On unbounded commuting Jacobi operators and some related issues

Abstract We consider the situations, when two unbounded operators generated by infinite Jacobi matrices, are self-adjoint and commute. It is found that if two Jacobi matrices formally commute, then two corresponding operators are either self-adjoint and commute, or admit a commuting self-adjoint extensions. In the latter case such extensions are explicitly described. Also, some necessary and sufficient conditions for self-adjointness of Jacobi operators are studied.


Introduction
In operator theory, the study of commuting operators remains an important topic [2]. We recall that two self-adjoint (possibly unbounded) operators (or, more generally, two normal operators) are said to commute if their spectral projections commute. In [1] E. Nelson showed an example of two essentially self-adjoint operators A and B in a Hilbert space H having a common dense domain D such that for all x in D, ABx = BAx, but such thatĀ andB do not commute (whereĀ stands for the closure of A). He proved the following su cient conditions for the commutativity of two operators:

Theorem, Nelson . Let A and B be symmetric operators in a Hilbert space H and let D be a dense linear manifold in H such that D is contained in the domain of A , B , AB and BA and such that ABx = BAx for all x ∈ D. If the restriction of A + B on D is essentially self-adjoint then A and B are essentially self-adjoint and A andB commute.
Another proof of Nelson's theorem (which does not use operator representations of Lie algebras) was o ered in the paper of A.E. Nussbaum [3], which contains some results on commutativity of two operators. In case of the Nelson's example, it was shown in [3] (Theorem 5) that for the commutativity ofĀ andB some additional conditions are required:

Theorem, Nussbaum. Let A and B be symmetric operators in a Hilbert space H and let D be a dense linear manifold in H such that D is contained in the domain of A and B, and such that (Ax,By)=(Bx,Ay) for all x and y in D. Let T be the closure of the restriction of A + B to D. If T is self-adjoint and its domain D(T) ⊂ D(Ā). then A and B are essentially self-adjoint andĀ andB commute.
In this paper, we mainly study the commutativity of Jacobi operators, i. e. the operators generated by in nite Jacobi matrices. Note that the commutativity of a certain class of block Jacobi operators (i. e. Jacobi operators with matrix elements) is important in the theory of orthogonal polynomials of several variables and multidi-mensional moment problem ( see [4,8,9]), and there are some open questions there (e. g. the commutativity of block Jacobi operators, which satisfy the Carleman type condition). We believe that our results may be useful for studying the commutativity of such operators and plan to address these issues in a further paper.
The paper is organized as follows. In the section to follow, we consider the general case of Jacobi operators and study the conditions for their self-adjointness, including some criteria. In Section 3, we consider the nonself-adjoint Jacobi operators and o er a procedure for their self-adjoint extension. In the nal section, we study the commutativity of a couple of Jacobi operators generated by commuting Jacobi matrices.
Some results about the self-adjointness of Jacobi operators.
First, we recall some facts from the theory of Jacobi operators, necessary for our further considerations [5][6][7]12]. An in nite Jacobi matrix is de ned to be matrix of the form: where an , bn ∈ R, an > , n ∈ Z+.
To the matrix J we assign the second-order di erence equation: l(y)n := a n− y n− + bn yn + an y n+ = λyn , λ ∈ C, a − = . (1) It has two linearly independent solutions (P i (λ)) ∞ i=− and (Q i (λ)) ∞ i=− with the initial conditions so any solution of (1) is a linear combination of them. The function P i (λ) is a polynomial of degree i in λ and is called a polynomial of the rst kind, while Q i (λ) is a polynomial of degree i − and is called a polynomial of the second kind. Note that P(λ) := (P i (λ)) ∞ i= is a solution of the equation Jy = λy, but Q(λ) := (Q i (λ)) ∞ i= is not: (JQ) = ≠ = λQ .
It follows from (2) that Wn(y, z) does not depend on n; two solutions of (1) are linearly independent if and only if their Wronskian is nonzero. Let y = y(λ) be a solution of (1) with the parameter λ, and z = z(µ) a solution of the same equation with the parameter µ. Then it is easy to verify the Green's formula: where m and N are arbitrary nonnegative integers. Our next aim is to pass from the matrix J to Jacobi operators. For this purpose we introduce the Hilbert space l [ , ∞) consisting of the complex sequences y = (yn) ∞ n= such that ∞ n= |yn| < ∞, with the inner product (y, z) = ∞ n= ynzn. Denote by (en) ∞ n= its standard orthonormal basis. Then we introduce the linear set D of all elements y ∈ l [ , ∞) such that Jy ∈ l [ , ∞). We de ne the operator L on D by the equality Ly = Jy. For y and z from D, similarly to (3) we have The operator L is a closed symmetric operator with de ciency indices (0,0) or (1,1) (it is said that the limitpoint or limit-circle case holds for L ), and the operator L is the adjoint of L . Thus, if the limit-point case holds, then L = L. In the limit-circle case the linear space of solutions Jy = λy from l [ , ∞) is at most onedimensional: y = y P(λ)). The operators L and L are called, respectively, the minimal and maximal (Jacobi) operators generated by the matrix J.
For the operator L , the function ρ(λ) = (E λ e , e ), where E λ is a (generalized) spectral function of L (more details and de nitions can be found in [5,6,17]) yields a solution of the Hamburger moment problem, and we have Thus, we can talk about the Hamburger moment problem associated with L (or with J ). The polynomials P i (λ) are orthogonal with respect to dρ(λ): and for Q i (λ) the following representation holds From the orthogonality of P(λ) and (1) follows that The above formulae give the reconstruction procedure for the matrix J (operator L ) from dρ(λ): namely, rst, applying the Gram-Schmidt orthogonalization process to the sequence (λ i ) ∞ i= , we get P(λ). Then we nd the elements of J by (7). Note that an and bn are the two leading Fourier coe cients (we call them principal ones) of the functions λPn(λ) with respect to the system P(λ).
If the de ciency indices of L are maximal ( (1,1) ), it means that for the Hamburger moment problem associated with L the indeterminate case holds (i. e. it has multiple solutions, see [6] for details). In this case both P(λ) and Q(λ) belong to l [ , ∞) for all complex values of λ. Conversely, if P(λ) and Q(λ) belong to l [ , ∞) for all λ ∈ C, then for all non-real λ the solution of equation JP(λ) = λP(λ) belongs to this space, therefore the de ciency subspaces of L are one-dimensional, which means that the de ciency indices of L are (1,1). This condition is ful lled if at least for one λ = λ ∈ C, both P(λ ) and Q(λ ) are from l [ , ∞) (for the operators generated by Jacobi block matrices (block Jacobi operators) the proof of a similar fact (that the de ciency indices of the block counterpart of L are maximal) is contained in [16], Theorem 2). To prove this "l -property" for P(λ), we rst check that (indeed, the right hand side of this equation is a polynomial of degree i − , and therefore is expandable into a linear combination of P (λ), ..., P i− (λ)). The coe cients K i,j are determined by the Fourier formulae [13]: here we used the representation (6) for Q(λ) and the fact that P i (λ) is orthogonal to polynomials of lower degree. Thus we get Substituting (9) into (6), we nd a similar representation for Q i (λ): The claimed property for P(λ) and Q(λ) then follows from Lemma 1.3.2 of [6]. In view of the above considerations, we obtain the following criterion of self-adjointness for L .
If (10) is ful lled, then As we see, the coe cients K i,j depend on λ . In what follows, unless the other is speci ed, we will set λ = .
For the (minimal) operators generated by block Jacobi matrices, the criterion of maximality for their deciency indices in terms of K i,j (which were de ned similarly to (8)) was established in [14,15]. Utilizing this result for the scalar Jacobi operators L ( whose de ciency indices are at most (1,1) ), we get another criterion of self-adjointness.

Theorem 2. The operator L is self-adjoint i there exists a sequence of intervals of positive integers [n k , m k ]
such that m k ≤ n k+ < m k+ , (k ∈ N) for which the following condition is satis ed: Proof. Assume L is self-adjoint. According to the previous theorem, this implies that As follows from (8) Take an arbitrary sequence [n k , m k ]. We have Taking the square root from both sides of the obtained inequality and summing over k, we get Hence, as follows from (12), the condition (11) is ful lled for [n k , m k ]. Thus the necessity is proved. Now assume that (11) is ful lled for a certain sequence [n k , m k ]. As follows from (13), the condition (12) is ful lled as well. By the previous theorem, the latter implies that L is self-adjoint.
This criterion leads to the following important result about self-adjointness of L . (for p = , we have an equality here). If the series in the right side of this inequality is divergent, then at least one of p series in the left side is divergent too; therefore L is self-adjoint.

Corollary. If for an arbitrary xed p ∈ N
Thus, for di erent values of p, we have a number of su cient conditions for the operator L to be self-adjoint. As noted in [14], some of these conditions coincide with the previously known results on the Jacobi operators (see also [11], where various su cient conditions are studied). For example, for p = we nd using (2) that so the condition (14) in this case becomes Thus, all elements of the martix J except b , as well as Kn+p,n for p > can be expressed through K n+ ,n , K n+ ,n , n ∈ Z+. If we de ne K ,− similarly to (8) we get the formula for b b = − K ,− K , .
As follows from (8) As follows from the above calculations, the set PFC(J,P(λ)) is equivalent to the set of elements of the matrix J : (bn , an) ∞ n= , which, as noted above, are in turn the principal Fourier coe cients of the functions λP(λ) with respect to the system P(λ). Therefore, we may associate the operator L with PFC(J,P(λ)). Now consider the matrix J with zero main diagonal: bn = , n ∈ Z+. We denote by J this matrix and by K i,j the corresponding elements K i,j . Also denote by L the operator L generated by J . It follows from (17)-(18) that K i,i− l = , l = , . . . , [i/ ]. In this case, using (15),(18), we nd the formulae for K i,j in terms of an If we take the matrix J such that |bn| < C < ∞, n ∈ Z+, then one may consider the corresponding operator L as a perturbation of L by a bounded symmetric (bn ∈ R) operator. Then, as known, the de ciency indices of L are the same as of L , and applying the Theorem 2, we come to the following conclusion.
Theorem 3. Assume that the diagonal elements of the matrix J are uniformly bounded. Then, the corresponding operator L is self-adjoint i there exists a sequence of intervals of positive integers [n k , m k ] such that m k ≤ n k+ < m k+ , (k ∈ N) for which the following condition is satis ed: where K i,j are calculated by (21).
Note that the claim of this theorem remains valid for the operators A = L +B, where B is an arbitrary bounded Hermitian operator in l [ , ∞) de ned on all the space.
Here we consider the non-self-adjoint case for L . As mentioned in the previous section, the de ciency indices of this operator are then equal to (1,1).
Since the numbers [Q(λ ), z]∞ and [P(λ ), z]∞ can be arbitrary, the latter equality is ful lled if and only if [y, P(λ )]∞ and [y, Q(λ )]∞ are equal to zero, and we come to the following conclusion. Note that for λ = this theorem was proved in [5] (Theorem 1.1). Now we recall the de nition of a space of boundary values (SBV) of an operator (see e. g. [10], page 155). In particular, this concept is used in study of extensions of the symmetric operators with equal de ciency indices. Namely, let A be a closed symmetric operator on H. Then, a triple (H, Γ , Γ ), whereH is a Hilbert space and Γ , Γ are linear mappings of D(A * ) intoH, is called a boundary value space of the operator A, if: It is known, that for any operator A with de ciency indices (n, n), n ≤ ∞ there exists a space of boundary values (H, Γ , Γ ), such that dimH = n (see [5], Chapter 3). Thus, for the operators L in the limit-circle case, one may takeH = C. It follows from (4)  Applying to our case the Theorem 1.6 of [10] (page 156), which describes the extensions of symmetric operators in terms of their SBV, we get the following result.
Theorem 5. If the de ciency indices of the operator L are (1,1), then for any λ ∈ R there exists a self-adjoint extension L ext (λ , h) of L , determined by the equality L ext (λ , h) = L on the elements y ∈ D satisfying the boundary condition: where h is a real number or in nity. In the latter case, the condition (24) takes the form y, P(λ ) ∞ = .
Also note that according to the above mentioned theorem, every self-adjoint extension of L can be described in terms of the Theorem 5 (see [10] for the details).

Commuting Jacobi operators
We start with two Jacobi matrices: As in the previous section, denote by L and L the minimal closed symmetric operators in l [ , ∞), generated by J and J respectively. Also, denote by P (λ) = (P i (λ)) ∞ i= , Q (λ) = (Q i (λ)) ∞ i= and P (λ) = (P i (λ)) ∞ i= , Q (λ) = (Q i (λ)) ∞ i= the systems of polynomials of the rst and the second kind, corresponding to J and J . The formal matrix commutativity J J = J J (which amounts to commutativity of L and L on the set D ) yields a n = ka n , b n = kb n − b; For the elements of PFC(J ,P (λ)) = ({K n,n− }, {K n,n− }) ∞ n= and PFC(J ,P (λ)) = ({K n,n− }, {K n,n− }) ∞ n= we nd, using (15), (17), that K n,n− = k K n,n− ; n ∈ N; K n,n− = k K n,n− + λ a n− a n− = k (K n,n− + λ K n,n− K n,n− ).
From (26) and the Theorem 1 follows, that the de ciency indices of L and L are the same, in particular, if L is self-adjoint, then so is L , and vice versa. In the latter case, the commutativity of L and L follows from Also note that if L and L are self-adjoint, then we nd from (25) that L = kL + bE, where E is the identity operator, and the commutativity of L and L can be easily derived from the spectral theorem for self-adjoint operators. Now consider the non-self-adjoint case for L and L . We assume that for J and J the conditions (25) are satis ed. Further on it will be shown that L and L , admit a commuting self-adjoint extensions. As in the self-adjoint case, we have that L = kL + bE, In view of the above, we can conclude that L and L admit a commuting self-adjoint extensions. Thus we obtain the main result of this section. As to the theorem of Nussbaum, in the limit-circle case it cannot be applied directly since the operator T, de ned as the sum of L and L , is not self-adjoint. However, the proof of Nussbaum's theorem relies on the following proposition from the same paper [3]: Thus, all conditions of the above proposition are ful lled, so it is applicable for our case.