A criterion for essential self-adjointness of a symmetric operator defined by some infinite hermitian matrix with unbounded entries

We shall consider a double infinite, hermitian, complex entry matrix $A=[a_{x,y}]_{x,y\in\mathbb Z}$, with $a_{x,y}^*=a_{y,x}$, $x,y\in\mathbb Z$. Assuming that the matrix is almost of a finite bandwidth, i.e. there exists an integer $n>0$ and exponent $\gamma\in[0,1)$ such that $ a_{x,x+z}=0$ for all $z>n\langle x\rangle^{\gamma}$ and the growth of the $\ell_1$ norm of a row is slower than $|x|^{1-\gamma}$ for $|x|\gg1$, i.e. $\lim_{|x|\to+\infty}| x|^{\gamma-1}\sum_{y}|a_{xy}|=0$ we prove that the corresponding symmetric operator, defined on compactly supported sequences, is essentially self-adjoint in $\ell_2(\mathbb Z)$. In the case $\gamma=0$ (the so called $(nJ)$-matrices) we prove that there exists $c_*>0$, depending only on $n$, such that the condition $\limsup_{|x|\to+\infty}| x|^{-1}\sum_{y}|a_{xy}|\le c_*$ suffices to conclude essential self-adjointness.


Introduction
We shall consider a double infinite, hermitian, complex entry matrix A = [a x,y ] x,y∈Z , with a * x,y = a y,x , x, y ∈ Z.Here a * denotes the complex conjugate of a ∈ C. We assume furthermore that the matrix is almost of a finite bandwidth, i.e. there exists an integer n ≥ 1 and exponent γ ∈ [0, 1) such that a x,x+z = 0 for all z > n x γ . (1.1) Here, for given a we let a := (1 + |a| 2 ) 1/2 .With the help of matrix A we can define a symmetric operator on the subset c 0 (Z) of the complex Hilbert space ℓ 2 (Z) -the space consisting of all double infinite sequences f = (f x ) equipped with the norm Here c 0 (Z) is the subspace containing all compactly supported f .The operator is given by (Af ) x := y a xy f y , x ∈ Z, f ∈ c 0 (Z). (1.2) According to Theorem 4, p. 102 of [1], assumption (1.1) implies that the operator is closable.Denote its closure by Ā : D( Ā) → ℓ 2 (Z).In our principal result, see Theorem 2.1 below, we formulate a sufficient condition, in terms of the growth of |a xy |, see (2.4) below, for the operator Ā to be self-adjoint.The above means that the deficiency index of A : c 0 (Z) → ℓ 2 (Z) equals (0, 0), see [10].Note that in the particular case when γ = 0 we have and the definition coincides with the usual definition of (nJ)-matrices, see [13], (sometimes also called finite bandwidth matrices).When n = 1 they are called Jacobi matrices and play an important role in the theory of Hamburger moment problem.This case has been well studied in the literature, see e.g.[5,6,7,8,12] and the references contained therein, although also then our results formulated in Corollary 2.2 and Theorem 2.3 below seem to be new.

The statement of the main result
Since matrix A = [a x,y ] is hermitian the operator Ā is obviously symmetric, i.e. ( Here, •, • ℓ 2 (Z ) denotes the usual scalar product in ℓ 2 (Z).Let f ∈ ℓ 2 (Z) be such that the functional is bounded, i.e. for some C > 0 Self-adjointness of Ā means that any f , for which (2.3) holds, belongs to D( Ā) and, as a consequence, (2.1) is in force.For example, if there exists M > 0 such that y |a xy | ≤ M for all x ∈ Z then Ā is bounded on ℓ 2 (Z), see Example III.2.3, p. 143 of [3], therefore it is self-adjoint.Our main result can be stated as follows.(2.4) Then, operator A, given by (1.2), is essentially selfadjoint on ℓ 2 (Z).
Using the theorem for γ = 0 we immediately conclude the following.(2.6) Example.The condition (2.6) is in some sense optimal.Suppose that δ > 1 is arbitrary.Consider the Jacobi matrix with entries given by x δ , if x > 0 and z = 1.

Proof of Theorem 2.1
Recall the classical criterion for the essential self-adjointness of a symmetric operator, see Theorem 3 of Section 33.2 of [9], or Corollary 2.2 of [12].Adjusted to our settings it reads as follows: suppose that a closed operator Ā is symmetric and Then, it is self-adjoint.
(3.4) Combining this with (1.1) we conclude that a x,x+z = 0 for all |z| > c n x γ . (3.5) Furthermore, note that any f satisfying conditions (3.2) belongs to D( Ā).Indeed, consider f . Thanks to the second condition of (3.2) and (2.4) we can easily argue that (I − iA)f (N ) → g and f (N ) → f , strongly in ℓ 2 (Z), as N → +∞.Since Ā is the closure of A we conclude that f ∈ D( Ā) and (I − i Ā)f = g.Due to the fact that c 0 (Z) is dense in ℓ 2 (Z) and that the range R(I − i Ā) is closed we conclude that R(I − i Ā) = ℓ 2 (Z).The proof of the second equality in (3.1) goes along the same lines.What yet remains to be shown is therefore (3.2).
Proof of (3.2).Let For a fixed integer N define A (N ) as a bounded, symmetric operator corresponding to the hermitian matrix whose entries equal We show that for any positive integer k there exists a constant C > 0 such that Taking this claim for granted (its proof shall be shown momentarily) we finish the proof of (3.2).Using condition (3.7) with any k > 0 we conclude that the tails of the infinite sums defining the ℓ 2 (Z) norms of ( f (N ) ) are uniformly small in N.This proves that the sequence is strongly precompact in ℓ 2 (Z), see e.g.Theorem 4.20.1 of [2].In fact, observe that each f ) for an integer M ≥ 1.Of course f (N,M ) ∈ c 0 (Z) ⊂ D( Ā), and it converges to f (N ) strongly in ℓ 2 (Z), as M → +∞.On the other hand, from (3.7) for any k, c > 0 there exists C > 0 such that Using (3.5), we can estimate (3.9) Since γ ∈ [0, 1) condition |y − x| ≤ c n x γ implies that there exists c > 0 such that |y| ≥ c|x| for all x, y ∈ Z. Thanks to (3.8) the utmost right hand side of (3.9) can be estimated then by (3.10)This together with (2.4) imply that there exists C 1 > 0 such that (3.11) In consequence (A f (N,M ) ), M ≥ 1 is strongly precompact in ℓ 2 (Z), for a fixed N, and since Ā is the closure of A we obtain f (N ) ∈ D( Ā) and In addition, we also infer that and that for any k > 0 there exists a constant C > 0 From (3.7) and (3.13) we conclude that both sequences ( f (N ) ) and ( Ā f (N ) ) are strongly precompact in ℓ 2 (Z).Choosing a suitable subsequences if necessary we can assume with no loss of generality that f (N ) → f and Ā f (N ) converges to some h strongly in ℓ 2 (Z), as N → +∞.Then f ∈ D( Ā) and Āf = h.In light of (3.6), to finish the proof of (3.2) it suffices to show that lim Estimating as in (3.9) and (3.10) we obtain x as N → +∞.The passage to the limit on the utmost right hand side can be argued easily by virtue of the Lebesgue dominated convergence theorem.This ends the proof of (3.2), modulo the fact that estimate (3.7) still requires to be shown.Its proof is an adaptation to the present case of an argument used in [11], see also Section 2.7.4 of [4].Define a bounded operator T : ℓ 2 (Z) → ℓ 2 (Z) by the formula T f x := t x f x , where and 0 < X < Y are some constants to be determined later on.Directly from (3.15) it follows that Applying T to both sides of (3.6) and taking inner product against T f (N ) on both sides of the aforementioned equation we conclude that where [T, T is the commutator of T and A (N ) .Thanks to symmetry of A (N ) we have Here Re z and Im z denote the real and imaginary parts of a complex number z. Taking the real part of the expressions appearing on both sides of (3.17) we obtain The above expression can be bounded as follows (3.19) Applying Young's inequality we can estimate the right hand side of (3.19) by I 1 + I 2 , where Using condition (3.5) and the fact that t x is constant for |x| ≤ X, or Y ≤ |x| we conclude that where X := X − c n X γ and Ȳ := Y + c n Y γ .Thanks to (3.16) we can estimate Observe that Choose an arbitrary δ ∈ (0, 1).Note that for |m| ≤ n x γ there exist constants C, C ′ > 0 such that for all x.Combining (3.23) with (2.4) we conclude that for any δ ∈ (0, 1) there exists X, depending on parameters n, δ, k, γ, such that This together with (3.22) imply that for some constant C * > 0 that depends only on n, γ, k, we conclude that Choose X sufficiently large so that As a result we conclude Combining this with (3.25) we have Going back to (3.18) we obtain Now, let parameter Y , appearing in the definition of the operator T , tend to infinity.Since T g ℓ 2 (Z ) remains constant, starting with some sufficiently large Y (as g ∈ c 0 (Z)) we infer that (3.27) implies (3.7).This ends the proof of (3.2) finishing also the proof of Theorem 2.1.

Proof of Theorem 2.3
The argument used in the proof of Theorem 2.1 can be adapted to the present case, provided we are able to show that (3.2) holds for some fixed k 0 > 2. To prove this fact we repeat with no changes, except replacing c n by n (maintaining the notation from the previous section), the calculations made between (3.15) and (3.21).Instead of (3.23) we write that for some constant C > 0 1 + (x + m) 2  1 + x 2 Using the above together with (2.4) we conclude that for any ǫ > 0 there exists X (appearing in the definition of operator T ), depending on n, such that x k 0 y k 0 − 1 ≤ c * + ǫ, for |y| ≥ X.
Here X := X − n.This leads to an estimate