \\ & \leq |z-z_0|^{d-1}\|y\| + \left\|(z-z_0)^{d}A^{-1}(z)\left(\frac{A(z_0)y - A(z)y}{z-z_0}\right)\right\|. \label{lemeq002} \end{align}\tag{13}\] As \(z\) approaches \(z_0\), the right-hand side of is convergent. That is, is bounded for \(z\) close enough \(z_0\). From the uniform boundedness property of a convergent sequence see, e.g.[35], it is deduced from that \(|z-z_0|^{d-1}\|A^{-1}(z)P_1\|_{\mathcal{L}_{\mathcal{B}}}\) is uniformly bounded. This implies that \(A^{-1}(z)P_1\) has a pole of order at most \(d-1\). Combining this result with the fact that the second term in has a pole of order \(d-1\) at \(z=z_0\), we conclude that \(A^{[1]}(z)^{-1}\) has a pole of order \(d-1\) at \(z=z_0\). ◻
Proof of Proposition 2. By repeatedly applying Proposition 1, we may obtain the following: for \(j=1,\ldots,d\), \[A^{[j-1]}(z) = [P_{j} + (z-z_0)Q_{j}]A^{[j]}(z),\] where \(A^{[0]}(z) \mathrel{\vcenter{:}}= A(z)\), the existence of \(d<\infty\) is guaranteed by the analytic Fredholm theorem (e.g., Corollary 8.4 in [16]), and \(A^{[d]}(z)\) is holomorphic on \(U\) and does not have a singularity at \(z=z_0\). From Proposition 1, we can choose \(P_j\) as the projection on \(\mathop{\mathrm{ran}}A^{{[j-1]}}(z_0)\) for \(j=1,\ldots,d\). At \(z=z_0\), we have \[A^{{[j-1]}}(z_0) = P_{j} A^{[j]}(z_0), \quad j = 1,\ldots,d.\] Since \(P_j\) is a projection, we have \[\begin{align} \mathop{\mathrm{ran}}A^{{[j]}}(z_0) &= P_{j} \mathop{\mathrm{ran}}A^{[j]}(z_0) \oplus Q_{j} \mathop{\mathrm{ran}}A^{[j]}(z_0) \\ &= \mathop{\mathrm{ran}}A^{[j-1]}(z_0) \oplus Q_{j} \mathop{\mathrm{ran}}A^{[j]}(z_0), \end{align}\] Therefore, \(\mathop{\mathrm{ran}}A^{[j-1]}(z_0) \subset \mathop{\mathrm{ran}}A^{[j]}(z_0)\). Since \(\mathop{\mathrm{ran}}P_{j+1} = \mathop{\mathrm{ran}}A^{[j]}\), we conclude that holds. ◻
Proof of Proposition 3. directly follows from , and thus the detailed proof is omitted. Moreover, observing that \(P_{j}P_{j+\ell}=P_j\), \(P_{j}Q_{j+\ell}=0\), \(Q_{j}P_{j+\ell}=P_{j+\ell}-P_{j}\) and \(Q_{j}Q_{j+\ell} = Q_{j+\ell}\) for \(\ell\geq 1\), we find that \(\Pi_{j=1}^k(P_{j}+(z-z_0)Q_j) = \sum_{j=0}^{k}(P_{j+1}-P_{j})(z-z_0)^j\), where \(P_{k+1} = I\) and \(P_0=0\). Since \(A^{[k]}(z)\) is holomorphic, we know from Proposition 2 that \[\begin{align} A(z) = &\left[ \sum_{j=0}^{k}(P_{j+1}-P_{j})(z-z_0)^j \right]\left[ \sum_{j=0}^\infty A^{[k]}_{j,z_0}(z-z_0)^j \right], \label{lameqpf01} \end{align}\tag{14}\] where \(A^{[k]}_{j,z_0}\) is the coefficient associated with \((z-z_0)^j\) of the Taylor series of \(A^{[k]}(z)\) at \(z=z_0\). Since \(A(z)\) is holomorphic at \(z=z_0\), we have \[A(z) = \sum_{j=0}^{\infty}A_{j,z_0} (z-z_0)^j.\] For \(0\leq m \leq k\), we equate the coefficients of \((z-z_0)^m\) in and obtain \[A_{m,z_0} = \sum_{j=0}^m (P_{m+1-j} - P_{m-j}) A^{[k]}_{j,z_0}.\] Since \((P_{j+1}-P_j) (P_{\ell+1}-P_\ell) = 0\) if \(j \neq \ell\), we find that the following hold: \[\begin{align} &(P_1-P_0)A_{0,z_0} = (P_1-P_0)A^{[k]}_{0,z_0}, \notag \\ &(P_2-P_1)A_{1,z_0} = (P_2-P_1)A^{[k]}_{0,z_0}, \notag \\ &\vdots \notag \\ &(P_{k+1}-P_k)A_{k,z_0} = (P_{k+1}-P_k)A^{[k]}_{0,z_0}. \notag \end{align}\] Since \(\sum_{j=0}^{k}(P_{j+1}-P_j) = P_{k+1}-P_0 = I\), we find that \[\begin{align} A^{[k]}(z_0) = A^{[k]}_{0,z_0} = (P_1-P_0)A_{0,z_0}+ \cdots + (P_k-P_{k-1})A_{k-1,z_0} + (I-P_k)A_{k,z_0}, \end{align}\] as desired. ◻
Proof of Proposition 4. ([cor1a]) \(\Leftrightarrow\) ([cor1b]) is an immediate consequence of Proposition 2. We will show ([cor1b]) \(\Rightarrow\) both ([cor1c]) and ([cor1d]), ([cor1c]) \(\Leftrightarrow\) ([cor1d]), and both ([cor1c]) and ([cor1d]) \(\Rightarrow\) (ii).
First note that the invertibility of \(A^{[d]}(z_0)=\sum_{j=0}^d (P_{j+1}-P_{j}) A_{j,z_0}\) implies that \[\label{eqrequired1} \mathcal{B} =\mathop{\mathrm{ran}}\left(\sum_{j=0}^d (P_{j+1}-P_{j}) A_{j,z_0}\right)\tag{15}\] and \[\label{eqrequired2} \{0\} = \ker \left(\sum_{j=0}^d (P_{j+1}-P_{j}) A_{j,z_0}\right).\tag{16}\] Since \((P_{j+1}-P_j)(P_{\ell+1}-P_{\ell})=0\) unless \(j=\ell\) and \(\sum_{j=0}^d (P_{j+1}-P_j)=I\), \(\mathop{\mathrm{ran}}(P_{j+1}-P_j) = \mathop{\mathrm{ran}}(P_{j+1}-P_{j}) A_{j,z_0}\) and we find that is equivalent to the direct sum condition given in (iii). Moreover, from the fact that \((P_{j+1}-P_j)(P_{\ell+1}-P_{\ell})=0\) for \(j\neq \ell\), we find that \(x \in \ker (\sum_{j=0}^d (P_{j+1}-P_{j}) A_{j,z_0})\) must satisfy \(x \in \ker (P_{j+1}-P_{j}) A_{j,z_0}\) for all \(j=1,\ldots,d\). Combining this result with , we find that \[\ker \left(\sum_{j=0}^d (P_{j+1}-P_{j}) A_{j,z_0}\right)= \cap_{j=0}^d \ker (P_{j+1}-P_{j}) A_{j,z_0}.\] Thus (ii) \(\Rightarrow\) (iii) and (iv).
Note that \(A^{[d]}(z_0) = \sum_{j=0}^d (P_{j+1}-P_{j}) A_{j,z_0}\) is a Fredholm operator of index zero (see Proposition 2), and hence \(\dim(\mathop{\mathrm{coker}}A^{[d]}(z_0)) =\dim(\ker A^{[d]}(z_0))\). Thus if either of or is true, then the other is also true. Combining this with the fact that (resp. ) is equivalent to ([cor1b]) (resp.([cor1c])) by the properties of \(P_j\), we find that (iii) \(\Leftrightarrow\) (iv).
If \(A^{[d]}(z_0)\) is not invertible, then from the fact that \(A^{[d]}(z_0) = \sum_{j=0}^d (P_{j+1} - P_{j}) A_{j,z_0}\), it is straightforward to see that either or cannot hold. Thus, (iii) and (iv) \(\Rightarrow\) (ii). ◻
Proof of Proposition 5. From and , we observe that \[A(z) = \left[ \sum_{j=0}^d (P_{j+1}-P_{j})(z-z_0)^j \right]\left[ \sum_{j=0}^\infty G_{j,z_0}(z-z_0)^j \right]\] By equating the coefficients of \((z-z_0)^{k}\), we find the following: \[\begin{align} A_{k,z_0} = \begin{cases} \sum_{j=0}^k (P_{k+1-j} - P_{k-j}) G_{j,z_0} \quad &\text{ if 0\leq k \leq d},\\ \sum_{j=0}^d (P_{d+1-j} - P_{d-j}) G_{k-d+j,z_0} \quad &\text{ if k \geq d+1}.\\ \end{cases} \end{align}\] Since \((P_{j+1}-P_j) (P_{\ell+1}-P_\ell) = 0\) if \(j \neq \ell\), we find that the following hold for any \(k \geq 0\): \[\begin{align} &(P_1-P_0)A_{k,z_0} = (P_1-P_0)G_{k,z_0}, \notag \\ &(P_2-P_1)A_{k+1,z_0} = (P_2-P_1)G_{k,z_0}, \notag\\ &\vdots \notag\\ &(P_{d+1}-P_d)A_{k+d,z_0} = (P_{d+1}-P_d)G_{k,z_0}. \notag \end{align}\] Since \(\sum_{j=0}^d(P_{j+1}-P_j) = I\), we find that \(G_{k,z_0} = \sum_{j=0}^{d}(P_{j+1}-P_j) A_{k+j,z_0}\).
The characterization of \(H_{j,z_0}\) in can easily be deduced from the identity \(G(z)G(z)^{-1} = I\), and hence the detailed proof is omitted. ◻
Proof of Proposition 6. We know from Proposition 2 that \[A(z)^{-1} = G(z)^{-1}(P_d + (z-z_0)^{-1} Q_d)\cdots(P_1 + (z-z_0)^{-1} Q_1).\] We find that \(P_{j+\ell}P_{j}=P_j\), \(P_{j+\ell}Q_{j}=P_{j+\ell}-P_{j}\), \(Q_{j+\ell}P_{j}=0\) and \(Q_{j+\ell}Q_{j} = Q_{j+\ell}\) for \(\ell\geq 1\). Thus, we find that \[\begin{align} & A(z)^{-1} = G(z)^{-1} \sum_{j=0}^{d}(P_{j+1} - P_j)(z-z_0)^{-j} \notag \\ &= \left(\sum_{j=0}^{\infty} H_j (z-z_0)^{j}\right) \left( \sum_{j=0}^{d}(P_{j+1} - P_j)(z-z_0)^{-j} \right)\notag\\ &= \sum_{j=0}^{d-1} \left(\sum_{\ell=0}^{j}H_{j-\ell} (P_{d+1-\ell}-P_{d-\ell})\right) (z-z_0)^{j-d} +\sum_{j=0}^{\infty} \left(\sum_{\ell=0}^{d} H_{j+\ell} (P_{\ell+1}-P_{\ell})\right) (z-z_0)^{j}\notag\\ &= \sum_{j=-d}^{-1} \left(\sum_{\ell=0}^{j+d}H_{j+d-\ell} (P_{d+1-\ell}-P_{d-\ell})\right) (z-z_0)^{j} +\sum_{j=0}^{\infty} \left(\sum_{\ell=0}^{d} H_{j+\ell} (P_{\ell+1}-P_{\ell})\right) (z-z_0)^{j}. \notag \end{align}\] This proves the desired expression of \(A(z)^{-1}\). ◻
Proof of Proposition 7. To show (i), we apply the equivalent linear filter induced by \((1-z)A(z)^{-1}\) to both sides of with \(A(z)\) in (see e.g., the proofs of Theorems 3.1 and 4.1 of [15]). We then know from Propositions 5 and 6 that \(\Delta X_t:= X_t-X_{t-1} = -\Psi_{-1}\varepsilon_t + (\nu_t-\nu_{t-1})\), where \(\nu_t = \sum_{j=0}^\infty \Phi_j \varepsilon_{t-j}\) and \(\Phi_j\) is the Taylor coefficient around \(z=0\) associated with \((z-z_0)^j\) of the analytic part of the Laurent series of \(A(z)^{-1}\) (see the proof of Proposition 4.1 of [17]). Then the desired representation of \(X_t\) and the expression of \(\Psi_j\) given in (i) are deduced without difficulty.
To show (ii), we apply the linear filter induced by \((1-z)^2A(z)^{-1}\), and obtain the desired results in a similar manner. ◻
Email : won-ki.seo@sydney.edu.au↩︎