The Cauchy dual subnormality problem for cyclic $2$-isometries

The Cauchy dual subnormality problem asks whether the Cauchy dual operator of a $2$-isometry is subnormal. Recently this problem has been solved in the negative. Here we show that it has a negative solution even in the class of cyclic $2$-isometries.


Introduction
Very recently the Cauchy dual subnormality problem was solved negatively in the class of 2-isometric operators (see [2]). Originally formulated for completely hyperexpansive operators (see [10,Question 2.11]), the problem entails on determining whether the Cauchy dual operator of a member of the underlying class is subnormal. It is worth mentioning that there are relatively broad subclasses of 2-isometric or 2-hyperexpansive operators for which this problem has an affirmative solution (see [2] and [5] respectively). In the present paper we show that the Cauchy dual subnormality problem has a negative solution even in the class of cyclic 2-isometric operators. The counterexample is implemented with the help of a weighted composition operator on L 2 space over a directed graph with a circuit (see Theorem 4.4).
Apart from Introduction, the paper consists of three parts. The first one gives the theoretical background on weighted composition operators on L 2 spaces needed in this paper. In the next one, we construct a concrete class of weighted composition operators on an L 2 space and characterize the subnormality of their Cauchy duals. In the last part, by specifying the weights and using Hausdorff's moment problem technique along with subtle classical analysis, we get the required counterexample.
We will now provide the necessary concepts and facts related to the issues discussed, placing more emphasis on the Hausdorff moment problem. Given a complex Hilbert space H, we denote by B(H) the C * -algebra of all bounded linear operators on H. Let T ∈ B(H). We write |T | for (T * T ) 1/2 and call it the modulus of T. We say that T is cyclic if there exists a vector e 0 , called a cyclic vector of T, such that the linear span of the set {T n e 0 } ∞ n=0 is dense in H. We call T subnormal if there exist a complex Hilbert space K and a normal operator N ∈ B(K) such that H ⊆ K (an isometric embedding) and T h = N h for all h ∈ H. If T is left-invertible (or equivalently T is bounded from below), then T * T is an invertible element of B(H) and the operator T ′ := T (T * T ) −1 is called the Cauchy dual operator of T (abbreviated to: the Cauchy dual of T ). Finally, T is said to be a 2-isometry (or that T is 2-isometric) if We refer the reader to [11], [1] and [20] for more information on subnormal operators, 2-isometric operators and the Cauchy dual operation, respectively.
Hereafter C stands for the field of complex numbers. The ring of all polynomials in one complex variable z with complex coefficients is denoted by C[z]. Given a sequence {γ n } ∞ n=0 of complex numbers, we say that γ n is a polynomial in n (of degree d) if there exists p ∈ C[z] (of degree d) such that γ n = p(n) for all n ∈ Z + , where Z + := {0, 1, 2, . . .}. The following fact will be used later (cf. [12,Exercise 7.2]).
If γ n is a polynomial in n of degree d, then m n=0 (−1) n m n γ n = 0, m max{d + 1, 0}.
Clearly, any Hausdorff moment sequence is a Stieltjes moment sequence, but not conversely. Using Lebesgue's monotone convergence theorem, one can show the following.
Any bounded Stieltjes moment sequence is a Hausdorff moment sequence. (2) Recall that Hausdorff moment sequences can be characterized as follows (see [6,Proposition 6.11]).
In this paper, we adhere to the conventions: n j=m a j = 0 and n l=m a l = 1 whenever m > n and whatever a j 's are.
To simplify the notation, we write whenever f is a C-valued or a [0, ∞]-valued function on a set X.

Weighted composition operators on L 2 -spaces
Given a measure space (X, A , µ), we denote by L 2 (µ) the complex Hilbert space of all square µ-integrable A -measurable complex functions on X endowed with the standard inner product.
Definition 2.1. Let (X, A , µ) be a σ-finite measure space, φ : X → X be an A -measurable map and w : X → C be an A -measurable function. By a weighted composition operator in L 2 (µ) we mean a mapping C φ,w : We call φ and w the symbol and the weight of C φ,w , respectively.
As a matter of fact, C φ,w may not be well-defined. The well-definiteness of [µ] whenever f, g : X → C are Ameasurable functions such that f = g a.e.
Moreover, if C φ,w is well defined, u : X → C is A -measurable and u = w a.e.
To avoid repetition, we distinguish the following assumption, which we will often refer to in this article.
(X, A , µ) is a σ-finite measure space, φ : X → X is an A -measurable map and w : X → C is an A -measurable function such that µ w • φ −1 ≪ µ.
If the condition (6) Before going further, we make an important observation.
In view of [9,Proposition 8(v) and Theorem 18], the following is valid.
From now on, we assume that h φ,w takes finite values whenever C φ,w ∈ B(L 2 (µ)). This assumption is justified by (9).
(ii) Note that by (9) and (i), there exists c 1 , . By virtue of (9), we have where g : X → [0, ∞) is an A -measurable function such that g = 1 h φ,w on the set {h φ,w > 0}. It is easily seen that Now applying [3, Theorem 1.6.12] and (7), we obtain . This completes the proof.
Recall that (see [9, Lemma 26]) if (6) holds and C φ,w ∈ B(L 2 (µ)), then for any integer n 1, the nth power C n φ,w of C φ,w equals C φ n ,w [n] , where φ n denotes the n-fold composition of φ with itself (φ 0 is the identity map on X) and w [n] : X → C is the function given by [µ] and the following recurrence formula holds: where E φ,w (f ) stands for the conditional expectation of an A -measurable function f : X → [0, ∞) with respect to the σ-algebra φ −1 (A ) and the measure µ w ; we refer the reader to [9,Sect. 2.4] for the precise definitions of E φ,w (f ) and The above discussion and Proposition 2.4 yield the following.
: X → C is the function given by with w ′ as in (10).
The subnormality of the Cauchy dual C ′ φ,w of a left-invertible bounded weighted composition operator C φ,w can be characterized as follows (cf. [16,Theorem 4.5]).
We conclude this section by characterizing bounded 2-isometric weighted composition operators (see [15,Lemma 2.3] for the case of composition operators).
Proposition 2.7. Suppose that (6) holds and C φ,w ∈ B(L 2 (µ)). Then the following conditions are equivalent: Proof. Since C n φ,w = C φ n ,w [n] , we infer from (9) that C * n φ,w C n φ,w = M h φ n ,w [n] for any n ∈ Z + . Applying the definition of 2-isometricity, one can show that (i) and (ii) are equivalent. That (ii) and (iii) are equivalent follows from (13).

A family of weighted composition operators on ℓ 2 (Z + )
In this section we concentrate on a family of weighted composition operators coming from [7, Example 42] (see also [8,Section 3.2]).
Example 3.1. Denote by A the power set 2 Z+ of Z + and by µ the counting measure on 2 Z+ . Clearly, all selfmaps of Z + and complex functions on Z + are A -measurable. Note that (Z + , A , µ) is a σ-finite measure space. For n ∈ Z + , we denote by e n the element of L 2 (µ) given by Let w : Z + → C be any function. It is obvious that µ w • φ −1 ≪ µ and so C φ,w is well defined (cf. Proposition 2.2). It follows from (7) that which yields h φ,w (0) = α w and h φ,w (n) = |w(n + 1)| 2 for n 1, where α w := |w(0)| 2 + |w(1)| 2 .
Combined with (9), this implies that C φ,w ∈ B(L 2 (µ)) if and only if sup n 0 |w(n)| < ∞; if this is the case, then Moreover, by Proposition 2.4(i), C φ,w is bounded from below if and only if Concerning the cyclicity of C φ,w , one can make the following observation.
If C φ,w ∈ B(L 2 (µ)) and w(n) = 0 for all n 1, then C φ,w is cyclic with the cyclic vector e 0 .
This can be deduced from the equality (20) below which is a direct consequence of the definition.
Using Proposition 2.4, we can describe the Cauchy dual C ′ φ,w of a left-invertible C φ,w as follows.
It is well known that any 2-isometry is expansive (see [18,Lemma 1]), so we can consider its Cauchy dual operator. Below, we follow the conventions (4). Theorem 3.3. Let X, A , µ, φ and w be as in Example 3.1. Assume that C φ,w ∈ B(L 2 (µ)) and C φ,w is a 2-isometry. Then the following conditions hold: (i) the Cauchy dual C ′ φ,w of C φ,w is subnormal if and only if the sequence {h φ n ,w ′ [n] (0)} ∞ n=0 is a Hausdorff moment sequence, (ii) the value of h φ n ,w ′ [n] at 0 is given by the following explicit formula Proof. (i) The "only if" part is a direct consequence Proposition 2.6(ii). In view of Proposition 2.6(i) to prove the "if" part, it suffices to show that the sequence {h φ n ,w ′ [n] (k)} ∞ n=0 is a Stieltjes moment sequence for every k 1. Fix k 1. Arguing as in (15) with (φ n , w ′ [n] ) in place of (φ, w), we verify that Since the unilateral weighted shift on ℓ 2 (Z + ) with weights {|w(k + 1 + n)|} ∞ n=0 is 2-isometric (see the proof of Proposition 3.2), we deduce from 1 [4, Remark 4] that the unilateral weighted shift on ℓ 2 (Z + ) with weights 1 |w(k+1+n)| ∞ n=0 is subnormal, which by Berger-Gellar-Wallen theorem (see [13,14]) is equivalent to the fact that the sequence {h φ n ,w ′ [n] (k)} ∞ n=0 is a Stieltjes moment sequence. (ii) Arguing as in (i), we get It is a matter of simple verification that (27) holds for n = 0, 1 as well. 1 Recall that a 2-isometry is m-isometric for every integer m 2 (see [1, Paper I, §1]), and thus by [18, Lemma 1(a)] it is completely hyperexpansive. Corollary 3.4. Let X, A , µ, φ and w be as in Example 3.1. Assume that |w(0)| = |w(n)| = 1 for every integer n ≥ 2. Then C φ,w ∈ B(L 2 (µ)), C φ,w is a 2-isometry and C ′ φ,w is subnormal. Moreover, C ′ φ,w is an isometry if and only if C φ,w is an isometry or, equivalently, if and only if w(1) = 0.

Main example
The following example shows that the Cauchy dual subnormality problem has a negative solution even in the class of cyclic operators.  1 continued). Let X, A , µ, φ and w be as in Example 3.1. To achieve the main purpose of this paper, we will begin by specifying the weight w. Let x is any positive real number and let w be the weight function constructed as follows (for notational convenience the dependence of w on x will not be expressed explicitly). Set Then clearly α w (2 − w(0) 2 ) − 1 > 0 and Set w(n + 2) := ξ n (w(2)), n 1, where the functions ξ n are as in (22). It follows from the definition of w and (22) that sup n 0 w(n) < ∞, so by (17), C φ,w ∈ B(L 2 (µ)). Using Proposition 3.2, we deduce that C φ,w is a 2-isometry. It follows from (27) that , n ∈ Z + .
Now we compute the lth derivative ω  (i) if n ∈ Z + , l is a positive integer and x varies over Ω n , then (see (4)) Proof. (i) Applying the general Leibniz rule twice, we get which implies that (36) holds for l 2, n ∈ Z + and x ∈ Ω n . It is a matter of routine to verify that (36) holds for l = 1, n ∈ Z + and x ∈ Ω n as well. This yields (i).
(ii) Using (35) we verify that the factors in the square brackets appearing in (36) (the third one for i = 2, 3) when calculated at 0 are of the form:    This together with (36) implies that ω (l) n (0) is a polynomial in n of degree at most l whenever l ∈ {0, 1, 2, 3}. Therefore, (ii) is a direct consequence of (1) and (34).  Proof. It follows from (35) that that the third factor in the square brackets appearing in (36) for i = 4 when calculated at 0 is of the form 48S (2) n (0) + 16S   Now applying Lemma 4.2(ii) and Taylor's theorem to D m (see [19,Theorem 5.15] with n = 4 and α = 0) completes the proof.
Concerning Lemma 4.3, the reader is referred to Figure 1. Now we are ready to state the main result of the paper.