The Characteristic Function for Complex Doubly Infinite Jacobi Matrices

We introduce a class of doubly infinite complex Jacobi matrices determined by a simple convergence condition imposed on the diagonal and off-diagonal sequences. For each Jacobi matrix belonging to this class, an analytic function, called a characteristic function, is associated with it. It is shown that the point spectrum of the corresponding Jacobi operator restricted to a suitable domain coincides with the zero set of the characteristic function. Also, coincidence regarding the order of a zero of the characteristic function and the algebraic multiplicity of the corresponding eigenvalue is proved. Further, formulas for the entries of eigenvectors, generalized eigenvectors, a summation identity for eigenvectors, and matrix elements of the resolvent operator are provided. The presented method is illustrated by several concrete examples.


Introduction
In a recent paper [15], a method for the spectral analysis of a certain class of semi-infinite Jacobi matrices based on the so-called characteristic function was developed. The spectral analysis of semi-infinite Jacobi matrices is intimately related to classical branches of analysis such as orthogonal polynomials, moment problems, and continued fractions. There are many monographs that have focused on this, see [2,12,17] for several examples. On the other hand, the spectral theory of doubly infinite Jacobi matrices does not appear as often; a nice exposition of this theory can be found in [4,Chap. 7]. Some aspects of the spectral theory of the doubly and semi-infinite Jacobi matrices are quite similar (one may consult, for example, [10]); however, some are different. The main difference is the fact that, in the case of doubly infinite matrices, the space of the solutions to the corresponding second-order difference equation is two-dimensional.

F.Štampach IEOT
The main aim of this article is twofold. First, in analogy with the semiinfinite case treated in [15], we introduce the characteristic function associated with the doubly infinite Jacobi matrix where λ n , w n ∈ C and w n = 0 for all n ∈ Z. We also show how the characteristic function can be used to analyze spectral properties of a linear operator acting on 2 (Z) whose matrix representation with respect to the standard basis of 2 (Z) coincides with J . Second, we extend the method by proving a result concerning the algebraic multiplicity of eigenvalues and generalized eigenvectors.
The subclass of matrices J for which the characteristic function is well defined is determined by a simple convergence condition imposed on λ n and w n , see (8). The self-adjointness of an operator associated with J is not essential for the presented method; therefore, we treat the matrix J with complex entries, which might be of interest from the point of view of the spectral theory of non-self-adjoint operators -a currently very active and rapidly developing field [6,8,18]. The main results of this paper are stated in Theorems 14 and 20.
In more detail, we show [under a mild assumption additional to the essential convergence condition (8)] that the matrix J uniquely determines a densely defined closed (Jacobi) operator J whose spectral properties are related to the properties of the corresponding characteristic function. Namely, by being restricted to a certain subset of C, the spectrum of J in this set is discrete and coincides with the set of zeros of the characteristic function. Further, we provide formulas for eigenvectors, a summation formula for the entries of an eigenvector and an expression for the entries of the matrix representation of the resolvent operator. These results are worked out within Sect. 2 and represent a doubly infinite analog to the corresponding results derived in [15] for the case of semi-infinite Jacobi matrices. Section 3 is devoted to the connection between the order of zeros of the characteristic function and the algebraic multiplicity of the corresponding Jacobi operator's eigenvalues. In addition, we provide formulas for basis vectors of generalized eigenspaces. These results are of particular interest when questions on diagonalization of non-self-adjoint Jacobi operators are examined.
In Sect. 4, we impose some additional conditions on the diagonal sequence of J , which allows us to remove singularities of the characteristic function, with possibly one exception located at the origin, and introduce a regularized characteristic function. According to the type of the additional condition, we distinguish 3 different cases and illustrate the respective results on concrete examples. Moreover, in 2 cases concerning either a compact Vol. 88 (2017) Characteristic Function for Jacobi Matrix 503 operator or an operator with compact resolvent, we indicate a connection between the regularized characteristic function and the theory of regularized determinants.

The Characteristic Function and Doubly Infinite Jacobi Matrices
In this section, we introduce the characteristic function associated with the doubly infinite Jacobi matrix (1) and derive spectral results similar to those obtained in [15]. Where the verification of a result is completely analogous to the corresponding one given in [15] for the semi-infinite matrix, the proof is only indicated for the sake of brevity.

Function F
The main algebraic tool for the definition of the characteristic function is a function called F which appeared in [14] for the first time. The definition given below is a slight generalization of the original one from [14, Def. 1] and is consistent with the one mentioned in [16,Sec. 2]. where , where x k := 0, whenever k < n 1 or k > n 2 , and provided that {x k } ∞ k=−∞ ∈ Dom F. Conventionally, for n 1 , n 2 ∈ Z, we also put Remark 2. Note that the absolute value of the mth summand on the RHS of (2) is majorized by the expression cf. [14,Rem. 2]. Consequently, the function F is well defined on sequences from Dom F, and we have the estimate Recall several properties of F. In the following formulas we always assume that all the expressions are well defined, i.e., the vector in the argument of F (possibly with additional zeros) belongs to Dom F. First of all, we have the important relation [15,Eq. (19)] where n ∈ Z satisfies n 1 ≤ n ≤ n 2 and n 1 , n 2 ∈ Z ∪ {±∞}. Equivalently, (3) can be written as where we have used that (5) which is the special case of (3) with n 1 := n. Similarly, by putting n 2 := n+1 in (3), we obtain Second, one has the limit relations and which one verifies in the same way as [15,Lem. 2].

The Characteristic Function
For two given sequences satisfying a certain convergence condition, we will define a complex function defined on a subset of C in terms of F. This function will be called the characteristic function associated with the doubly infinite Jacobi matrix J since it plays a similar role to the characteristic polynomial in linear algebra, as it will be further demonstrated. First, let us introduce a notation we will use. For λ : Z → C a complex sequence, we put Ran(λ) := {λ n | n ∈ Z} and C λ 0 := C\Ran(λ), where Ran(λ) is the closure of Ran(λ). Further, we denote by der(λ) the set of all (finite) accumulation points of Ran(λ), i.e., der(λ) is the set of all limit points of all possible convergent subsequences of λ. Clearly, Ran(λ) = Ran(λ) ∪ der(λ).
For λ, w : Z → C, the characteristic function will be defined under the condition Vol. 88 (2017) Characteristic Function for Jacobi Matrix 505 that is valid for at least one z 0 ∈ C λ 0 . If this is true, then (8) remains valid for all z 0 ∈ C λ 0 and the convergence is locally uniform on C λ 0 , as it is straightforward to verify; cf. the proof of [15,Lem. 8]. In fact, one also readily shows that (8) remains valid also for z ∈ Ran(λ)\ der(λ) provided the finite number of terms where we would divide by zero are omitted in the series. Definition 3. Let λ : Z → C and w : Z → C\{0} be such that (8) holds for at least one z 0 ∈ C λ 0 . We define the characteristic function associated with the doubly infinite matrix J given by (1) by where γ : Z → C is any sequence satisfying the difference equation γ n γ n+1 = w n for all n ∈ Z.
Remark 4. Note that since the sequence in the argument of F in (9) fulfills (8) for all z 0 ∈ C λ 0 , it belongs to Dom F and the RHS of (9) is therefore well defined for all z ∈ C λ 0 . Further, since 0 / ∈ Ran w, the sequence γ always exists and is determined uniquely by specifying one value, for example, by setting γ 0 = 1. The definition of the characteristic function does not depend on a particular choice of the sequence γ.
By a simple modification of the argument used in the proof of [15,Lem. 8], one verifies that and the convergence is locally uniform on C λ 0 , provided the condition (8) holds for at least one z 0 ∈ C λ 0 . Consequently, F J is an analytic function on C λ 0 . Moreover, F J is meromorphic on C\ der(λ). Indeed, the singularity of F J at a point z = λ n , for some n ∈ Z such that λ n ∈ der(λ), is either a removable singularity or a pole of order less than or equal to r(z) where is the number of values of λ coinciding with z. To see this, for z ∈ Ran(λ)\ der(λ) fixed, take m, M ∈ Z, m ≤ M , such that λ n = z for all n ≤ m and all n ≥ M . Using the rule (4) twice, one derives, for u ∈ C λ 0 , that where we temporarily denote can have a singularity at z which can be either a removable singularity or a pole of order at most r(z). The remaining terms on the RHS of (12) are functions analytic at z.

The Jacobi Operator
Let us recall the standard procedure of prescribing densely defined and closed operators associated with J . On the algebraic level, the formal doubly infinite matrix J can be understood as a linear mapping acting on the space of complex sequences x (indexed by Z) by a formal matrix multiplication, i.e., Let {e n | n ∈ Z} stands for the standard basis of 2 (Z), i.e., (e n ) m = δ m,n for m, n ∈ Z. Define an auxiliary operator J 0 as The operator J 0 , as an operator on the Hilbert space 2 (Z), need not be closed in 2 (Z), but is always closable, cf. [3, Subsec. 2.1]. Hence, one can introduce the so-called minimal operator J min as the operator closure of J 0 . On the other hand, it is natural to define the maximal operator J max by putting Here, the expression J x is to be understood as in (13).
One can show that J min ⊂ J max , but the equality does not hold in general. The operators J min and J max are related via their adjoints: where C stands for the complex conjugation operator acting on complex sequences as (Cx) n := x n , n ∈ Z. The verification of formulas in (14) is a matter of straightforward use of the definition of the adjoint operator; see also [3,Lem. 2.1] for an analogous proof for operators associated with a semi-infinite complex Jacobi matrix. It follows from (14) that J max is closed. In addition, any closed linear operator A acting on 2 (Z) such that {e n | n ∈ Z} ⊂ Dom A, whose matrix representation with respect to the standard basis coincides with J , satisfies J min ⊂ A ⊂ J max . If J min = J max , the Jacobi matrix J determines uniquely the Jacobi operator. In this case, the subscripts can be omitted and we simply write J := J min = J max .

Spectral Properties of the Jacobi Operator via the Characteristic Function
Note that P(z) : Z → C\{0} satisfies the equations Further, for λ : Z → C and z ∈ C, we introduce quantities and for n ∈ Z, where γ : Z → C is as in Definition 3 and P n (z) given by (15). Further, we extend the definition of f (z) and g(z) for z ∈ Ran(λ)\ der(λ) by formulas where r ± (z) are defined by (17).
Note that the sequences f (z) and g(z) are well defined by Definition 5 for all z ∈ C\ der(λ). In addition, for z ∈ C\ der(λ) and n > 0 such that n ≥ max{k ∈ Z | λ k = z}, one has Similarly, for n ≤ 0 such that n ≤ min{k ∈ Z | λ k = z}, one has 508 F.Štampach IEOT Proposition 7. Let λ : Z → C and w : Z → C\{0} be such that (8) holds for at least one z 0 ∈ C λ 0 . Then f (z) and g(z) are solutions of the eigenvalue equation J u = zu for all z ∈ C\ der(λ). In addition, for their Wronskian Proof. We verify the statement for z ∈ C λ 0 only. The extension to all z ∈ C\ der(λ) is to be treated readily with the aid of limit formulas (20).
By using the definition relations (18), (19) and taking into account the recurrence (16), the verification of the equation J u = zu, i.e., for u = f (z) and u = g(z), is a straightforward application of the identities (5) and (6).
Further, for n ∈ Z and z ∈ C λ 0 arbitrary, we have According to (3), the RHS of the above equality coincides with F J (z) which completes the proof.
For the spectral analysis of a Jacobi operator associated to J , we will need to know the asymptotic behavior of the sequences f (z) and g(z) from Definition 5, for z ∈ C\ der(λ), as the index approaches +∞ and −∞, respectively. The following auxiliary result will be used for this purpose.
Proof. We verify the convergence of the first series. The second one is to be treated similarly. Let z ∈ C\ der(λ) be fixed and denote by M (z) ∈ N an index such that Vol. 88 (2017) Characteristic Function for Jacobi Matrix 509 and hence, without loss of generality, we may assume that Next, with the aid of (23), one obtains where Since z = λ n for all n ≥ M (z) and z / ∈ der(λ), hence, the RHS of the inequality (24) is the convergent majorant for the first series from the statement.
Next, for the purpose of the main theorem of this section, we introduce an extended zero set of the characteristic function F J .
Remark 10. Note that Z(J ) decomposes into the union of the set of all zeros of F J located in C λ 0 (since r(z) = 0 for all z ∈ C λ 0 ) and the set of those points from Ran(λ)\ der(λ) which are not the poles of F J of order r(z), i.e., they are either removable singularities or poles of order strictly less then r(z). Proposition 11. Let λ : Z → C and w : Z → C\{0} be such that (8) holds for at least one z 0 ∈ C λ 0 . Then one has and, for z ∈ Z(J ), the corresponding eigenvector of J max can be chosen as f (z).
Proof. For z ∈ C\ der(λ) fixed, we have, by (7), (21), and (22), that and is a summable sequence at +∞ and (wα(z)) −1 is a summable sequence at −∞. Consequently, for all z ∈ C\ der(λ), f (z) is square summable at +∞ and g(z) square summable at −∞. Assume z ∈ Z(J ). Then, according to Proposition 7, f (z) and g(z) are two solutions of the eigenvalue equation J u = zu, which are linearly dependent since their Wronskian vanishes. Hence, by the above discussion, f (z) and g(z) belong to 2 (Z). Particularly, we have f (z) ∈ 2 (Z) and J max f (z) = zf (z). Thus, if f (z) = 0, then z is an eigenvalue of J max and f (z) the corresponding eigenvector. Assume, on the contrary, that f (z) = 0. Then, by (21), one has for all n ∈ N sufficiently large. However, according to (7), the LHS of the above equality tends to 1, as n → ∞, which is a contradiction.
Next, we derive a summation formula which, for real w and λ, turns out to be the 2 -norm of an eigenvector of J max corresponding to an eigenvalue located in C λ 0 .
where A(z) is given by the formula with an arbitrary n ∈ Z such that the denominator does not vanish.
Proof. According to Proposition 7, f (z) is a solution of the second-order difference equation By application of the Green formula, one obtains . With the aid of the asymptotic formula (26), one verifies that Vol. 88 (2017) Characteristic Function for Jacobi Matrix 511 for x, y ∈ C λ 0 and n → ∞. Note that The RHS in the above estimate tends to 0, as n → ∞, which follows from the assumption (8). Consequently, the product on the RHS of (30) tends to 0, as n → ∞, and hence for all x, y ∈ C λ 0 . Thus, by sending n → ∞ in (29), one gets Finally, with the aid of (26) and similarly to what occurs in the proof of Lemma 8, one verifies the sum on the LHS of (31) converges locally uniformly in y on C λ 0 with x ∈ C λ 0 fixed (we omit details). Thus, by sending y → x in (31) we may interchange the limit and the summation getting Analogously, one proves that For z ∈ C λ 0 the zero of F J (z), the sequences f (z) and g(z) are linearly dependent by Proposition 7. Hence there exists A(z) = 0 such that Since f (z) is an eigenvector of J max by Proposition 11, A(z) = 0, indeed. By differentiating both sides of the equality F J (x) = W (f (x), g(x)) and making use of (34), one obtains Further, by substituting from (32) and (33) in the above equality, one gets Finally, by sending n → −∞ in the above formula, one arrives at (27). To obtain the expression (28) for A(z), it suffices to use (34) and Definition 5.
Remark 13. Note that if the denominator on the RHS of (28) vanishes for some n ∈ Z, then it is not the case for n + 1. Indeed, if which one deduces from the recurrence (6). However, this would contradict the second limit relation in (7).
is given by the formula for all z ∈ ρ(J)\ der(λ), where m := min(i, j) and M := max(i, j). (If z ∈ Ran(λ)\ der(λ), the RHS of (35) is to be understood as the corresponding limit value.) Proof. We divide the proof into 2 parts. 1) Let z / ∈ Z(J ). Such z exists due to the assumption that F J = 0 on C λ 0 . For the sake of simplicity, we will further assume that this z belongs to C λ 0 . The purpose of this assumption is to avoid complicated expressions caused by the necessary regularization if z ∈ Ran(λ)\ der(λ). However, the idea of the proof remains completely the same.
Let G(z) be the doubly infinite matrix whose elements are given by the RHS of (35). First, by employing ideas similar to those used in the proof of Lemma 8, one shows that Vol. 88 (2017) Characteristic Function for Jacobi Matrix 513 for some constant C 1 (z) > 0 and all i, j ∈ Z, where m := min(i, j) and M := max(i, j). Next, the assumptions also guarantee that the expression where the constant C(z) = C 1 (z)C 2 (z) > 0 is independent of the indices i and j. Now, we may introduce the operator R(z) defined as where R(z; s) is the bounded operator acting on 2 (Z) determined by its matrix entries R i,j (z; s) := δ i,j+s G i,j (z), for i, j ∈ Z. By (36), the norm of R(z; s) satisfies Consequently, the series (37) converges in the operator norm and R(z) is well defined bounded operator on 2 (Z) whose matrix in the standard basis coincides with G(z). Note that where f (z) and g(z) are given in Definition 5. Since, according to Proposition 7, f (z) and g(z) are solutions of the eigenvalue equation J u = zu, one readily verifies that on the level of formal matrix product By inspection of domains, the above equalities yield the operators J max − z and R(z) are mutually inverse and hence Consequently, z ∈ ρ(J max ) and we have shown that spec(J max )\ der(λ) ⊂ Z(J ). Taking also into account (25), we get Consequently, if we show that the claim (i) holds true, i.e., J min = J max , the theorem is proved.
2) We show that the assumption of the existence of z ∈ C λ 0 such that F J (z) = 0 implies J min = J max , indeed. It follows from (14) that and 514 F.Štampach IEOT for all z ∈ C.
Let z ∈ C λ 0 be such that F J (z) = 0. Then z ∈ ρ(J max ), as has already been proved in the first part of the proof. Further, by using the second equation in (38), one verifies that Hence J min − z is injective and its left-inverse is R(z). Further, it follows from (41) that Ran(J min − z) is a closed subspace, see [9, Chp. IV, Thm. 5.13], and one has The second equality in the above equation holds since Ker((J min −z) * ) = {0}, which follows from (40) and the injectivity of J max − z. Thus, J min − z is an invertible operator with a bounded inverse that has to coincide with its leftinverse R(z). Taking into account (39), we obtain which implies, in particular, that Dom(J min ) = Dom(J max ).
Remark 15. Theorem 14 has been derived under two assumptions: (i) The convergence condition (8) is fulfilled for at least one z 0 ∈ C λ 0 . (ii) The function F J (z) does not vanish identically on C λ 0 . The assumption (i) is necessary for the definition of the characteristic function F J and is essential. The assumption (ii) guarantees that the matrix J determines the unique Jacobi operator. It might seem hard to decide if the assumption (ii) is fulfilled or not. Let us point out that (ii), if assumed jointly with (i), is not very restrictive and, in applications, it is usually satisfied since der(λ) is typically an empty, 1-point, or 2-point set.
Indeed, (ii) is automatically fulfilled if (i) holds and the Ran λ is contained in a sector z 0 + {z ∈ C | |arg z − θ 0 | ≥ c > 0} for some z 0 ∈ C and θ 0 ∈ [0, 2π). Then the half-line L = z 0 + {re iθ0 | r > 0} is contained in C λ 0 and 1/|λ k − z| tends to 0 monotonically, for all k ∈ Z, as z → ∞, z ∈ L. Similarly as in Remark 2, one derives the estimate It follows that F J (z) → 1 as z → ∞, z ∈ L, and hence (ii) is satisfied. Note also that Ran λ is contained in a sector, if it is contained in a half-plane or, in particular, if it is real.
for some n 0 ∈ N, then (J − z) −1 belongs to the Schatten-von Neumann class S p .
Let z ∈ ρ(J) = C\Z(J ) be such that z ∈ C λ 0 . Such z exists since F J does not vanish identically on C λ 0 by assumptions. By a slight refinement of the estimate (36), one derives that This implies that, for R(z; s) defined in (37), R(z; s) i,j → 0 as i, j → ±∞ with i − j = s ∈ Z being fixed. Consequently, R(z; s) is compact for all s ∈ Z and, since the series (37) converges in the operator norm, R(z) is also compact. By assuming (43) additionally, we show that R(z) ∈ S p . The operator R(z; s)(R(z; s)) * is a diagonal operator with diagonal elements |G i,i+s (z)| 2 , i ∈ Z. Hence, the numbers |G i,i+s (z)|, i ∈ Z, are singular values of the compact operator R(z; s). Taking into account (44) and using the Cauchy-Schwarz inequality, one obtains the estimation for the p-th power of the p-th Schatten-von Neumann norm of R(z; s) in the form: The assumption (43) guarantees the series on the RHS of the above estimate converges. Moreover, it also follows that the series (37) converges in S p and hence R(z) ∈ S p .

The Characteristic Function and the Algebraic Multiplicity
The main aim of this section is to prove that the order of a zero of the characteristic function F J coincides with the algebraic multiplicity of the corresponding eigenvalue. It turns out that the geometric multiplicity of an eigenvalue of the Jacobi operator J under investigation is always one. Consequently, knowledge regarding the algebraic multiplicities of eigenvalues is straightforwardly connected to a possible similarity of J to a diagonal operator. Namely, simplicity of all zeros of F J is a necessary condition for the possible diagonalizability of J.

Preliminaries
If A is a closed operator acting on a Hilbert space H and z ∈ spec p (A), we denote by ν g (z) := dim Ker(A − z) the geometric multiplicity of A. Further, if z is an isolated point of spec(A), the algebraic multiplicity of z is defined as ν a (z) := dim Ran P z , where Proof. First, we show that dim M 1 ≤ 1. There is a standard result that goes back to Wall, see [19,Thm. 22.1], which shows that the following claims are equivalent: (i) The second-order difference equation (J − z)u = 0 has two linearly independent solutions belonging to 2 (Z) for one z ∈ C.
(ii) The second-order difference equation (J − z)u = 0 has two linearly independent solutions belonging to 2 (Z) for all z ∈ C. Note that dim M 1 ≤ 2. If dim M 1 = 2, then by the above equivalence, dim Ker(J − z) = 2 for all z ∈ C which contradicts the assumption ρ(J) = ∅. Hence dim M 1 ≤ 1.
Further, we will need the following purely algebraic statement.

Lemma 18. Let {f (m) } m∈N0 be a sequence of elements of the space of complex sequences such that
and f (0) = 0. Then {f (m) } m∈N0 is linearly independent. In addition, if {g (m) } m∈N0 is another sequence satisfying the equations (45) (with f being replaced by g) and where C n (f, g) := f n g n+1 − f n+1 g n , n ∈ Z, then the implication holds for any k ∈ N 0 .
The last preliminary result is a generalization of Lemma 8 with the spectral parameter z restricted to C λ 0 . It will be used later to show that, for the sequences f (z) and g(z) given in Definition 5, all the derivatives of f (z) are square summable at +∞ and all the derivatives of g(z) are square summable at −∞. Lemma 19. Let λ : Z → C and w : Z → C\{0} be such that (8) holds for at least one z 0 ∈ C λ 0 . Then, for all z ∈ C λ 0 and k ∈ N 0 , one has where P n (z) is given by (15).
Proof. We prove the convergence of the first series only. The verification of the convergence of the second one is analogous.
Let z ∈ C λ 0 . First, the same arguments as in the proof of Lemma 8 show that for all n sufficiently large, where D(z) > 0 is a constant independent of n. Further, note that Vol. 88 (2017) Characteristic Function for Jacobi Matrix 519 where ξ n (z) = n k=1 1 λ k − z .
One readily verifies by mathematical induction in k ∈ N that the kth derivative of P n can be expressed as where p k is a polynomial in ξ n , ξ n . . . , ξ (k−1) n of the form with coefficients m α ∈ Z. Since where δ := dist(z, Ran(λ)) > 0, one deduces from (50), (51), (52), and (49) that there exists C k (z) > 0 such that P (k) n (z) ≤ C k (z)n k 2 −n , for all n sufficiently large. The above estimate gives a summable majorant for the first series from the statement for arbitrary k ∈ N 0 .

The Multiplicity Theorem
Theorem 20. Let the assumptions of Theorem 14 be fulfilled. Then the following claims hold true.
(ii) Suppose additionally that the set C\ der(λ) is connected. Then the set spec(J) ∩ C λ 0 consists of isolated eigenvalues and, if z ∈ C λ 0 ∩ spec(J), then ν a (z) coincides with the order of z as a zero of F J . Moreover, the space of generalized eigenvectors is spanned by vectors f (z), f (z), . . . , f (νa(z)−1) (z).
Next, the fact that spec(J) ∩ C λ 0 contains eigenvalues only follows from the statement (ii) of Theorem 14. If C\ der(λ) is connected, then C λ 0 is clearly connected as well. Further, to arrive at a contradiction, assume that spec(J)∩ C λ 0 has an accumulation point. Then by part (ii) of Theorem 14, the set of zeros of the function F J , which is analytic on C λ 0 , has an accumulation point in C λ 0 . Thus F J has to vanish identically on C λ 0 , a contradiction with the assumption.
Further, let z 0 ∈ C λ 0 ∩ spec(J), then, by Theorem 14 again, F J (z 0 ) = 0. Let us denote by n 0 the order of z 0 as the zero of F J . From the formula (35), one observes that any zero of F J is a pole (or removable singularity) of the Green function of order less or equal to the order of the zero. Thus, any matrix element of the resolvent operator (J −z) −1 has a pole at z 0 of order at IEOT most n 0 . Consequently, for any φ, ψ ∈ 2 (Z), the function z → φ, (J −z) −1 ψ has a pole at z 0 of order at most n 0 . It follows from the last assertion that Indeed, if ν a (z 0 ) > n 0 , then there exists a Jordan chain of J − z 0 of the length at least n 0 + 1, i.e., there are nonzero vectors φ 0 , φ 1 , . . . , φ n0 ∈ Dom J such that From the above equations, one deduces that where z is supposed to be in a neighborhood of z 0 belonging to the resolvent set of J that exists since z 0 is an isolated eigenvalue of J. By putting k = n 0 in (54), one observes that φ 0 , (J − z) −1 φ n0 has a singularity at z 0 of order n 0 + 1, a contradiction.
In the remaining part of the proof, we prove that {f (z 0 ), f (z 0 ), . . . , f (n0−1) (z 0 )} is a linearly independent set of generalized eigenvectors of J corresponding to the eigenvalue z 0 which, together with (53), will conclude the proof of the assertion (ii).
First, we show that f (j) (z) is a square summable sequence at +∞ for arbitrary z ∈ C λ 0 and j ∈ N 0 . To this end, it suffices to note that for all n ∈ N and 0 ≤ i ≤ j, with some K(z) > 0. This holds true since and the convergence is local uniform in z on C λ 0 , as one deduces from the inequality which, in its turn, is obtained similarly as the one from Remark 2. Recalling (18) and using (55), one gets Now, by applying Lemma 19, one concludes that f (j) (z) is a summable and hence also square summable sequence at +∞. Analogously, with the aid of Lemma 19, one verifies that g (j) (z) is a square summable sequence at −∞ for all z ∈ C λ 0 and j ∈ N 0 .
Vol. 88 (2017) Characteristic Function for Jacobi Matrix 521 Further, recall Proposition 7. By differentiating the equation J f (z) = zf (z) with respect to z, one obtains equalities In addition, f (z 0 ) = 0 since it is an eigenvector of J. Clearly, the same holds true if f is replaced by g. Thus, the couple of sequences {f (j) (z 0 )} j∈N0 and {g (j) (z 0 )} j∈N0 fulfills the assumptions of Lemma 18 where J is replaced by J − z 0 . According to this Lemma, the set {f (z 0 ), f (z 0 ), . . . , f (n0−1) (z 0 )} is linearly independent. Further, since F J (z) = W (f (z), g(z)) = w n C n (f (z), g(z)), ∀z ∈ C λ 0 , for n ∈ Z arbitrary, one obtains by differentiation that n (f (z), g(z)), ∀j ∈ N 0 , ∀n ∈ Z, and ∀z ∈ C λ 0 , (57) where C (j) (f (z), g(z)) is as in (46). Since z 0 is a zero of F J of order n 0 , F (j) J (z 0 ) = 0 for 0 ≤ j < n 0 , and hence, by (57), C (j) (f (z 0 ), g(z 0 )) = 0 for 0 ≤ j < n 0 . Thus, Lemma 18 yields Since all the sequences from the span on the LHS of (58) are square summable at +∞ and all the sequences from the span on the RHS of (58) are square summable at −∞ one concludes that Remark 21. Note that, under the assumptions of Theorem 20, any spectral point of J located in C λ 0 is an isolated eigenvalue whose algebraic multiplicity is finite. Moreover, the proof of Theorem 20 together with Lemma 17 actually show that, for z 0 ∈ C λ 0 a zero of F J of order n 0 , it holds where {f (z 0 ),f (z 0 ), . . . ,f (n0−1) (z 0 )} is the set of vectors obtained by the application of the Gram-Schmidt orthogonalization procedure to the set {f (z 0 ), f (z 0 ), . . . , f (n0−1) (z 0 )}.
The following corollary gives a necessary condition for J to be diagonalizable, i.e., similar to a diagonal operator. The notion of diagonalizability of a non-self-adjoint operator deserves a more detailed explanation. Usually, the operator of the similarity transformation is required to be bounded with 522 F.Štampach IEOT a bounded inverse. This yields nontrivial questions concerning basiness of the set of eigenvectors. However, these questions are out of the scope of the current paper, and we do not address them here. Let us only mention that the coincidence of algebraic and geometric multiplicity of all eigenvalues is a necessary condition for an operator to be diagonalizable in any reasonable sense.
Corollary 22. Let the assumptions of Theorem 14 be fulfilled and let C\ der(λ) be connected. If there exists an eigenvalue z ∈ C λ 0 of J such that the corre- Proof. By Theorem 20, ν g (z) = 1. Thus, v(z) = cf (z) for some c = 0. By applying Theorem 14 and Proposition 12, we obtain F J (z) = F J (z) = 0. Consequently, Theorem 20 implies that ν a (z) ≥ 2.
Corollary 23. Let λ : Z → R and w : Z → R\{0} be such that (8) holds for at least one z 0 ∈ C λ 0 and der(λ) = R. Then all the zeros of F J are real and simple.
Proof. Since Ran λ ⊂ R, F J does not vanish identically on C λ 0 , see Remark 15. Further, according to Theorem 14, J max = J min =: J. Since w is also assumed to be real, J is self-adjoint.
Next, note that, if der(λ) = R, then C\ der(λ) is connected. Let z 0 ∈ C λ 0 be a zero of F J . By Theorems 14 and 20, z 0 is an isolated eigenvalue of the self-adjoint operator J and therefore z 0 ∈ R. Moreover, by the self-adjointness of J and the claim (i) of Theorem 20, one has ν a (z 0 ) = ν g (z 0 ) = 1. Hence, according to the claim (ii) of Theorem 20, z 0 is a simple zero of F J .

Diagonals Admitting Global Regularization and Connections with Regularized Determinants IEOT
functions f and g being replaced by their regularized extensionsf andg. For this reason and the sake of brevity, the proof is only indicated.
Proposition 24. Let λ : Z → C and w : Z → C\{0} be such that w ∈ 2 (Z) and λ ∈ p (Z), for some p ∈ N. Then In addition, the algebraic multiplicity ν a (z) of a nonzero eigenvalue z of J coincides with the order of z as a zero ofF J and the space of generalized eigenvectors is spanned by vectorsf (z),f (z), . . . ,f (νa(z)−1) (z).
To verify (64), it suffices to realize that J is compact and hence 0 ∈ spec(J). Indeed, one easily shows that the Jacobi operator J is compact if and only if lim n→±∞ λ n = lim n→±∞ w n = 0.
The above equalities are guaranteed by the assumptions w ∈ 2 (Z) and λ ∈ p (Z). The remaining part of the statement is to be derived by the same way as Theorem 14 where the solutions f and g are replaced by their regularized extensionsf andg.
Let us remark that although 0 ∈ spec(J), 0 need not be an eigenvalue of J. Note also that the matrix elements of the resolvent, see Theorem 14 part (iii), can be now written as for all z ∈ ρ(J). Further, following the same steps as in the proof of Proposition 12 replacing everywhere f (z), g(z), and F J (z) byf (z),g(z), andF J (z), respectively, one arrives at the summation formula for any z = 0 such thatF J (z) = 0, whereÃ(z) =f n (z)/g n (z) for any n ∈ Z such thatg n (z) = 0.
Remark 25. There is a close connection between the regularized characteristic functionF J and the theory of regularized determinants, see [13,Chp. 9]. First, note that J can be decomposed as J = Λ + UW + W U * , where Λe n = λ n e n , W e n = w n e n , and Ue n = e n+1 for all n ∈ Z. The assumptions λ ∈ p (Z) and w ∈ 2 (Z) imply that Λ ∈ S p and W ∈ S 2 . Consequently, J ∈ S p + S 2 ⊂ S max(2,p) since S p ⊂ S q for 1 ≤ p ≤ q ≤ ∞. Thus, if p ≥ 2, the Vol. 88 (2017) Characteristic Function for Jacobi Matrix 525 regularized determinant det p (1 − zJ) is well defined and is an entire function of z. We will show that F J (z) = det p (1 − z −1 J), ∀z ∈ C\{0}.
Let P N stand for the orthogonal projection on the space span{e n | |n| ≤ N }. Without loss off generality, we may assume that (59) holds with p ≥ 2. Then, by the above discussion, J ∈ S p and one has P N JP N → J in S p , as N → ∞. Further, with the aid of the formula for the determinant of a tridiagonal matrix [15,Eq. (13) Now, it suffices to send N → ∞ in the above formula to verify (67), where one has to take into account that det p (1 + ·) is a continuous functional on S p , see [13, Thm. 9.2(c)], and the formula (10). Having the formula (67) at hand, claims of Proposition 24, with the exception of the one about vectors spanning the generalized eigenspace, may be deduced from general results of the theory of regularized determinants [13].
Finally, we illustrate the results derived within this subsection on a concrete example. We follow the standard notation for hypergeometric series, the Bessel function of the first kind, the gamma, and digamma function as it is used, for example, in [1].