On a criterion for the determinate-indeterminate dichotomy of the moment problem



<span>5in</span>

<span>**Abstract**</span>

When the classical Hamburger moment problem has solutions, it has either exactly one solution or infinitely many solutions. Correspondingly, the moment problem is said to be either determinate or indeterminate. In terms of Jacobi operators, this dichotomy translates into the operator being either selfadjoint or symmetric nonselfadjoint. In this work, we present a new criterion for the determinate-indeterminate classification which hinges on bases of representation (in Akhiezer-Glazman terminology) for Jacobi operators so that the corresponding matrices have a certain structure.

Introduction {#sec:intro}
============

The classical Hamburger moment problem has played a central role in the development of modern mathematical analysis. It consists in finding a Borel measure $\mu$ such that $$  s_{k}=\int_{{\mathbb{R}}}t^{k}d\mu\,.$$ for a given real sequence of numbers $\{s_{k}\}_{k=0}^{\infty}$. To exclude from our consideration the trivial, degenerate solutions to the moment problem, let us always assume that the solutions have infinite support. This deceptively simple problem leads to fundamental questions in various fields of analysis and reveals unexpected connections between seemingly unrelated theories and notions. When the classical Hamburger moment problem has solutions, it has either exactly one solution or infinitely many solutions. In the first case, the moment problem is said to be determinate, while in the second case it is said to be indeterminate. This dichotomy is crucial within the moment problem theory; on the one hand, the criteria shed light on the intricacies of the theory and interconnections between fields of analysis, and on the other hand, the determinate and indeterminate cases lead to two different facets of the theory.

Due to the inherent richness of the moment problem, one can approach the determinate-indeterminate dichotomy from different viewpoints using different mathematical notions and, consequently, there are numerous criteria for finding out whether the moment problem is determinate or indeterminate. There is a nonexhaustive list of these criteria at the end of Section \[sec:matr-repr-clos\], although this is not the main point in this section, but rather the consequences of the one-to-one correspondence between Jacobi matrices and sequences of moments for which the corresponding moment problems admit solutions. On the basis of this correspondence, the determinate-indeterminate dichotomy is transformed into the selfadjoint-nonselfadjoint dichotomy for Jacobi operators. For the passing from matrices to operators, the concept of matrix representation for unbounded closed symmetric operators @MR1255973 [Sec.47] is essential.

The criterion presented in this paper is actually an if-and-only-if criterion for resolving the selfadjoint-nonselfadjoint dichotomy for Jacobi operators, however it does not rely on the operator theory techniques nor on the function theoretic methods for establishing selfadjointness or nonselfadjointness (<span>*cf.*</span> @MR0184042 [Chs.3 and 4], @MR0222718 [Ch.7 Sec.1], @MR1627806 and @MR2961944 [Ch.2]). Instead, we use the so-called bases of representation for Jacobi operators (Definition \[def:basis-representation\]) and the result can be stated exclusively in terms of these bases, namely:

-   If for a Jacobi operator there is more than one basis of matrix representation so that the corresponding matrix representation is a Jacobi matrix, then the operator is selfadjoint.

-   If for a Jacobi operator there is only one basis of matrix representation so that the corresponding matrix representation is a Jacobi matrix, then the operator is nonselfadjoint.

Apart from presenting a new criterion for the selfadjoint-nonselfadjoint dichotomy, the aim of this work is to shed light on the relationship between sequences of moments, Jacobi operators, matrices of representation, and measures. As a byproduct, necessary and sufficient conditions for a basis to be a basis of representation for a Jacobi operator are provided. Furthermore, it is shown how to construct a basis of matrix representation so that the corresponding measure has arbitrary index of determinacy.

Let us outline how the material of this work is presented. Section \[sec:matr-repr-clos\] introduces the main objects and the corresponding notation. This section is expository and presents classical results on the Hamburger moment problem and its relation to the theory of Jacobi matrices. Section \[sec:self-simpl-oper\] is a review of the theory of selfadjoint simple operators and tackles the problem of constructing bases of matrix representation for these operators. The index of determinacy and the connection to an algorithm to construct bases of matrix representation for selfadjoint Jacobi operators are given in Section \[sec:index-determ-bases\]. Finally, Section \[sec:non-self-jacobi\] deals with the case of nonselfadjoint Jacobi operators. This section uses Krein representation theory of symmetric operators @MR0011170 [@MR0011533; @MR0012177; @MR0048704] and de Branges theory on Hilbert spaces of entire functions @MR0229011.

Jacobi matrices and the Hamburger moment problem {#sec:matr-repr-clos}
================================================

Let us introduce the notions relevant to this paper and lay out the notation. Consider a closed symmetric operator $A$ in a Hilbert space $\mathcal{H}$ and an orthonormal basis $\{\delta_{k}\}_{k=1}^{\infty}$ of $\mathcal{H}$. If the domain of $A$, denoted by $\operatorname{dom}A$, coincides with the whole space $\mathcal{H}$ (which implies that $A$ is bounded since we have assumed it to be closed), then the operator can be uniquely recovered from the numbers $$\label{eq:matrix-entries-from-operator}
  a_{kj}:={\left\langle\delta_{k},A\delta_{j}\right\rangle}\,;$$ here and henceforth the inner product is considered to be antilinear in its first argument. If $\operatorname{dom}A\varsubsetneq\mathcal{H}$, then the operator $A$ is not reconstructed uniquely from even when $\delta_{k}\in\operatorname{dom}A$ for any $k\in{\mathbb{N}}$ (${\mathbb{N}}$ denotes the set of positive integers). For this reason, one needs the following:

\[def:basis-representation\] An orthonormal basis $\{\delta_{k}\}_{k=1}^{\infty}$ is said to be a basis of representation for the closed operator $A$ when

(a) $\delta_{k}\in\operatorname{dom}A$ for all $k\in{\mathbb{N}}$;

(b) if there is a closed operator $B$ such that $B\delta_{k}=A\delta_{k}$, then $B\supset A$.

When $\{\delta_{k}\}_{k=1}^{\infty}$ is a basis of representation for $A$, the matrix $$\label{eq:general-infinite-matrix}
    [A]=
      \begin{pmatrix}
        a_{11} & a_{12} & a_{13}  &  a_{14}  &  \cdots\\[1mm]
        a_{21} & a_{22} & a_{23} & a_{24} & \cdots \\[1mm]
        a_{31}  &  a_{32}  & a_{33}  & a_{34} &  \\
        a_{41} & a_{42} & a_{43} & a_{44} & \ddots\\
        \vdots & \vdots &  & \ddots & \ddots
  \end{pmatrix}\,,$$ with entries given by , is the matrix representation of $A$ with respect to $\{\delta_{k}\}_{k=1}^{\infty}$.

In @MR1255973 [Sec.47, Thm.3], it is established that any closed symmetric operator has a basis of representation. Conversely, if the matrix is Hermitian and satisfies $$\label{eq:condition-on-matrix-for-operator}
   \sum_{j=1}^{\infty}{\left|a_{jk}\right|}^{2}<+\infty\,, \text{ for all }k\in{\mathbb{N}}\,,$$ then there is a unique closed symmetric operator $A$ such that $[A]$ is its matrix representation with respect to a given orthonormal basis $\{\delta_{k}\}_{k=1}^{\infty}$ of a Hilbert space $\mathcal{H}$.

Let $\{q_{k}\}_{k=1}^{\infty}$ be a sequence of real numbers and $\{b_{k}\}_{k=1}^{\infty}$ be a sequence of positive numbers. An infinite matrix of the form $$\label{eq:jacobi-matrix}
   [J]=
     \begin{pmatrix}
    q_1 & b_1 & 0  &  0  &  \cdots
\\[1mm] b_1 & q_2 & b_2 & 0 & \cdots \\[1mm]  0  &  b_2  & q_3  &
b_3 &  \\
0 & 0 & b_3 & q_4 & \ddots\\ \vdots & \vdots &  & \ddots
& \ddots
  \end{pmatrix}$$ is said to be an infinite Jacobi matrix, or more specifically a semi-infinite Jacobi matrix to emphasize that the diagonals are enumerated by ${\mathbb{N}}$ rather than $\mathbb{Z}$. Since this matrix satisfies , upon fixing an orthonormal basis $\{\delta_{k}\}_{k=1}^{\infty}$ of a Hilbert space $\mathcal{H}$, there is a unique closed symmetric operator $J$, called Jacobi operator, having $[J]$ as its matrix representation with respect to $\{\delta_{k}\}_{k=1}^{\infty}$. Usually, one takes $\mathcal{H}=l_{2}({\mathbb{N}})$ and $\{\delta_{k}\}_{k=1}^{\infty}$ being the so-called canonical basis of $l_{2}({\mathbb{N}})$, <span>*i.e.*</span> $\delta_{k}$ is in turn the sequence $\{\delta_{jk}\}_{j=1}^{\infty}$, where $\delta_{jk}$ is the Kronecker delta. Henceforth, we assume that these choices for the space and the orthonormal basis are always made.

Thus, the operator $J$ is the closure of the operator $J_{0}$ whose domain is $l_{\rm fin}({\mathbb{N}})$ (the space of sequences with a finite number of nonzero elements) and satisfies $$\label{eq:difference-expr}
  \begin{split}
      (J_{0} \phi)_1&:=q_1 \phi_1 + b_1 \phi_2\,,\\
     (J_{0} \phi)_k&:= b_{k-1}\phi_{k-1} + q_k \phi_k + b_k\phi_{k+1}\,,
  \quad k \in \mathbb{N} \setminus \{1\},
\end{split}$$ for any $\phi\in l_{\rm fin}({\mathbb{N}})$. Also, one verifies that $J^{*}=J_{0}^{*}$ is the operator defined on the maximal domain, <span>*i.e.*</span>, $$\label{eq:domain-maximal}
  \operatorname{dom}J^{*}=\{\phi\in l_{2}({\mathbb{N}}): \sum_{k=2}^{\infty}{\left|
    b_{k-1}\phi_{k-1} + q_k \phi_k + b_k\phi_{k+1}\right|}^{2}<+\infty \}\,.$$

By setting $\pi_{1}:=1$, a solution to the equations $$\label{eq:difference-equation}
  \begin{split}
      z\pi_1&:=q_1 \pi_1 + b_1 \pi_2\,,\\
     z\pi_k&:= b_{k-1}\pi_{k-1} + q_k \pi_k + b_k\pi_{k+1}\,,
  \quad k \in \mathbb{N} \setminus \{1\},\quad z\in{\mathbb{C}}\,,
\end{split}$$ can be found uniquely by recurrence. This solution, $\pi(z)=\{\pi_{k}(z)\}_{k=1}^{\infty}$, is a sequence of polynomials of $z$ called the polynomials of the first kind generated by $[J]$.

\[rem:on-kernel-adjoint-at-zeta\] Since the polynomials' coefficients are real, if $\pi(z)\in l_{2}({\mathbb{N}})$, then $\pi({\overline{z}})\in l_{2}({\mathbb{N}})$. Also, it follows from  and that $\pi(z)\in l_{2}({\mathbb{N}})$ if and only if $\pi(z)\in\ker(J^{*}-zI)$. This means on the one hand that the deficiency indices of the symmetric operator $J$ are always equal to each other, <span>*i.e.*</span> $n_{+}(J)=n_{-}(J)$ and, on the other hand, if $\pi(z)\in l_{2}({\mathbb{N}})$ for one nonreal $z$, then this is true for any nonreal $z$. When $\pi(z)\in l_{2}({\mathbb{N}})$, the deficiency indices are equal to one because any other solution of coincides with $\pi(z)$ modulo a multiplicative constant (see @MR0184042 [Ch.4 Sec.1.2]). Thus, either $n_{+}(J)=n_{-}(J)=0$ or $n_{+}(J)=n_{-}(J)=1$. Since $J$ is closed by definition, the case when $n_{+}(J)=n_{-}(J)=0$ corresponds to $J$ being selfadjoint.

The *second order difference* expression (<span>*i.e.*</span> the matrix ) may be either in the limit point case or in the limit circle case. The asymptotic behavior of the sequence of Weyl circles determines the occurrence of one of these two possibilities since either the circles degenerate into a single point or a limit circle @MR0184042 [Ch.1 Sec.3]. For the class of *second order differential* expressions pertaining to the Sturm-Liouville operator, the same dichotomy between the limit point and limit circle cases takes place @MR0069338 [Ch.9]. Actually, the theory behind the Weyl circles originated in the context of differential equations.

It turns out that the limit point case corresponds to the selfadjoint case, <span>*i.e.*</span> $n_{+}(J)=n_{-}(J)=0$, while the limit circle case occurs when $n_{+}(J)=n_{-}(J)=1$. This correspondence is evident from the following expression @MR0184042 [Eq.1.21] $$\label{eq:radii-weyl-circles}
  \left({\left|z-{\overline{z}}\right|}\sum_{k=1}^{n}{\left|\pi_{k}(z)\right|}^{2}\right)^{-1}\,,$$ which gives the $n$-th Weyl circle's radius for $z\in{\mathbb{C}}\setminus{\mathbb{R}}$. Indeed, by von Neumann extension theory and Remark \[rem:on-kernel-adjoint-at-zeta\], selfadjointness of $J$ is equivalent to the radius vanishing as $n\to\infty$ in since $\pi(z)\not\in l_{2}({\mathbb{N}})$ for $z\in{\mathbb{C}}\setminus{\mathbb{R}}$, while nonselfadjointness of $J$ means that the limit of the sequence of radii is not zero since, in this case, $\pi(z)\in l_{2}({\mathbb{N}})$ for $z\in{\mathbb{C}}\setminus{\mathbb{R}}$.

Let us now turn to the moment problem posed at the beginning of Section \[sec:intro\]. A necessary and sufficient condition for a solution to the Hamburger moment problem to exist @MR0184042 [Thm.2.1.1] is that $$\label{eq:positivity-of-sequence}
  \det
  \begin{pmatrix}
    s_0&s_1&\dots&s_k\\
    s_1&s_2&\dots&s_{k+1}\\
    \vdots&\vdots&\ddots&\vdots\\
    s_k&s_{k+1}&\dots&s_{2k}
  \end{pmatrix}>0$$ for all $k\in{\mathbb{N}}\cup\{0\}$.

For a sequence $\{s_{n}\}_{n=0}^{\infty}$ satisfying there is either one solution or more than one solution to the Hamburger moment problem. In the first case, the moment problem is said to be determinate, while in the second case, it is called indeterminate.

As is customary, it is assumed in this paper that the sequence of moments $\{s_{n}\}_{n=0}^{\infty}$ is normalized, <span>*i.e.*</span> $s_{0}=1$. This involves no loss of generality since the general case reduces to the normalized one by dividing the sequence of moments and its solution by $s_{0}$.

There is a one-to-one correspondence between Jacobi matrices and normalized sequences $\{s_{n}\}_{n=0}^{\infty}$ satisfying (see @MR0184042 [Ch.1]). Moreover, this bijection pairs every limit point Jacobi matrix with a sequence for which the Hamburger moment problem is determinate and every limit circle Jacobi matrix with a sequence for which the Hamburger moment problem is indeterminate @MR0184042 [Thm. 2.1.2 and Cor. 2.2.4].

Let us briefly describe how the above mentioned one-to-one correspondence is realized. First, consider the starting point to be an operator $J$ having the matrix representation with respect to the orthonormal basis $\{\delta_{k}\}_{k=1}^{\infty}$. Since $$  \begin{split}
      J\delta_1&=q_1 \delta_1 + b_1 \delta_2\,,\\
     J\delta_k&= b_{k-1}\delta_{k-1} + q_k \delta_k + b_k\delta_{k+1}\,,
  \quad k \in \mathbb{N} \setminus \{1\}\,,
\end{split}$$ it is verified that $$\label{eq:polynomials-show-simplicity}
  \delta_{k}=\pi_{k}(J)\delta_{1}\,.$$ This means that $\delta_{1}$ is in the domain of any power of the Jacobi operator $J$. Thus, if one defines $s_{k-1}:={\left\langle\delta_{1},J^{k-1}\delta_{1}\right\rangle}$ for all $k\in{\mathbb{N}}$, then a solution to the corresponding moment problem is given by the measure $$\label{eq:extremal-measure}
  \mu(\cdot):={\left\langle\delta_{1},E(\cdot)\delta_{1}\right\rangle}\,,$$ where $E$ is either the spectral measure of $J$ if it is selfadjoint or the spectral measure of any of the canonical selfadjoint extensions[^1] of $J$ otherwise. Hence, $\{s_{k}\}_{k=0}^{\infty}$ is a sequence of moments and the nonselfadjoint case yields different solutions to the corresponding moment problem. This conclusion is complemented in the classical moment problem theory by showing, on the one hand that if $J$ is selfadjoint, then $\mu$ is the unique solution of the moment problem @MR0184042 [Cor.2.2.4] and, on the other hand, that there are other solutions apart from the ones given by the canonical selfadjoint extensions of the nonselfadjoint Jacobi operator (see @MR0184042 [Ch.2 Secs. 2 and 3] and @MR1627806 [Thm.4]).

Now, let the starting point be any normalized sequence of moments. In this case it is known that one can construct from this sequence a unique Jacobi matrix using the determinantal formulae (see @MR0184042 [Ch.1 Sec.1] and @MR1627806 [Thm.A.2]). The corresponding Jacobi operator $J$ in $l_{2}({\mathbb{N}})$ turns out to be such that $s_{k-1}={\left\langle\delta_{1},J^{k-1}\delta_{1}\right\rangle}$ for all $k\in{\mathbb{N}}$. According to the contraposition of @MR0184042 [Th.2.1.2], if the sequence of moments gives rise to a determinate moment problem, then the Jacobi matrix is in the limit point case. On the other hand, the contraposition of @MR0184042 [Cor.2.2.4] asserts that when the moment problem is indeterminate, the Jacobi matrix is in the limit circle case.

Thus, the Hamburger moment problem is determinate if and only if the corresponding Jacobi matrix is in the limit point case, which in turn means that the Jacobi operator $J$ is selfadjoint. Complementarily, the fact that the Hamburger moment problem is indeterminate is equivalent to the corresponding Jacobi matrix being in the limit circle case, <span>*i.e.*</span> $J$ is not selfadjoint.

Other if-and-only-if criteria are: (a) the finite difference analogue @MR0184042 [Thm.1.3.1] of the Weyl alternative for Sturm-Liouville operators @MR0069338 [Ch.9] (related to the limit circle/point dichotomy and the presence/absence of uniqueness of the Weyl $m$-coefficient), (b) the Hamburger criterion (given in terms of the moment sequence $\{s_{k}\}_{k=0}^{\infty}$) @MR0184042 [Addenda and problems of Ch.2]. This list is not exhaustive, but all the criteria found in the literature boil down directly or indirectly to the properties of the sequence $\pi(z)$.

For the Stieltjes moment problem @MR1627806 [Pag.83], the determinate/indeterminate dichotomy reduces to the existence of one/multiple nonnegative selfadjoint extensions of the corresponding Jacobi operator @MR1627806 [Thms.2 and 3.2]. This paper is not concerned with the Stieltjes moment problem.

This section concludes with an overview of the isometry map associated with the moment problem which will be of use in the next sections. By @MR0184042 [Thms.2.3.3 and 4.1.4], one has the following classical result.

\[prop:density-polynomials\] The polynomials are dense in $L_{2}({\mathbb{R}},\mu)$ if and only if $\mu$ is the measure given by .

If one assumes that $\mu$ is given by , then the sequence of monomials $\{t^{k-1}\}_{k=1}^{\infty}$ is total in $L_{2}({\mathbb{R}},\mu)$. So, till the end of this section, $\mu$ is assumed to be given this way. Now, the so-called Gram-Schmidt procedure applied to $\{t^{k-1}\}_{k=1}^{\infty}$ yields an orthonormal basis $\{P_{k-1}(t)\}_{k=1}^{\infty}$ in $L_2({\mathbb{R}},\mu)$. The Gram-Schmidt procedure is assumed in this paper to be defined as in @MR1192782 [Ch.2 Sec.2 Thm. 5] which implies that $P_{k}$ is a polynomial of degree $k$ with positive leading coefficient. The orthonormal sequence of polynomials is uniquely determined by these properties @MR3729411 [Prop.5.1]. One has the well-known three-term relation theorem (see @MR2085852 [Sec.3.1.3], @MR3729411 [Prop.5.6], @MR1627806 [Pag.92]):

\[prop:three-term-relation\] If $\{P_{k-1}(t)\}_{k=1}^{\infty}$ is the orthonormal sequence of polynomials defined above, then $$\label{eq:difference-equation-polynomials}
  \begin{split}
      tP_0(t)&:=q_1 P_0(t) + b_1 P_1(t)\,,\\
     tP_k(t)&:= b_{k}P_{k-1}(t) + q_{k+1} P_k(t) + b_{k+1}P_{k+1}(t)\,,
  \quad k \in \mathbb{N}\,,\quad t\in{\mathbb{R}}\,,
\end{split}$$ where the sequences $\{q_{k}\}_{k=1}^{\infty}$ and $\{b_{k}\}_{k=1}^{\infty}$ are obtained from the moments $\{s_{k-1}\}_{k=1}^{\infty}$ by means of the determinantal formulae mentioned above.

\[rem:coincides-with-first-kind\] The coefficients of the three-term recurrence relation form the matrix . By comparing with , one concludes that the polynomials of the first kind generated by coincide with the polynomials obtained by orthonormalization of $\{t^{k-1}\}_{k=1}^{\infty}$ in $L_{2}({\mathbb{R}},\mu)$.

\[def:map-U\] Let $\{P_{k-1}\}_{k=1}^{\infty}$ be the orthonormal sequence of polynomials in $L_{2}({\mathbb{R}},\mu)$ given above. Define the map $U:L_{2}({\mathbb{R}},\mu)\to l_{2}({\mathbb{N}})$ such that $$  UP_{k-1}=\delta_{k}\,,\qquad k\in{\mathbb{N}}\,,$$ where $\{\delta_{k}\}_{k=1}^{\infty}$ is the canonical basis in $l_{2}({\mathbb{N}})$.

By definition $U$ realizes an isometric isomorphism between $L_{2}({\mathbb{R}},\mu)$ and $l_{2}({\mathbb{N}})$.

\[def:operator-multiplication\] Let $\mu$ be an arbitrary Borel measure. Denote by $M_{\mu}$ the operator of multiplication by the independent variable in $L_{2}({\mathbb{R}},\mu)$ defined in its maximal domain, <span>*i.e.*</span>, $$    \operatorname{dom}(M_{\mu}):=\{f\in L_{2}({\mathbb{R}},\mu):
    \int_{R}t^{2}{\left|f(t)\right|}^{2}d\mu(t)<+\infty\}\,.$$

Note that $M_{\mu}$ is completely determined by the measure $\mu$. It is important to bear in mind that in the definition of $M_{\mu}$, the measure $\mu$ is not necessarily a solution to the moment problem. This will be relevant in the next section.

Returning to the case in which $\mu$ is a solution to a Hamburger moment problem, if this problem is determinate, then the operator $J$ in $l_{2}({\mathbb{N}})$ whose matrix representation is coincides with the selfadjoint operator implicit in . In this case, on the basis of Proposition \[prop:three-term-relation\], one concludes that $U^{-1}JU=M_{\mu}$ with $\{P_{k-1}(t)\}_{k=1}^{\infty}$ being a basis of representation of it and the corresponding matrix representation. Now, if the moment problem is indeterminate, then $J$ (whose matrix representation is ) is not selfadjoint and the operator implicit in is a selfadjoint extension of it. Lets denote this selfadjoint extension by ${\widetilde}{J}$. It turns out that $U^{-1}{\widetilde}{J}U=M_{\mu}$, but $\{P_{k-1}(t)\}_{k\in{\mathbb{N}}}$ is no longer a basis of representation of it since the minimality condition (b) of Definition \[def:basis-representation\] is not satisfied.

Selfadjoint simple operators {#sec:self-simpl-oper}
============================

Let $A$ be a selfadjoint operator in a separable Hilbert space $\mathcal{H}$ and $E$ be its spectral measure given by the spectral theorem. For any real Borel set $\partial$ and $h\in\mathcal{H}$, denote by $$\label{eq:def-mu-f}
  \mu_{h}(\partial):={\left\langleh,E(\partial)h\right\rangle}$$ the corresponding nonnegative measure. Thus, the spectral theorem allows one to define the operator $$  \phi(A):=\int_{{\mathbb{R}}}\phi dE\,,
  \qquad\operatorname{dom}\phi(A):=\{h\in\mathcal{H}:\phi\in L_{2}({\mathbb{R}},\mu_{h})\}\,.$$

\[def:generating-element\] An element $g\in\mathcal{H}$ is called a generating element of the selfadjoint operator $A$ if the span over all Borel sets $\partial\subset{\mathbb{R}}$ of $E(\partial)g$ is dense in $\mathcal{H}$. The operator $A$ is said to be simple when it has a generating element.

For any simple operator $A$ and any of its generating elements $g$, there is a unitary map $\Psi_{g}$ from $L_{2}({\mathbb{R}},\mu_{g})$ onto $\mathcal{H}$ given by $$\label{eq:psy-def}
 \phi\overset{\Psi_{g}}\mapsto\phi(A)g$$ such that the operator of multiplication $M_{\mu_{g}}$ (see Definition \[def:operator-multiplication\]) is transformed into the operator $A$. The unitary map $\Psi^{*}_{g}$ realizes the canonical representation of the simple operator $A$ with respect to $g$.

For any Borel measure $\mu$, the operator of multiplication $M_{\mu}$ is a selfadjoint simple operator. Any function $\eta\in L_{2}({\mathbb{R}},\mu)$ such that $\eta(t)\ne 0$ for $\mu-$a.e. $t$ is a generating element of $M_{\mu}$.

\[def:cyclic-vector\] A vector $f$ is a cyclic vector of $A$ when $f\in\operatorname{dom}A^{k}$ for all $k\in{\mathbb{N}}$ and $$    \operatorname{clos}\operatorname*{span}_{k\in{\mathbb{N}}\cup\{0\}}A^{k}f=\mathcal{H}\,.$$

A cyclic vector is a generating element @MR1255973 [Sec.69 Thm.1], but the converse is not necessarily true. However, one can always construct a cyclic vector from a generating element. This is done below.

By Definition \[def:cyclic-vector\], a function $\eta$ is a cyclic vector of the operator of $M_{\mu}$ if only if $$\label{eq:dense-span-cyclic}
    \operatorname{clos}\operatorname*{span}_{k\in{\mathbb{N}}\cup\{0\}}t^{k}\eta(t)=L_{2}({\mathbb{R}},\mu)\,.$$ Therefore, a straightforward consequence of the canonical representation of simple operators is the following lemma.

\[lem:creation-of-cyclic\] Assume $\mu=\mu_{g}$, with $g$ being a generating element of a simple operator $A$. For the vector $\eta(A)g$ to be a cyclic vector of $A$ it is necessary and sufficient that $\eta$ satisfies .

The next statements are used to establish results pertaining to the existence of cyclic vectors. They are based on a reasoning used to prove @MR0184042 [Thm.4.2.3]. Although they are known, we present the proofs below for the sake of completeness.

\[lem:auxiliary-for-stone\] Let $\mu$ be a $\sigma$-finite Borel measure on ${\mathbb{R}}$ and $f$ a function in $L_{2}({\mathbb{R}},\mu)$. Define $$\label{eq:auxiliary-lemma-g}
    \mathcal{G}(t):=\int_{-\infty}^{t}f(s)d\mu(s) + C\,,
    \qquad C\in{\mathbb{C}}\,.$$ There exist a constant $C_{0}\in{\mathbb{C}}$ such that, under the assumption that $C=C_{0}$ in , if $$\label{eq:conditions-g-lemma}
    \int_{{\mathbb{R}}}t^{k}e^{-\frac12t^{2}}d\mathcal{G}=0$$ for all $k\in{\mathbb{N}}\cup\{0\}$, then $\mathcal{G}(t)=0$ for a.e. t in ${\mathbb{R}}$.

If one defines $$    C_{0}:=\frac{-1}{\sqrt{2\pi}}
\int_{{\mathbb{R}}}\left(\int_{-\infty}^{t}f(s)
        d\mu(s)\right)e^{-t^{2}}dt\,,$$ then $$\label{eq:first-zero}
    \int_{{\mathbb{R}}}\mathcal{G}(t)e^{-\frac12t^{2}}dt=0\,.$$ Integrating by parts, one arrives at $$\label{eq:integration-by-parts}
    \int_{{\mathbb{R}}}\left(kt^{k-1}-t^{k+1}\right)
    \mathcal{G}(t)e^{-\frac12t^{2}}dt=0\,.$$ Substituting $k=0$ in this equation, one obtains $$\label{eq:k-is-zero}
    \int_{{\mathbb{R}}}\mathcal{G}(t)te^{-\frac12t^{2}}dt=0\,.$$ Using and , it follows from by recurrence that $$\label{eq:lemma-result}
    \int_{{\mathbb{R}}}\mathcal{G}(t)t^{k}e^{-\frac12t^{2}}dt=0\,,\quad
    \forall\, k\in{\mathbb{N}}\cup\{0\}\,.$$ By the closure of the Chebyshev-Hermite functions in $L_{2}({\mathbb{R}})$ (see @MR0372517 [Thm.5.7.1] and @MR1255973 [Sec.11.C]), one concludes from that $\mathcal{G}(t)=0$ for a.e. $t\in{\mathbb{R}}$.

\[lem:cyclic-every-measure\] Let $\mu$ be an arbitrary finite Borel measure. If $\eta(t)=\exp(-\alpha t^{2})$ with $\alpha\ge 1/2$, then $\eta$ satisfies .

Since $\mu$ is finite, $t^{k}\eta(t)$ is in $L_{2}({\mathbb{R}},\mu)$ for any $k\in{\mathbb{N}}\cup\{0\}$. Suppose that $\phi$ in $L_{2}({\mathbb{R}},\mu)$ is orthogonal to all functions $t^{k}\eta(t)$, <span>*i.e.*</span>, for all $k\in{\mathbb{N}}\cup\{0\}$, $$\begin{aligned}
        0&=\int_{{\mathbb{R}}}{\overline{\phi(t)}}t^{k}e^{-\alpha t^{2}}d\mu(t)\\
    &=\int_{{\mathbb{R}}}t^{k}e^{-\frac12t^{2}}d\mathcal{G}(t)\,,
  \end{aligned}$$ where $$    \mathcal{G}(t)=\int_{-\infty}^{t}{\overline{\phi(s)}}
    e^{-(\alpha-\frac12)s^{2}}d\mu(s) + C$$ with $C$ being an arbitrary constant. By Lemma \[lem:auxiliary-for-stone\], one obtains that $\mathcal{G}(t)=0$ for a.e. $t\in{\mathbb{R}}$. Thus, $$  {\left\|\phi\right\|}^{2}=
  \int_{{\mathbb{R}}}e^{(\alpha-\frac12)t^{2}}\phi(t)d\mathcal{G}(t)=0\,.$$

The conclusion of Lemma \[lem:cyclic-every-measure\] motivates the following definition:

\[def:stone-vectors\] Let $g$ be a generating element for the selfadjoint operator $A$. For any $\alpha\ge 0$, define $$\eta(\alpha,g):=\exp(-\alpha A^{2})g\,.$$ We refer to $\eta(\alpha,g)$ as to the Stone vector of order $\alpha$ obtained from the generating element $g$.

The combination of Lemmas \[lem:creation-of-cyclic\] and \[lem:cyclic-every-measure\] yields the following assertion which is the first part of a slight generalization of Stone classical result (see @MR0184042 [Thm. 4.2.3]).

\[cor:stonecyclic\] For any generating element $g$ of a simple selfadjoint operator $A$, any Stone vector $\eta(\alpha,g)$ is a cyclic vector of $A$ for all $\alpha\ge 1/2$.

\[rem:delta-1-cyclic\] Let $J$ be the operator whose matrix representation is with respect to the canonical basis $\{\delta_{k}\}_{k=1}^{\infty}$. If $J$ is selfadjoint, then it follows from that $J$ is simple and $\delta_{1}$ is a cyclic vector of it. If $J\varsubsetneq J^{*}$, then $\delta_{1}$ is a cyclic vector for each of the selfadjoint extensions of $J$ (and therefore each selfadjoint extension is simple).

The next proposition amounts, in a certain sense, to the converse of the assertion in the preceding remark.

\[prop:gram-schmidt-basis\] Let $\delta$ be a cyclic vector of $A$ and $\{\delta_{k}\}_{k=1}^{\infty}$ be the orthonormal basis obtained from applying the Gram-Schmidt procedure to the sequence $\{A^{k-1}\delta\}_{k=1}^{\infty}$. If $B$ is the minimal closed operator such that $B\delta_{k}=A\delta_{k}$ for all $k\in{\mathbb{N}}$ (<span>*cf.*</span> Definition \[def:basis-representation\]), then the matrix representation of $B$ with respect to $\{\delta_{k}\}_{k=1}^{\infty}$ is a semi-infinite Jacobi matrix (see ).

First note that, by the Gram-Schmidt algorithm (see @MR1192782 [Ch.2 Sec.2 Thm. 5]), one has $\delta_{1}=\delta/{\left\|\delta\right\|}$, so $\delta_{1}$ is a normalized cyclic vector. Taking this into account, the proof reduces to a well known assertion @MR2085852 [Sec.3.1.3] on orthogonal polynomials by means of the canonical representation of $A$ with respect to $\delta_{1}$. Indeed, using the map introduced in , one has $$\Psi^{*}_{\delta_{1}}(A^{k-1}\delta_{1})=t^{k-1}$$ for any $k\in{\mathbb{N}}$. The unitarity of the map given in and Definition \[def:cyclic-vector\] imply that the sequence $\{t^{k-1}\}_{k=1}^{\infty}$ is total in $L_2({\mathbb{R}},\mu_{\delta_{1}})$. Therefore, the Gram-Schmidt procedure applied to $\{t^{k-1}\}_{k=1}^{\infty}$ yields an orthonormal basis $\{P_{k-1}(t)\}_{k\in{\mathbb{N}}}$ in $L_2({\mathbb{R}},\mu_{\delta_{1}})$ (see the end of Section \[sec:matr-repr-clos\]). Clearly, $\delta_{k}=\Psi_{\delta_{1}}P_{k-1}$. Therefore, on the basis of Proposition \[prop:three-term-relation\], one concludes that the numbers $$\label{eq:obtained-matrix}
  a_{jk}:={\left\langle\delta_{j},A\delta_{k}\right\rangle}=
  {\left\langleP_{j-1},tP_{k-1}\right\rangle}_{L_{2}({\mathbb{R}},\mu_{\delta_{1}})}\,,\quad
j,k\in{\mathbb{N}}\,,$$ generate a semi-infinite Jacobi matrix. To finish the proof notice that, by Definition \[def:basis-representation\], $B$ is the operator whose matrix representation has the entries .

\[rem:example\] In the assertion of Proposition \[prop:gram-schmidt-basis\], it could be that $B\varsubsetneq A$, <span>*i.e.*</span> the orthonormal basis obtained from the Gram-Schmidt procedure applied to $\{A^{k-1}\delta\}_{k=1}^{\infty}$ is not necessarily a basis of representation for $A$. An example of this has already appeared at the end of Section \[sec:matr-repr-clos\]. Indeed, let $J$ be the operator whose matrix representation is with respect to the canonical basis $\{\delta_{k}\}_{k=1}^{\infty}$ and assume that $J\varsubsetneq J^{*}$. By Remark \[rem:delta-1-cyclic\], $\delta_{1}$ is a cyclic vector of ${\widetilde}{J}$, a fixed selfadjoint extension of $J$. Moreover, it follows from and Remark \[rem:coincides-with-first-kind\] that the basis $\{\delta_{k}\}_{k=1}^{\infty}$ is obtained from the Gram-Schmidt procedure applied to $\{{\widetilde}{J}^{k}\delta_{1}\}$ since ${\widetilde}{J}\supset J$. Note that $\{\delta_{k}\}_{k=1}^{\infty}$ is the basis of representation for $J$, but not for ${\widetilde}{J}$.

The next assertion is a slight generalization of a classical result by Stone on simple operators (see @MR0184042 [Thm.4.2.3]).

\[prop:basis-representation\] For any simple selfadjoint operator, there is an uncountable set of bases of matrix representation such that the corresponding matrix representation of the operator with respect to each of the bases is a Jacobi matrix.

Let $A$ be a simple operator and $g$ a generating element of it. If $\alpha\ge\frac12$, then $\eta(\alpha,g)$ given in Definition \[def:stone-vectors\] is a cyclic vector of $A$. Due to Proposition \[prop:gram-schmidt-basis\], if $\{\delta_{k}(\alpha,g)\}_{k=1}^{\infty}$ is the orthonormal basis obtained from applying the Gram-Schmidt procedure to the sequence $\{A^{k-1}\eta(\alpha,g)\}_{k=1}^{\infty}$, then $$    {\left\langle\delta_{j}(\alpha,g),A\delta_{k}(\alpha,g)\right\rangle}$$ is a Jacobi matrix which will be denoted by $[A](\alpha,g)$.

It remains to prove that $[A](\alpha,g)$ is the matrix representation of $A$ with respect to the orthonormal basis $\{\delta_{k}(\alpha,g)\}_{k=1}^{\infty}$. According to Definition \[def:basis-representation\], this boils down to showing that $A$ is the minimal closed operator associated with the matrix $[A](\alpha,g)$.

Let $B$ be the operator whose matrix representation is $[A](\alpha,g)$ (on account of what is said in the paragraph below such operator is univocally determined by the matrix and this operator is symmetric). Assume that $h:=\phi(A)g$ is orthogonal to $(B-{i}I)\delta_{k}(\alpha,g)$ for all $k\in{\mathbb{N}}$, then $$\begin{aligned}
  0&={\left\langleh,(B-{i}I)\delta_{k}(\alpha,g)\right\rangle}
  ={\left\langleh,(A-{i}I)\delta_{k}(\alpha,g)\right\rangle}\\
   &={\left\langle\phi(A)g,(A-{i}I)P_{k-1}(A)e^{-\alpha A^{2}}g\right\rangle}\\
  & =\int_{{\mathbb{R}}}{\overline{\phi(t)}}(t-{i})
    P_{k-1}(t)e^{-\alpha t^{2}}d\mu_{g}(t)\,,\end{aligned}$$ where the second equality holds since $B\subset A$. In the third equality, one uses Definition \[def:stone-vectors\] and the fact that $\delta_{k}(\alpha,g)=\Psi_{\eta(\alpha,g)}(P_{k-1})$ (see the proof of Proposition \[prop:gram-schmidt-basis\]). In the last equality, one recurs to the isometric property of $\Psi_{g}$. Thus, for any $k\in{\mathbb{N}}$, one has $$\begin{aligned}
  0&=\int_{{\mathbb{R}}}{\overline{\phi(t)}}(t-{i})
     t^{k-1}e^{-\alpha t^{2}}d\mu_{g}(t)\\
  &=\int_{{\mathbb{R}}}t^{k}e^{-\frac12t^{2}}d\mathcal{F}\,,\end{aligned}$$ where $$  \mathcal{F}(t):=\int_{-\infty}^{t}{\overline{\phi(t)}}(s-{i})
  e^{-(\alpha-\frac12)s^{2}}d\mu_{g}(s)+ C$$ with $C$ being an arbitrary constant. By Lemma \[lem:auxiliary-for-stone\], $\mathcal{F}(t)=0$ for a.e. $t\in{\mathbb{R}}$. Therefore $$\begin{aligned}
  0&=\int_{{\mathbb{R}}}(t-{i}){\overline{\phi(t)}}d\mathcal{F}\\
  &=\int_{{\mathbb{R}}}{\left|t+{i}\right|}^{2}e^{-\frac12t^{2}}{\left|\phi(t)\right|}^{2}d\mu_{g}(t)\,.\end{aligned}$$ This implies that ${\left\|\phi\right\|}_{L_{2}({\mathbb{R}},\mu_{g})}=0$. Thus, one concludes that the deficiency space of $B$ on the upper-half plane is trivial and therefore $B$ is maximal which, in turn, means that it does not have proper symmetric extensions.

\[rem:special-cyclic\] For any generating element $g$ of $A$ and $\alpha\ge\frac12$, the definition of Stone vectors $\eta(\alpha,g)$ by means of a Gaussian function guarantees not only cyclicity, but also the fact that $\{\delta_{k}(\alpha,g)\}_{k=1}^{\infty}$ is a basis of representation for $A$ (<span>*cf.*</span> Remark \[rem:example\]).

The following assertion gives necessary and sufficient conditions for a cyclic vector $\delta$ of $A$ to generate, through the Gram-Schmidt procedure applied to the sequence $\{A^{k-1}\delta\}_{k=1}^{\infty}$, a basis of representation for $A$.

\[prop:necessary-and-sufficient\] Let $A$ be a simple operator and $\delta$ a cyclic vector of it. The Gram-Schmidt procedure applied to the sequence $\{A^{k-1}\delta\}_{k=1}^{\infty}$ yields a basis of representation for $A$ if and only if $$\label{eq:special-cyclic}
    \operatorname{clos}\operatorname*{span}_{k\in{\mathbb{N}}}\{(A-{i}I)A^{k-1}\delta\}=\mathcal{H}\,.$$

Let $\{\delta_{k}\}_{k=1}^{\infty}$ be the orthonormal basis obtained by the Gram-Schmidt procedure applied to the sequence $\{A^{k-1}\delta\}_{k=1}^{\infty}$. Thus, $\delta_{k}=P_{k-1}(A)\delta$, where $P_{k}$ is a polynomial of degree $k$ (see the proof of Proposition \[prop:gram-schmidt-basis\]). Denote by $B$ the minimal closed operator so that $B\delta_{k}=A\delta_{k}$. Assume first that holds. If the vector $h$ is such that ${\left\langleh,(B-{i}I)\delta_{k}\right\rangle}$ vanishes for all $k\in{\mathbb{N}}$, then $$\label{eq:sufficiency-aux}
    {\left\langleh,(A-{i}I)A^{k-1}\delta\right\rangle}=0\,,\quad \forall\,k\in{\mathbb{N}}\,.$$ Therefore, it follows from and that $h=0$. Since $\operatorname{ran}(B-{i}I)$ contains $\operatorname*{span}_{k\in{\mathbb{N}}}(B-{i}I)\delta_{k}$, one concludes that the closed symmetric operator $B$ is maximal and therefore $B=A$.

Now suppose that $B=A$ and does not hold, <span>*i.e.*</span> there is a nonzero vector $h$ so that $h\perp (A-{i}I)A^{k-1}\delta$ for all $k\in{\mathbb{N}}$. This implies that $$\label{eq:condition-h-delta-k}
    {\left\langleh,(B-{i}I)\delta_{k}\right\rangle}=0\,,\quad \forall\,k\in{\mathbb{N}}\,,$$ since $\delta_{k}$ is a polynomial of $A$ applied to $\delta$. By Proposition \[prop:gram-schmidt-basis\], $B$ is a Jacobi operator so one can denote the entries of the corresponding matrix as in . Therefore, by writing $h=\sum_{k=1}^{\infty}h_{k}\delta_{k}$, one obtains from that $$  \begin{split}
      {i}h_1&:=q_1 h_1 + b_1 h_2\,,\\
     {i}h_k&:= b_{k-1}h_{k-1} + q_k h_k + b_kh_{k+1}\,,
  \quad k \in \mathbb{N} \setminus \{1\}\,.
\end{split}$$ By the assumption that $B=A$, $B$ is selfadjoint and therefore $$  \sum_{k=1}^{\infty}{\left|h_{k}\right|}^{2}=+\infty\,.$$ This contradicts the fact that $h$ is a nonzero element of the space.

To close up this section, we put its results in the context of Jacobi operators in the selfadjoint and nonselfadjoint cases.

A straightforward consequence of Proposition \[prop:basis-representation\] and Remark \[rem:delta-1-cyclic\] is the following:

\[prop:representations-jacobi-operator\] Let $J$ be the operator whose matrix representation is with respect to the canonical basis $\{\delta_{k}\}_{k=1}^{\infty}$. If is in the limit point case, then, for any $\alpha\ge \frac12$, the basis $\{\delta_{k}(\alpha,g)\}_{k=1}^{\infty}$, constructed from any generating element $g$, is a basis of matrix representation for $J$ and the corresponding matrix $[J](\alpha,g)$ is a Jacobi matrix in the limit point case.

\[rem:relation-moments-matrices\] If the hypotheses of the preceding proposition hold, then, for each generating element $g$ of $J$ and $\alpha\ge\frac12$, the Jacobi matrix $[J](\alpha,g)$ generates a sequence of moments $\{s_{k}(\alpha,g)\}_{k=0}^{\infty}$ so that the solution to the corresponding moment problem is unique. By @MR1192782 [Ch.5 Sec.3 Lem.3], this unique solution is given by $$\label{eq:sol-moment-alpha}
    \mu_{\eta(\alpha,g)}(\partial)=\int_{\partial}e^{-2\alpha
      t^{2}}d\mu_{g}(t)$$ for any Borel set $\partial$.

According to Remark \[rem:delta-1-cyclic\], if $J$ is a nonselfadjoint Jacobi operator, then, for each selfadjoint extension ${\widetilde}J$ and $\alpha\ge\frac12$, the Jacobi matrix $[{\widetilde}J](\alpha,\delta_{1})$ is associated with a determinate moment problem whose unique solution is $$\label{eq:sol-moment-alpha}
    \mu_{\eta(\alpha,\delta_{1})}(\partial)=\int_{\partial}e^{-2\alpha
      t^{2}}d{\widetilde}\mu_{\delta_{1}}(t)$$ for any Borel set $\partial$, where ${\widetilde}\mu_{\delta_{1}}$ is given by with $h=\delta_{1}$ and $E$ being the spectral measure of ${\widetilde}J$.

Index of determinacy and bases of representation {#sec:index-determ-bases}
================================================

In the previous section, bases of matrix representation for a selfadjoint Jacobi operator $J$ were constructed from an arbitrary generating element of it by means of the Stone vectors (Definition \[def:stone-vectors\]). The matrices representing $J$ with respect to these bases were Jacobi matrices. The function involved in Definition \[def:stone-vectors\] guarantee not only that the Stone vector is a cyclic vector, but also that the basis obtained from it is a basis of matrix representation.

In this section, an alternative method is used for the construction of bases of matrix representation of a selfadjoint Jacobi operator so that the corresponding matrix is a Jacobi matrix. This method is related to the so-called index of determinacy @MR1308001 of a solution to the moment problem.

Let $\mu$ be a Borel measure. For any Borel set $\partial$, denote $$\label{eq:leser-determinacy-measure}
    \mu_{n}(\partial):=\int_{\partial}(1+x^{2})^{n}d\mu(x)\,,\quad n\in{\mathbb{N}}\,.$$ Note that $\mu_{n}$ is obtained by applying the transformation with $n=1$ to the measure $\mu_{n-1}$.

Two classical results pertaining to the density of polynomials in $L_{2}$ spaces are Propositions \[prop:density-polynomials\] and

\[prop:riez-index\] The measure $\mu$ is the solution to a determinate Hamburger moment problem if and only if the polynomials are dense in $L_{2}({\mathbb{R}},\mu_{1})$.

The proof of this assertion is found in @mriesz1923 (see also @MR1308001 [Lem.A] and @MR3729411 [Cor.6.11]).

\[def:index-determinacy\] The index of determinacy of a solution $\mu$ to a Hamburger moment problem is $$    \operatorname{ind}\mu:=\sup\{n\in{\mathbb{N}}: \text{ the polynomials are dense in }
    L_{2}({\mathbb{R}},\mu_{n})\}.$$ It is not excluded that $\operatorname{ind}\mu$ could be $\infty$, which takes place when the polynomials are dense in $L_{2}({\mathbb{R}},\mu_{n})$ for any $n\in{\mathbb{N}}$.

This definition differs from the one in @MR1308001. If $\mu$ has index $n$ according to @MR1308001 [Eq.1.1], then $\operatorname{ind}\mu=n+1$ by Definition \[def:index-determinacy\]. The index of determinacy in this paper is so that any determinate measure has positive index of determinacy. Indeed, by Proposition \[prop:riez-index\], the index of determinacy makes sense only for solutions to determinate moment problems and for any such solution $\mu$, $\operatorname{ind}\mu\ge1$. Note that the index of determinacy decreases one unit each time the transformation with $n=1$ is applied. Also, it follows from Propositions \[prop:density-polynomials\] and \[prop:riez-index\] that if $\operatorname{ind}\mu=1$, then $\mu_{1}$ is given by with $E$ being the spectral measure of a canonical selfadjoint extension of a nonselfadjoint Jacobi operator and $\mu_{2}$ is such that the polynomials are no longer dense in $L_{2}({\mathbb{R}},\mu_{2})$.

By reverting the transformation , one can increase the index of determination of a given measure. Indeed, let $J$ be a nonselfadjoint Jacobi operator in $l_{2}({\mathbb{N}})$ and $[J]$ its matrix representation with respect to the canonical basis $\{\delta_{k}\}_{k=1}^{\infty}$. Fix a canonical selfadjoint extension ${\widetilde}J$ of $J$ and denote by $\mu$ the measure given by with $E$ being the spectral measure of ${\widetilde}J$. Now, for any Borel set $\partial$ and $n\in{\mathbb{N}}$, define $$  \nu_{n}(\partial):=\frac{1}{C}\int_{\partial}(1+x^{2})^{-n}d\mu(x)\,,\quad{
  where }\quad C:=
  \int_{{\mathbb{R}}}(1+x^{2})^{-n}d\mu(x)\,.$$ Due to Propositions \[prop:density-polynomials\] and \[prop:riez-index\], one verifies that according to Definition \[def:index-determinacy\] $\operatorname{ind}\nu_{n}=n$ for any $n\in{\mathbb{N}}$.

Let $\{R_{k-1}\}_{k=1}^{\infty}$ be the orthonormal sequence of polynomials in $L_{2}({\mathbb{R}},\nu_{1})$ obtained from monomials by the Gram-Schmidt procedure. Define $$\label{eq:orthonormal-functions}
  f_{k-1}(t):=\frac{1}{\sqrt{C}(t-{i})}R_{k-1}(t)\,,\quad
  \text{ for all }\quad k\in{\mathbb{N}}\,.$$ Thus, $$\label{eq:orthonormality-f}
  {\left\langlef_{k},f_{j}\right\rangle}_{L_{2}({\mathbb{R}},\mu)}=
  \frac{1}{C}\int_{{\mathbb{R}}}(1+t^{2})^{-1}{\overline{R_{k}(t)}}R_{j}(t)d\mu=
  \int_{{\mathbb{R}}}{\overline{R_{k}(t)}}R_{j}(t)d\nu_{1}=\delta_{jk}$$ so that $\{f_{k-1}\}_{k=1}^{\infty}$ is orthonormal in $L_{2}({\mathbb{R}},\mu)$. This orthonormal system is also complete due to the fact that if, for any $k\in{\mathbb{N}}$, $$  0={\left\langlef_{k-1},h\right\rangle}_{L_{2}({\mathbb{R}},\mu)}=
  \frac{1}{\sqrt{C}}\int_{{\mathbb{R}}}(t-{i})^{-1}R_{k-1}(t)h(t)d\mu(t)\,,$$ then $h(t)=0$ for $\mu$-a.e. $t\in{\mathbb{R}}$ since the polynomials are dense in $L_{2}({\mathbb{R}},\mu)$ and $(t-{i})^{-1}$ never vanishes on the real line. It is equally straightforward to establish that $f_{k-1}$ is in the domain of the multiplication operator $M_{\mu}$ (see Definition \[def:operator-multiplication\]) for any $k\in{\mathbb{N}}$ since $R_{k-1}$ is in the domain of $M_{\nu_{1}}$ for any $k\in{\mathbb{N}}$.

Thus, the orthonormal basis $\{f_{k-1}\}_{k=1}^{\infty}$ in $L_{2}({\mathbb{R}},\mu)$ satisfies (a) of Definition \[def:basis-representation\] with respect to the operator of multiplication $M_{\mu}$.

\[prop:f-is-basis-representation\] If $\mu$ is a solution to an indeterminate moment problem, then the orthonormal basis $\{f_{k-1}\}_{k=1}^{\infty}$ defined above is a basis of matrix representation for the operator $M_{\mu}$.

Item (a) of Definition \[def:basis-representation\] has already been established. Let us show that the sequence $\{f_{k-1}\}_{k=1}^{\infty}$ is obtained by the Gram-Schmidt procedure applied to the sequence $\{t^{k-1}(\sqrt{C}(t-{i}))^{-1}\}_{k=1}^{\infty}$ in $L_{2}({\mathbb{R}},\mu)$. Indeed, proceeding as in , one has for the first step of the Gram-Schmidt algorithm @MR1192782 [Ch.2 Sec.2 Thm.5] $$\begin{aligned}
    &(\sqrt{C}(t-{i}))^{-1}[t-
      {\left\langlet(\sqrt{C}(t-{i}))^{-1},(\sqrt{C}(t-{i}))^{-1}\right\rangle}_{L_{2}({\mathbb{R}},\mu)}]
    \nonumber\\
    =&(\sqrt{C}(t-{i}))^{-1}[t-
      {\left\langlet,1\right\rangle}_{L_{2}({\mathbb{R}},\nu_{1})}]\,.\label{eq:first-step-gs}
  \end{aligned}$$ The expression in the square brackets of is the first step of Gram-Schmidt procedure applied to the sequence $\{t^{k-1}\}_{k=1}^{\infty}$ in $L_{2}({\mathbb{R}},\nu_{1})$. By induction, taking into account , one verifies that $\{f_{k-1}\}_{k=1}^{\infty}$ is the result of orthonormalizing the sequence $\{t^{k-1}(\sqrt{C}(t-{i}))^{-1}\}_{k=1}^{\infty}$. In particular, this shows that $f_{0}=(\sqrt{C}(t-{i}))^{-1}$ is a cyclic vector of $M_{\mu}$ since $\{f_{k-1}\}_{k=1}^{\infty}$ is total in $L_{2}({\mathbb{R}},\mu)$.

Now, for any $k\in{\mathbb{N}}$, one has $$    [(M_{\mu}-{i}I) M_{\mu}^{k-1}f_{0}](t)=t^{k-1}\,,$$ which, on the basis of Proposition \[prop:density-polynomials\], implies that holds for $M_{\mu}$. Thus, Proposition \[prop:necessary-and-sufficient\] leads to the desired conclusion.

As a consequence of Propositions \[prop:gram-schmidt-basis\] and \[prop:f-is-basis-representation\], the matrix representation of $M_{\mu}$ with respect to $\{f_{k-1}\}_{k=1}^{\infty}$ is a Jacobi matrix. This matrix can be found by observing that the sequence $\{f_{k-1}\}_{k=1}^{\infty}$ satisfies the same three-term recurrence relation that the sequence of polynomials $\{R_{k-1}\}_{k=1}^{\infty}$ does. Since $\operatorname{ind}\nu_{1}=1$, the coefficients of the recurrence relation form a Jacobi matrix in the limit point case, which is denoted by $[{\widehat}J]$. Hence, the matrix representation of $M_{\mu}$ with respect to $\{f_{k-1}\}_{k=1}^{\infty}$ is $[{\widehat}J]$. Note that $U^{-1}{\widetilde}JU=M_{\mu}$, where $U$ is the map given in Definition \[def:map-U\]. Thus, if one defines $$    \omega_{k}:=Uf_{k-1}\,,\quad\text{ for all}\quad k\in{\mathbb{N}}\,,$$ then $\{\omega_{k}\}_{k=1}^{\infty}$ is a basis of matrix representation for ${\widetilde}J$ and the corresponding matrix is $[{\widehat}J]$. As has been said before (see Remark \[rem:example\]), $\{\delta_{k}\}_{k=1}^{\infty}$ is not a basis of representation for ${\widetilde}J$. Note that the sequence $\{\omega_{k}\}_{k=1}^{\infty}$ is not in $\operatorname{dom}J$ since otherwise the minimality condition (b) of Definition \[def:basis-representation\] for ${\widetilde}J$ is violated.

It is worth mentioning that the measure $\nu_{1}$, which gives rise to the matrix $[{\widehat}J]$, has the smallest index that a determinate measure could have. Furthermore, for any $n\in{\mathbb{N}}$, by using $\nu_{n}$ and modifying accordingly , one can construct a basis of matrix representation for ${\widetilde}J$ so that the corresponding matrix is a Jacobi matrix.

In contrast to the construction given above, the measures appearing in Section \[sec:self-simpl-oper\] have infinite index of determinacy. This is asserted in the following proposition.

\[prop:infinite-index\] For any $\alpha>0$, the measure $\mu_{\eta(\alpha,g)}$ given in has infinite index of determinacy.

Denote by $\chi_{t}$ the measure $$    \chi_{t}(\partial):=
    \begin{cases}
      1 & t\in\partial\\
      0 & t\not\in\partial\,,
    \end{cases}$$ where $\partial\subset{\mathbb{R}}$ is a Borel set.

If one assumes that $\mu_{\eta(\alpha,g)}$ has finite index, then the measure is discrete @MR1308001 [Cor.3.4] and according to @MR1308001 [Thm. 3.9] there is a finite collection of real numbers $t_{1},\dots,t_{n}$ out of the support of $\mu_{\eta(\alpha,g)}$ so that, for any positive numbers $a_{1},\dots,a_{n}$, $$    {\widetilde}\mu := \mu_{\eta(\alpha,g)}+\sum_{k=1}^{n}a_{k}\chi_{t_{k}}$$ is indeterminate. But, one verifies $$  \int_{{\mathbb{R}}}e^{{\left|t\right|}}d{\widetilde}\mu(t)=\int_{{\mathbb{R}}}e^{{\left|t\right|}-2\alpha
  t^{2}}d\mu_{g}(t)+\sum_{k=1}^{n}a_{k}e^{{\left|t_{k}\right|}}\chi_{t_{k}}<+\infty\,.$$ Whence, by @MR0481888 [Thm.5.2], one concludes that ${\widetilde}\mu$ is determinate which is a contradiction.

Non-selfadjoint Jacobi operators {#sec:non-self-jacobi}
================================

This section begins with an account on some of the remarkable properties of the class of entire operators @MR0012177 to which the class of nonselfadjoint Jacobi operators belongs. One of these properties leads to the fact that a nonselfadjoint Jacobi operator has a unique matrix representation being a Jacobi matrix.

\[def:regularity\] A closed operator $A$ in a Hilbert space ${\mathcal{H}}$ is said to be regular when for any $z\in{\mathbb{C}}$, there is a constant $C>0$ (which could depend on $z$) such that $$  {\left\|(A-zI)\phi\right\|}\ge C{\left\|\phi\right\|}$$ for all $\phi\in\operatorname{dom}A$.

The fact that an operator is regular means that the its spectral kernel is empty, therefore every regular symmetric operator is completely nonselfadjoint (<span>*i.e.*</span> there is no invariant subspace of the operator in which it induces a selfadjoint operator). Indeed, since any part of an operator with empty spectral kernel has empty spectral kernel, this part cannot be selfadjoint. It is noteworthy that there are completely nonselfadjoint operators which are not regular.

Completely nonselfadjointness of a closed symmetric operator $A$ means that @MR1764951 [Thm. 1.2.1] $$\label{eq:completely-nonselfadjoint}
  \bigcap_{z\in{\mathbb{C}}\setminus{\mathbb{R}}}\operatorname{ran}(A-z I)=\{0\}\,,$$ since the l.h.s of is the maximal invariant subspace in which $A$ is selfadjoint @MR1764951 [Thm.1.2.1].

\[def:involutive-conjugation\] An antilinear map $\mathcal{I}$ of $\mathcal{H}$ onto itself being an involution (<span>*i.e.*</span> $\mathcal{I}^{2}=\mathcal{J}$) and such that $$\label{eq:inversion-inner-conjugation}
    {\left\langle\mathcal{I}\phi,\mathcal{I}\psi\right\rangle}={\left\langle\psi,\phi\right\rangle}\quad\text{
    for any }\phi,\psi\in\mathcal{H}$$ is called a *conjugation* (see @MR2953553 [Sec.13.1] and @MR566954 [Eq.8.1]).

A procedure for constructing a conjugation commuting (see @MR566954 [Eq.8.1]) with all canonical selfadjoint extensions of a symmetric completely nonselfadjoint operator $A$ with one-dimensional deficiency spaces is presented in @MR3002855 [Prop.2.3]. This conjugation commutes with $A$ (see the proof of @MR2953553 [Prop.13.25(ii)]). Conversely, if a conjugation commutes with a symmetric operator $A$ with deficiency indices $n_{+}(A)=n_{-}(A)=1$, then the conjugation commutes with all canonical selfadjoint extensions of $A$ (see @MR2953553 [Prop.13.25 (iv)] and @MR1627806 [Cor.2.5]).

The following statement is motivated by Krein's representation theory of symmetric operators @MR0011170 [@MR0011533; @MR0012177; @MR0048704]. The assertion's constructive proof can be found in @MR3002855 [Prop.2.12].

\[prop:function-xi\] If $A$ is a regular, symmetric operator such that $n_{+}(A)=n_{-}(A)=1$ and $\mathcal{I}$ is a conjugation that commutes with $A$, then there is a vector function $\xi_{A}:{\mathbb{C}}\to\mathcal{H}$ with the following properties:

(a) $\xi_{A}$ is entire and zero-free.

(b) $\xi_{A}(z)\in\ker(A^{*}-z I)$ for each $z\in{\mathbb{C}}$.

(c) For all $z\in{\mathbb{C}}$, $\mathcal{I}\xi_{A}(z)=\xi_{A}({\overline{z}})$.

Having fixed the involution $\mathcal{I}$, the function $\xi_{A}$ is uniquely determined modulo a multiplicative scalar factor being an entire, zero-free function which turns out to be real (see @MR3002855 [Rem.2.13] and @zbMATH06526214 [Lem.3]). Recall that a complex valued function $f$ of complex variable satisfying $$\label{eq:real-function}
  {\overline{f(z)}}=f({\overline{z}})$$ is called *real*; thus a real function is real on the real line.

\[rem:function-xi\] It is worth mentioning that if, for a closed symmetric operator $A$ with $n_{+}(A)=n_{-}(A)=1$, the equality holds and there is a function $\xi_{A}$ satisfying (a)–(c) of Proposition \[prop:function-xi\], then the operator is regular. This is proven by means of the functional model given in @MR3002855 [Sec.2.3] and @MR3050164 [Sec.4] taking into account the properties of the operator of multiplication in a de Branges space @MR0229011.

For any regular, symmetric operator $A$ with $n_{+}(A)=n_{-}(A)=1$, there is $\mu$ in ${\mathcal{H}}$ such that $${\mathcal{H}}= \operatorname{ran}(A-zI)\dot{+}\operatorname*{span}\{\mu\}$$ for all $z\in{\mathbb{C}}\setminus S_{\mu}$, where $\operatorname{card}S_{\mu}\le\operatorname{card}{\mathbb{N}}$ (see @MR1466698 [Sec.2.2], @MR2607264 [Sec.2]). The set $S_{\mu}$ turns out to be at most countable since it is the zero set of the analytic function ${\left\langle\xi_{A}(\,{\overline{\,\boldsymbol{\cdot}\,}}\,),\mu\right\rangle}$, which does not vanish identically due to the fact that $\mu\ne 0$ and $\{\xi_{A}(z)\}_{z\in{\mathbb{C}}\setminus{\mathbb{R}}}$ is a total set in ${\mathcal{H}}$ (<span>*cf.*</span> ). The vector $\mu$ is said to be a gauge of $A$.

The gauge $\mu$ can be chosen in such a way so that the exceptional set $S_{\mu}$ lies entirely on the real line @MR2607264 [Lem.2.1] or completely outside the real line @MR2607264 [Thm.2.2]. This last assertion was first stated without proof in @MR0011533 [Thm.8].

\[def:entire-operator\] A regular, symmetric operator $A$ such that $n_{+}(A)=n_{-}(A)=1$ is said to be *entire* if there exists a gauge $\mu$ so that $S_{\mu}=\emptyset$. In this case $\mu$ is an *entire gauge* of $A$.

A straightforward consequence of this definition is that if $A$ is an entire operator and $\mu$ its entire gauge, then the entire function $$\label{eq:t-function}
    t(\boldsymbol{\cdot}):=
  {\left\langle\xi_{A}(\,{\overline{\,\boldsymbol{\cdot}\,}}\,),\mu\right\rangle}$$ is a zero free function. Another direct deduction is the following assertion.

\[lem:real-gauge\] The function $t$ given in is real if and only if $\mathcal{I}\mu=\mu$.

One has $$\label{eq:splited-equalities}
  \begin{split}
    {\left\langle\mu,\xi_{A}({\overline{z}})\right\rangle}&={\left\langle\xi_{A}(z),\mu\right\rangle}\\
                            &={\left\langle\mathcal{I}\xi_{A}({\overline{z}}),\mu\right\rangle}\\
                           &={\left\langle\mathcal{I}\mu,\xi_{A}({\overline{z}})\right\rangle}\,,
  \end{split}$$ where the first equality is actually , the second one follows from Proposition \[prop:function-xi\](c), and in the third one, and the involutive property of the conjugation are used. Thus, on the basis of , the assertion follows from .

It is established in @MR1466698 [Ch.2 Sec.4.1] that, for any entire operator $A$, the vector-valued function $\xi_{A}$ and the gauge $\mu$ can be chosen so that the scalar function $t$ given in is a real constant (see also @MR0012177 [Sec.2]). Below, it will be shown that the "natural" choice of the gauge $\mu$ and the function $\xi$ for an entire operator is the one for which $t$ is a real constant. However, having done this choice, we are also interested in the behavior of the zero-free entire function ${\left\langle\xi_{A}({\overline{z}}),{\widetilde}\mu\right\rangle}$ where the entire gauge $\mu$ has been substituted by another entire gauge ${\widetilde}{\mu}$. To this end, let us recall two notions related to the theory of growth of entire functions.

A function of at most exponential type is a function of at most order one and normal type @MR589888 [Ch.1 Sec.20]. The dependence of the growth of a function $f$ of exponential type on the direction in which the independent variable tends to infinity is given by the function $$    h_{f}(\theta):=\limsup_{r\to\infty}\frac{\log{\left|f(re^{i\theta})\right|}}{r}
    \qquad \theta\in [0,2\pi)$$ which is the so-called *indicator function* of the function of exponential type $f$ (see @MR589888 [Ch.1 Sec.15] and @MR0444912 [Ch.II.9 Sec.45]).

In @MR0048704 [Sec.8] (see also @MR1466698 [Ch.2 Sec.5]), the following assertion is established.

\[prop:krein-growth-functions\] Let $A$ be an entire operator and pick the corresponding function $\xi_{A}$ and gauge $\mu$ so that the function $t$ given in is a real constant. Then, for any $\phi\in \mathcal{H}$, the function $f(\boldsymbol{\cdot}):={\left\langle\xi_{A}(\,{\overline{\,\boldsymbol{\cdot}\,}}\,),\phi\right\rangle}$ is at most of exponential type and its indicator function obeys $$\label{eq:indicator}
    h_{f}(\theta)=
    \begin{cases}
      h_{f}(\frac{\pi}{2})\sin\theta & \text{if}\quad 0\le\theta\le\pi\\
      -h_{f}(-\frac{\pi}{2})\sin\theta & \text{if}\quad \pi<\theta\le 2\pi\,.
    \end{cases}$$

The proof of the first part of Proposition \[prop:krein-growth-functions\] is found in the paragraph preceding @MR0048704 [Lem.8.1] (see also @MR1466698 [Eq. 5.1]), where implicitly it is used that is a constant. As regards the second part see the proof of @MR0048704 [Lem.8.1] or the proof of @MR1466698 [Ch.2 Lem.5.1]).

The following assertion exhibits a property of entire operators which is crucial for this section. It is related to @MR0012177 [Thm.1] whose proof can be found in @MR2607264 [Prop.4.6].

\[prop:uniqueness-entire-gauge\] Let $A$ be an entire operator. If there are two entire gauges of $A$ and two functions $\xi_{A}^{(1)}$, $\xi_{A}^{(2)}$ satisfying (a)–(c) of Proposition \[prop:function-xi\] so that ${\left\langle\xi_{A}^{(1)}(\,{\overline{\,\boldsymbol{\cdot}\,}}\,),\mu_{1}\right\rangle}$ and ${\left\langle\xi_{A}^{(2)}(\,{\overline{\,\boldsymbol{\cdot}\,}}\,),\mu_{2}\right\rangle}$ are real constants, then there is a real constant $C$ such that $\mu_{1}=C\mu_{2}$.

Lemma \[lem:real-gauge\] yields that $\mathcal{I}\mu_{1}=\mu_{1}$ and $\mathcal{I}\mu_{2}=\mu_{2}$. Thus, the zero-free function $f(\boldsymbol{\cdot}):=
  {\left\langle\xi_{A}^{(1)}(\,{\overline{\,\boldsymbol{\cdot}\,}}\,),\mu_{2}\right\rangle}$ is real and has the form $\exp(g(\boldsymbol{\cdot}))$, where $g$ is a real entire function. Furthermore, since ${\left\langle\xi_{A}^{(1)}(\,{\overline{\,\boldsymbol{\cdot}\,}}\,),\mu_{1}\right\rangle}$ is a real constant, the first part of Proposition \[prop:krein-growth-functions\] implies that the function $f$ is a function of at most exponential type and therefore the function $g$ is a polynomial of the first degree, whence ${\left\langle\xi_{A}^{(1)}({\overline{z}}),\mu_{2}\right\rangle}=C\exp((a+{i}b)z)$ for all $z\in{\mathbb{C}}$.

On the one hand, it follows from the second part of Proposition \[prop:krein-growth-functions\] that the indicator function of $f$ is ; on the other hand, the indicator function of $\exp((a+{i}b)\,\boldsymbol{\cdot}\,)$ has the form: $$\label{eq:trigonometric-indicator}
    h(\theta)=a\cos\theta - b\sin\theta\,.$$ Comparing with , one arrives at the conclusion that $a=0$. Finally, it follows from the reality of the function ${\left\langle\xi_{A}^{(1)}(\,{\overline{\,\boldsymbol{\cdot}\,}}\,),\mu_{2}\right\rangle}$ that $b=0$ and $C$ is real.

Any nonselfadjoint Jacobi operator is regular. This is a classical result of the moment problem (or Jacobi operator) theory. It is shown by establishing that the spectra of its selfadjoint extensions do not intersect (see the proof of @MR0184042 [Thm.4.2.4] and @MR1627806 [Thm. 5]). Recall that the spectral kernel of an operator is contained in the spectral kernel of its extension, thus if a point is in the spectral kernel of a symmetric operator, then this point is in the spectrum on any of its selfadjoint extensions.

\[prop:jacobi-entire\] Any nonselfadjoint Jacobi operator is an entire operator.

It has been established that any nonselfadjoint operator $J$ having the matrix representation with respect to the orthonormal basis $\{\delta_{k}\}_{k=1}^{\infty}$ has deficiency indices $n_{+}(J)=n_{-}(J)=1$ and it is regular. As mention in Remark \[rem:on-kernel-adjoint-at-zeta\], the vector-valued function $\pi$ satisfies (b) of Proposition \[prop:function-xi\]. Also, it follows from the fact that the zeros of polynomials of the first kind interlace @MR0184042 [Thm.1.2.2] that $\pi$ complies with (a) of Proposition \[prop:function-xi\]. The property (c) is a consequence of the reality of the polynomials' coefficients. For finishing the proof it only remains to note that ${\left\langle\pi(\,{\overline{\,\boldsymbol{\cdot}\,}}\,),\delta_{1}\right\rangle}\equiv 1$.

\[prop:only-one-basis\] There is only one basis of representation (modulo reflection[^2]) with respect to which any nonselfadjoint Jacobi operator has a Jacobi matrix as its matrix representation. The Jacobi matrix representing a nonselfadjoint Jacobi operator is unique.

Let $J$ be nonselfadjoint and have the matrix representation with respect to the orthonormal basis $\{\delta_{k}\}_{k=1}^{\infty}$. As has been shown, $\delta_{1}$ determines all the elements of the orthonormal basis and, consequently, the entries of the Jacobi matrix. Indeed, the vectors $\delta_{2},\delta_{3},\dots$ are obtained by applying the Gram-Schmidt procedure to the sequence $\{J^{k-1}\delta_{1}\}_{k=1}^{\infty}$ (see the proof of Proposition \[prop:basis-representation\]). Likewise, as asserted in Remark \[rem:relation-moments-matrices\], the entries of the matrix can be obtained from the moments ${\left\langle\delta_{1},J^{k-1}\delta_{1}\right\rangle}$, $k\in{\mathbb{N}}$. Now, suppose that for $J$ there is another orthonormal basis $\{{\widetilde}\delta_{k}\}_{k=1}^{\infty}$ with respect to which $J$ has a Jacobi matrix representation. As shown in the proof of Proposition \[prop:jacobi-entire\], the vector ${\widetilde}\delta_{1}$ is an entire gauge of $J$ satisfying the hypothesis of Proposition \[prop:uniqueness-entire-gauge\]. Therefore ${\widetilde}\delta_{1}=C\delta_{1}$, where $C\in{\mathbb{R}}$. Since ${\left\|\delta_{1}\right\|}=\|{\widetilde}\delta_{1}\|=1$, one concludes that $C$ is either $1$ or $-1$.

Acknowledgments {#acknowledgments .unnumbered}
---------------

D.H.B. is supported with a postdoctoral fellowship by brett75@example.net. L.O.S. has been partially supported by CONACyT Ciencia de Frontera (949)411-0516. The authors thank Professor R. Szwarz for his comments and interest in this work.

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[^1]: A canonical selfadjoint extensions of a symmetric operator is a selfadjoint restriction of its adjoint.

[^2]: Reflection means that every element of the orthonormal basis is multiplied by $-1$.

{'source': 'arXiv_out/2107'}