Manuscript: September, 20 1996 MEASURES WITH FINITE INDEX OF DETERMINACY

OR A MATHEMATICAL MODEL

FOR DR. JEKYLL AND MR. HYDE

Christian Berg^{1} and Antonio J. Duran^{2}

1 Københavns Universitet 2 Universidad de Sevilla

Abstract. In this note measures with finite index of determinacy (i.e. determinate measures μ for which there exists a polynomial p such that |p|^{2}μ is indeterminate), are characterizated in terms of the operator associated to its Jacobi matrix. Using this characterization, we show that such determinate measures with finite index of determinacy (Jekyll) turn out to be indeterminate (Hyde) when considered as matrices of measures.

1. Results.

By M^{∗} we denote the set of positive measures μ on ℝ having moments of every order and infinite support. A measure μ ∈ M^{∗} is determinate if no other measure has the same moments as those of μ, otherwise μ is indeterminate. With μ ∈ M^{∗} we can associate the sequence (p_{n})_{n} of orthonormal polynomials. We always assume that p_{n} is of degree n with positive leading coefficient, and this condition together with orthonormality determines (p_{n})_{n} uniquely from μ.

The sequence of polynomials (p_{n})_{n} satisfies the so-called three-term recurrence formula

tp_{n}(t) = a_{n+1}p_{n+1}(t) + b_{n}p_{n}(t) + a_{n}p_{n−1}(t),

where a_{n} > 0, b_{n} ∈ ℝ and p_{−1}(t) = 0. The well-known theorem of Favard establishes that this algebraic relation characterizes the orthonormality of (p_{n})_{n} with respect to a positive measure.

The so-called Jacobi matrix is defined from the three-term recurrence relation as follows:

J = ( \begin{matrix}b_{0} & a_{1} & 0 & 0 & 0 \\ a_{1} & b_{1} & a_{2} & 0 & 0 \\ 0 & a_{2} & b_{2} & a_{3} & 0 \\ & & ... & ... & ... \end{matrix} ) .

It is the matrix representation of the operator of multiplication by t in the space ℙ of complex polynomials with respect to the orthonormal basis (p_{n}).

Typeset by AMS-TEX

This work has been partially supported by DGICYT ref. PB93-0926

1991 Mathematics Subject Classification. 42C05, 44A60.

The Jacobi matrix J as well as its powers J^{k} define symmetric operators (possibly unbounded) in ℓ^{2} with domain equal to the space of sequences with only finitely many non-zero terms.

It is well-known that the deficiency index of J is either (0, 0) or (1, 1) corresponding to the determinate resp. indeterminate case. We recall that a densely defined symmetric operator is called essentially self-adjoint, if its closure is self-adjoint. This is equivalent to the deficiency index being (0, 0).

The index of determinacy of a determinate measure μ was introduced and studied by the authors in [BD1]. This index checks the determinacy under multiplication by even powers of |t − z| for z a complex number, and it is defined as

ind_{z}(μ) = sup{k ∈ ℕ | |t − z|^{2k}μ is determinate}.

Using the index of determinacy, determinate measures can be classified as follows:

If μ is constructed from an N-extremal measure ν (i.e. ν is indeterminate and the linear space of polynomials is dense in L^{2}(ν)) by removing the mass at k + 1 points in the support of ν, then μ is determinate with

ind_{z}(μ) = \{ \begin{matrix}k, & for z ∉ supp(μ) \\ k + 1, & for z ∈ supp(μ). \end{matrix}  (1.1)

For an arbitrary determinate measure μ the index of determinacy is either infinite for every z, or finite for every z. In the latter case the index has the form (1.1), and μ is derived from an N-extremal measure by removing the mass at k + 1 points.

Using that the index of determinacy is constant at complex numbers outside of the support of μ, we define the index of determinacy of μ by

ind(μ) := ind_{z}(μ), z ∉ supp(μ).

We stress that a measure μ of finite index is discrete and ind(μ) + 1 is the smallest degree of a polynomial p such that |p|^{2}μ is indeterminate.

The purpose of this note is to prove the following characterization of a measure μ with finite index of determinacy in terms of its associated Jacobi matrix J:

Theorem 1. Let μ be a determinate measure. Then the following conditions are equivalent:

(1) μ has finite index of determinacy ind(μ) = k.

(2) The operators J, J^{2}, …, J^{k+1} are essentially self-adjoint and the operator J^{k+2} is not essentially self-adjoint.

(3) For any a ∈ ℝ, the operators J + aI, J^{2} + aI, …, J^{k+1} + aI are essentially self-adjoint and the operator J^{k+2} + aI is not essentially self-adjoint.

If the conditions are fulfilled then J^{n} + aI is not essentially self-adjoint for n ≥ k + 2 and a ∈ ℝ.

We now show that a determinate measure with finite index of determinacy may turn out to be indeterminate when considered as a matrix of measures.

Let us put B_{μ} for the inner product defined by μ ∈ M^{∗}, i.e.

B_{μ}(p, q) = ∫ p(t)\overline{q(t)} dμ.

This inner product has other representations using positive definite matrices of measures. Indeed, for N ∈ ℕ we define the matrix of measures W_{μ} = (μ_{i,j})_{i,j=0}^{N−1} by μ_{i,j} = ψ_{N}(t^{i+j}μ), the image measure under ψ_{N}(t) = t^{N}, i.e.

μ_{i,j}(A) = ∫_{ψ_{N}^{−1}(A)} t^{i+j}d μ(t) for A ∈ 𝔹,

the Borel subsets of ℝ. It is easy to see that W_{μ} is positive definite, i.e. for any A ∈ 𝔹 the numerical matrix W_{μ}(A) is positive semidefinite. To any positive definite N × N matrix of measures W = (μ_{i,j})_{i,j=0}^{N−1} we associate the (possibly degenerate) inner product B_{W} defined by

B_{W} (p, q) = ∑_{m,m′=0}^{N−1} ∫ R_{N,m}(p)\overline{R_{N,m′}(q)} dμ_{m,m′} for p, q ∈ ℙ,  (1.2)

where the operators R_{N,m} : ℙ → ℙ, 0 ≤ m ≤ N − 1, are defined by

R_{N,m}(p) = ∑_{n} \frac{p^{(nN+m)}(0)}{(nN + m)!}t^{n},

(i.e. for every m, the operator R_{N,m} takes from the polynomial p just those powers t^{k}, for which k ≡ m(modN) and then changes t^{nN+m} to t^{n}).

If W = W_{μ} with μ ∈ M^{∗} then from the definition of R_{N,m}, it is easy to see that

∫ p(t)\overline{q(t)} dμ = ∑_{m,m′=0}^{N−1} ∫ R_{N,m}(p)\overline{R_{N,m′}(q)} d μ_{m,m′},  (1.3)

i.e. B_{μ} = B_{W}. (We notice that the construction of the matrix of measures W_{μ} from μ was incorrect in the introduction of [D2]).

Inner products B_{W} defined from a positive definite matrix of measures W as in (1.2) are characterized by the operator of multiplication by t^{N} being symmetric for them, and so their sequence of orthonormal polynomials satisfies a (2N + 1)- term recurrence relation instead of the three-term recurrence formula satisfied by orthonormal polynomials with respect to a positive measure (see [D1] and [D2]). These inner products yield the important link between orthogonal matrix polynomials and scalar polynomials satisfying higher order recurrence relation (see [DV]).

A surprising consequence of Theorem 1 is the following:

Theorem 2. Let μ be a determinate measure with finite index of determinacy ind(μ) = k.

Then, for N = 1, 2, …, k + 1 the matrix of measures W_{μ} is the only positive definite N × N matrix of measures, such that the inner product B_{μ} is of the form (1.2). For N ≥ k + 2 there is an infinite convex set of positive definite N × N matrices of measures, such that the inner product B_{μ} is of the form (1.2).

To see this we use results linking determinacy of N × N matrices of measures and self-adjoint extensions of J^{N}. These results are implicitly in [D2], but for the convenience of the reader we shall state them and give complete proofs.

For each N the matrix J^{N} is symmetric with zeros outside the 2N + 1 main diagonals. It is the matrix of the operator of multiplication with t^{N} in ℙ with respect to the orthonormal basis (p_{n}) in the inner product B_{μ}.

Let us consider the more general situation of an inner product B on ℙ with the following properties for p, q ∈ ℙ

(a) B(\overline{p}, \overline{q}) = \overline{B(p, q)},

(b) B(t^{N}p, q) = B(p, t^{N}q).

We denote by W (N) the convex set of positive definite N × N matrices of measures W = (μ_{i,j})_{i,j=0}^{N−1} representing B in the sense of (1.2). It is of course assumed that every polynomial is integrable with respect to every μ_{i,j}. For this it is necessary and sufficient that the trace τ(W) = ∑ μ_{i,i} has moments of every order.

By (b) the operator M : ℙ → ℙ given by Mp = t^{N}p is symmetric in the Hilbert space ℙ̂, the completion of ℙ with respect to B. Because of (a) the deficiency spaces Δ^{+} = [(t^{N} + i)ℙ]^{⊥} and Δ^{−} = [(t^{N} − i)ℙ]^{⊥} are isometric, so M has equal deficiency indices. This means that M has self-adjoint extensions in ℙ̂, and this implies, as we shall see below, that the set W (N) is non-empty. The following result proves Theorem 2.

Theorem 3. The operator M is essentially selfadjoint in ℙ̂ if and only if W (N) is a singleton.

Proof: For each self-adjoint extension T of M in ℙ̂ we consider the (projectionvalued) spectral measure E of T and define W_{T} = (μ_{i,j})_{i,j=0}^{N−1} by

μ_{i,j}(A) = ⟨E(A)t^{i}, t^{j}⟩, i, j = 0, …, N − 1, A ∈ 𝔹(ℝ),  (1.4)

where ⟨·, ·⟩ is the inner product on ℙ̂ extending B. Then W_{T} is a positive definite matrix of measures because

∑_{m,m′=0}^{N−1} μ_{m,m′}(A)c_{m}\overline{c_{m′}} = ⟨E(A)p, p⟩ ≥ 0

with p(t) = ∑ c_{m}t^{m}.

Since ℙ ⊂ D(T^{n}) for all n, we see that each diagonal measure μ_{m,m} has moments of every order and furthermore that for any Borel set A

⟨T^{n}E(A)t^{m}, t^{m′}⟩ = ∫_{A} s^{n} dμ_{m,m′}(s), m, m′ = 0, …, N − 1.  (1.5)

Finally, if

p(t) = ∑_{n≥0} ∑_{m=0}^{N−1} a_{n,m}t^{nN+m}, q(t) = ∑_{n≥0} ∑_{m=0}^{N−1} b_{n,m}t^{nN+m}

we find using (1.5)

B(p, q) = ∑_{n,n′≥0} ∑_{m,m′=0}^{N−1} a_{n,m}\overline{b_{n′,m′}}B(t^{nN+m}, t^{n′N+m′})

= ∑_{n,n′≥0} ∑_{m,m′=0}^{N−1} a_{n,m}\overline{b_{n′,m′}}⟨T^{n+n′}t^{m}, t^{m′}⟩

= ∑_{m,m′=0}^{N−1} ∫ R_{N,m}(p)\overline{R_{N,m′}(q)} dμ_{m,m′},

showing that W_{T} ∈ W (N), so in particular W (N) is non-empty.

We next claim that different self-adjoint extensions of M lead to different matrices of measures. In fact, by (1.5) we have

∫_{A} s^{n+n′} dμ_{m,m′}(s) = ⟨E(A)t^{nN+m}, t^{n′N+m′}⟩,

and since the polynomials t^{nN+m}, n ≥ 0, m = 0, …, N − 1, span ℙ, which is dense in ℙ̂, it follows that the spectral measure E and hence T is uniquely determined from the matrix W_{T}.

This shows that W (N) is an infinite set if M is not essentially self-adjoint.

Assume finally that M is essentially self-adjoint, so the closure \overline{M} is the only self-adjoint extension of M in ℙ̂. We shall show that W\overline{M} is the only element in W (N).

Let W = (μ_{i,j}) be an arbitrary element in W (N). With W is associated a Hilbert space L^{2}(W) of equivalence classes of ℂ^{N}-valued Borel functions f = (f_{0}, …, f_{N−1}) in the following way: all the measures μ_{i,j} in W are absolutely continuous with respect to the positive trace measure τ(W) = ∑_{i=0}^{N−1} μ_{i,i}. Denoting ϕ_{i,j} the Radon-Nikodym derivative of the measure μ_{i,j} with respect to the trace measure τ(W), then |ϕ_{i,j}| ≤ 1, 0 ≤ ϕ_{i,i}, and a vector valued function f = (f_{i})_{i=0}^{N−1} belongs to L^{2}(W) if and only if the function ∑_{i,j=0}^{N−1} f_{i}\overline{f_{j}}ϕ_{i,j} is integrable with respect to the trace measure τ(W). The inner product is given by

⟨f, f⟩ = ∫ ∑_{i,j=0}^{N−1} f_{i}(t)\overline{f_{j}}(t)ϕ_{i,j}(t) dτ(W)(t).

(There is a misprint in the first line of p. 96 in [D2] which has been put right in the definition of L^{2}(W) given here).

Let us consideer the following operator of multiplication T in L^{2}(W):

D(T) = {f ∈ L^{2}(W) | tf ∈ L^{2}(W)}, T f = tf = (tf_{0}, …, tf_{N−1}),  (1.6)

which is easily seen to be self-adjoint.

The linear mapping j : ℙ → L^{2}(W) defined by

j(p) = (R_{N,0}(p), …, R_{N,N−1}(p))  (1.7)

is an isometry, so it has a unique linear isometric extension ĵ of ℙ̂ into L^{2}(W). We claim that it is onto. We remark first that ĵ is intertwining for \overline{M} and T, i.e.

ĵ(\overline{M}(ξ)) = T (ĵ(ξ)) for ξ ∈ D(\overline{M}).  (1.8)

This is simply because

j(M(p)) = (R_{N,0}(t^{N}p), …, R_{N,N−1}(t^{N}p)) = (tR_{N,0}(p), …, tR_{N,N−1}(p)).

By the self-adjointness of \overline{M} we know that (\overline{M} −λI)(D(\overline{M})) = ℙ̂ for all λ ∈ ℂ\ℝ. For each such λ and m = 0, …, N − 1 there exists an element ξ_{λ,m} ∈ D(\overline{M}) ⊂ ℙ̂ so that (\overline{M} − λI)(ξ_{λ,m}) = t^{m}, hence by (1.8)

(T − λI)(ĵ(ξ_{λ,m})) = ĵ((\overline{M} − λI)(ξ_{λ,m})) = j(t^{m}) = (0, …, 1, …, 0)

with 1 at the m’th place. If we put ĵ(ξ_{λ,m}) = (φ_{0}, …, φ_{N−1}), we see that

φ_{m′}(t) = \{ \begin{matrix}1/(t − λ) for m′ = m \\ 0 for m′ ≠ m. \end{matrix}

Suppose now that f ∈ L^{2}(W) is orthogonal to the closed subspace ĵ(ℙ̂). In particular f ⊥ ĵ(ξ_{λ,m}), hence

∫ ∑_{m′=0}^{N−1} \frac{f_{m′}(t)ϕ_{m′,m}(t)}{t − \overline{λ}} dτ(W)(t) = 0,

for λ ∈ ℂ\ℝ and m = 0, …, N−1. By the uniqueness of the Stieltjes transformation we get ∑_{m′=0}^{N−1} f_{m′}ϕ_{m′,m}τ(W) = 0 for each m, and in particular

∫ ∑_{m′=0}^{N−1} f_{m′} \overline{f_{m}}ϕ_{m′,m} dτ(W) = 0,

which implies that

⟨f, f⟩ = ∫ ∑_{m′,m=0}^{N−1} f_{m′} \overline{f_{m}}ϕ_{m′,m} dτ(W) = 0,

thereby showing that f = 0 in L^{2}(W), so ĵ is surjective. The self-adjoint operators \overline{M} and T are unitarily equivalent under ĵ, and in particular their spectral measures satisfy

ĵ ∘ E\overline{M}(A) = E_{T} (A) ∘ ĵ

for any Borel set A. The spectral measure for T is given as the multiplication operator E_{T} (A)f = 1_{A}f, and we therefore have for m, m′ = 0, …, N − 1

μ_{m,m′}(A) = ⟨E_{T} (A)ĵ(t^{m}), ĵ(t^{m′})⟩ = ⟨ĵ(E\overline{M}(A)t^{m}), ĵ(t^{m′})⟩
= ⟨E\overline{M}(A)t^{m}, t^{m′}⟩,

which shows that W (N) only consists of the matrix of measures associated with the self-adjoint extension \overline{M}.

Remark. When M is not essentially self-adjoint, say of deficiency index (k, k) with 0 < k (and necessarily k ≤ N), then the self-adjoint extensions T of M in ℙ̂ are parametrized by the unitary group U(k). The corresponding matrices of measures W_{T} do not fill up the whole set W (N). To obtain them all one has to consider triples (j, H, T) consisting of a Hilbert space H, an isometric linear injection j : ℙ → H and a self-adjoint operator T in H with domain D(T) ⊃ j(ℙ) such that T ∘j = j∘M. For such a tripel the matrix W = (μ_{m,m′}) given by

μ_{m,m′}(A) = ⟨E_{T} (A)j(t^{m}), j(t^{m′})⟩

belongs to W (N). (Here E_{T} is the spectral measure of T.) Any matrix of measures W ∈ W (N) can be realized in this way by using H = L^{2}(W), T and j given by (1.6) and (1.7) respectively. For the matrix of measures W_{T} associated with a selfadjoint extension T of M in ℙ̂ we have ĵ(ℙ) = L^{2}(W), as is easily seen by the proof in Theorem 3. On the other hand, if for W ∈ W (N) we have that j(ℙ) is dense in L^{2}(W), then the operator of multiplication T in L^{2}(W) is unitarily equivalent with a self-adjoint extension of M in ℙ̂.

2. Proof of Theorem 1.

We start with the following simple observation:

Lemma. Let μ ∈ M^{∗}, p_{0} ∈ ℙ for which p_{0}(t) ≠ 0 if t ∈ supp(μ). Then p_{0}ℙ is dense in L^{2}(μ) if and only if ℙ is dense in L^{2}(|p_{0}|^{2}μ).

Proof: The operator A : L^{2}(|p_{0}|^{2}μ) → L^{2}(μ) defined by Af = p_{0}f is an isometry which is onto and A(ℙ) = p_{0}ℙ. Therefore ℙ is dense in L^{2}(|p_{0}|^{2}μ) if and only if p_{0}ℙ is dense in L^{2}(μ).

We now prove Theorem 1:

We first prove (1) ⇔ (2).

Let us consider the Hilbert space ℓ^{2} of complex square summable sequences and the Jacobi matrix J associated to the measure μ. We also let J denote the operator defined by this matrix on ℓ^{2} with domain equal to the subspace F of sequences with only finitely many non-zero terms, i.e.

J : F → ℓ^{2}, J((a_{n})_{n}) = (a_{0}, a_{1}, …)J.

Then J is a symmetric operator with dense domain. As usual (e_{n})_{n} will denote the canonical basis of ℓ^{2}.

In the space of polynomials ℙ, we consider the inner product B_{μ} defined by the measure μ. Then the linear operator G : ℙ → ℓ^{2} defined by G(p_{n}) = e_{n} is an isometry from ℙ endowed with the inner product B_{μ} to F, and from the three-term recurrence relation it follows that G^{−1} ∘ J ∘ G = T is the operator of multiplication by t defined in ℙ. Therefore it is equivalent to prove Theorem 1 for the operator T instead of J. Note that ℙ is dense in L^{2}(μ), so ℙ̂ is isometrically isomorphic to L^{2}(μ).

We stress that the operator T^{N} is essentially self-adjoint if and only if (t^{N} + i)ℙ is dense in L^{2}(μ).

A measure μ has ind(μ) = k if and only if (1 + t^{2})^{j}μ is determinate for j = 0, 1, …, k but not for j = k+1. Taking into account that a measure μ is determinate if and only if the polynomials are dense in L^{2} ((1 + t^{2})μ) (see [R]), it follows that ind(μ) = k if and only if ℙ is dense in L^{2} ((1 + t^{2})^{j+1}μ) for j = 0, 1, …, k but not for j = k + 1. Since

1 ≤ \frac{(1 + x^{2})^{j}}{1 + x^{2j}} ≤ 2^{j−1}, j ≥ 1,

we see that ℙ is dense in L^{2} ((1 + t^{2})^{j}μ) if and only if it is dense in L^{2} ((1 + t^{2j})μ). Therefore ind(μ) = k if and only if ℙ is dense in L^{2} ((1 + t^{2j})μ) for j = 1, …, k + 1 but not for j = k + 2. From the Lemma, we get that ind(μ) = k if and only if (t^{j} + i) ℙ is dense in L^{2}(μ) for j = 1, …, k +1 but not for j = k +2, i.e. ind(μ) = k if and only if T^{j} is essentially self-adjoint for j = 1, …, k + 1 but not for j = k + 2.

We next prove (2) ⇔ (3) . For this it suffices to remark that for any a ∈ ℝ and j ∈ ℕ, there exist two constants C, c > 0 for which

c ≤ \frac{(a + x^{j})^{2} + 1}{1 + x^{2j}} ≤ C for all x ∈ ℝ.

We finally note that if (1 + t^{2})^{k+1}μ is indeterminate, then (1 + t^{2})^{n}μ is indeterminate for n ≥ k + 1, and (a + t^{n} + i)ℙ is not dense in L^{2}(μ) for n ≥ k + 2.

References

[BC] Berg, C. and Christensen, J.P.R., Density questions in the classical theory of moments, Ann. Inst. Fourier 31,3 (1981), 99-114.

[BD1] Berg, C. and Duran, A.J., The index of determinacy for measures and the ℓ^{2}-norm of orthonormal polynomials, Trans. Amer. Math. Soc. 347 (1995), 2795–2811.

[BD2] Berg, C. and Duran, A.J., When does a discrete differential perturbation of a sequence of orthonormal polynomials belong to ℓ^{2}?, J. Funct. Anal. (to appear).

[D1] Duran, A.J., A generalization of Favard’s theorem for polynomials satisfying a recurrence relation, J. Approx. Theory 74 (1993), 83-109.

[D2] Duran, A.J., On Orthogonal polynomials with respect to a positive definite matrix of measures, Can. J. Math. 47 (1995), 88-112.

[DV] Duran, A.J. and van Assche, W., Orthogonal matrix polynomials and higher order recurrence relations, Linear Algebra Appl. 219 (1995), 261-280.

[R] Riesz, M., Sur le problème des moments. Troisième Note., Arkiv för Mat., astr. och fys. 17 (1923), no. 16.

Matematisk Institut, Københavns Universitet, Universitetsparken 5, DK-2100 København Ø, Denmark; Departamento de Análisis Matemático, Universidad de Sevilla, Apdo. 1160. 41080-Sevilla, Spain.