m\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varvec{m}$$\end{document}–Isometric Composition Operators on Directed Graphs with One Circuit

The aim of this paper is to investigate m–isometric composition operators on directed graphs with one circuit. We establish a characterization of m–isometries and prove that complete hyperexpansivity coincides with 2–isometricity within this class. We discuss the m–isometric completion problem for unilateral weighted shifts and for composition operators on directed graphs with one circuit. The paper is concluded with an affirmative solution of the Cauchy dual subnormality problem in the subclass with circuit containing one element.


Introduction
The notion of m-isometric operator was introduced by Agler in [4] and initially investigated by Agler and Stankus in [5]. Recently, there have been published many papers devoted to problems related to m-isometric operators (see [1,6,[10][11][12]18,23]). In particular, much attention was paid to misometric unilateral and bilateral weighted shifts. Bermúdez et al. characterized m-isometric unilateral weighted shifts in [12,Theorem 3.4]. Later on, Abdullah and Le provided convenient characterizations of m-isometric unilateral and bilateral weighted shifts that bind their sequences of weights with certain real polynomials in one variable of degree at most m − 1 (see [1,Theorem 2.1,Theorem 5.2]).
Lately, more general classes of operators such as (m, p)-isometries (see [11,18]) have been investigated. Both papers [11] and [18] deal with composition operators on p for p ≥ 1. Gu provided a characterization of (m, p)isometric composition operators defined on p that ties cardinality of preimages φ −n (j) for j ∈ N with certain polynomials of degree at most m − 1 50 Page 2 of 26 Z. J. Jab loński, J. Kośmider IEOT (see [18,Theorem 2.9]). The operators considered in the above papers can be seen as composition operators on discrete measure spaces with counting measures. As opposed to them, we are focused on composition operators on discrete measure spaces with arbitrary discrete measures (see Sect. 2). The paper is devoted to investigation of classes of m-isometric operators which are more rich in examples than the classes of unilateral and bilateral weighted shifts. We focus on the class of composition operators on directed graphs with one circuit. It was used in [8,Section 3] in connection with the study of unbounded subnormality. Its weighted counterpart appeared much earlier in [14]. Moreover, these composition operators have been very popular lately. For example, Anand, Chavan and Trivedi utilized weighted composition operators of this type to provide an interesting example of analytic 3-isometric cyclic operator without the wandering subspace property (see [2,Example 3.1]). Although the operators belonging to this class can be, in general, unbounded and densely defined, we restrict ourselves to their subclass containing only bounded operators.
As shown in the paper, the class of m-isometric composition operators on directed graphs with one circuit is rich in strict m-isometries for m ≥ 2. Surprisingly, it turns out that the only completely hyperexpansive composition operators in this class are 2-isometric (see Corollary 2.15). This phenomenon is a consequence of the fact that, in particular, 2-expansiveness implies 2-isometricity within this class. It is worth to mention that 3-isometries have been very popular recently (see e.g. [2,10,23]). Beyond the results already mentioned from [2], these operators were, for instance, utilized by Bermúdez et al. as examples to shed some light on relations of certain ergodic and dynamical properties of m-isometric operators (see [10] for more details). That is why we provide an explicit description of 3-isometric composition operators on directed graphs with one circuit that involves real polynomials in one variable (see Theorem 3.5).
The paper is organized as follows. Section 2 begins with setting up preliminary notation and definitions. Later on, discrete measure spaces and composition operators on directed graphs with one circuit are introduced (see the standing assumption (2.8)). Next several technical results required in the further proofs are given. Theorem 2.11, which is the main result of this section, characterizes m-isometric composition operators on directed graphs with one circuit. As a consequence, Corollary 2.12 provides a characterization of m-isometric operators in the subclass of these operators with the circuit containing only one element. Finally, Corollary 2.15 shows that the notion of complete hyperexpansivity coincides with the notion of 2-isometricity in this class.
In Sect. 3 we state and investigate the m-isometric completion problem for unilateral weighted shifts and composition operators on directed graphs with one circuit. At the beginning we provide a complete description of the existence of a solution of the m-isometric completion problem for unilateral weighted shifts (see Proposition 3.2). Proposition 3.4, whose proof essentially relies on the properties of circulant matrices, shows that, under some additional conditions on the length of the circuit, there exists a solution of the m-isometric completion problem, if the measure of elements located on branches of φ is a priori given by polynomials of degree at most m − 1.
Using this fact we establish a characterization of 2-isometric and 3-isometric composition operators on directed graphs with one circuit having arbitrary number of elements in the circuit (see Theorem 3.5). This result is used in the proof of Proposition 3.7 which characterizes the existence of a solution of another 2-isometric completion problem, in which we assume that measure of elements located in the circuit is given. Section 4 is devoted to the study of subnormality of the Cauchy dual operators of 2-isometric composition operators on directed graphs with one circuit. We begin with a characterization of analyticity of composition operators on directed graphs with one circuit (see Proposition 4.2) and prove that m-isometric operators within this class are analytic (see Corollary 4.3). Theorem 4. 6 gives the affirmative answer to the Cauchy dual subnormality problem for 2-isometric operators with circuit having one element. We conclude this section with two results related to 2-isometric composition operators on directed graphs with one circuit, namely, we state a characterization of Δ C φ -regularity of these operators (see Proposition 4.7) and show that they never satisfy the kernel condition (see Theorem 4.9).

Characterization of m-isometries
In this paper we use the following notation. Symbols N, Z + , Z and R + stand for the sets of positive integers, nonnegative integers, integers and nonnegative reals, respectively. The fields of real and complex numbers are denoted by R and C, respectively.
Define a linear transformation : Relations 0 γ = γ and j+1 γ = j γ for j ∈ Z + inductively define j for all j ∈ Z + . Using Newton's binomial formula, we can easily prove that We say that γ n is a polynomial in n (of degree k) if there exists a polynomial p in one indeterminate x with real coefficients (of degree k) such that p(n) = γ n for all n ∈ Z + . As a consequence of the Fundamental Theorem of Algebra we see that, if exists, such p is unique.
The following well-known lemma plays a key role in our considerations (for more details see [21,Section 2]). we denote its positive square root. As usual, we write T * for the adjoint of T . Recall that an operator Given m ∈ N, we say that T is an m-isometry if B m (T ) = 0. Following [13], we say that an m-isometry T is strict if T is an isometry or m ≥ 2 and For an operator T ∈ B(H) and f ∈ H we define the sequence Observe that The following fact is a consequence of (2.3), (2.1) and Lemma 2.1.

Proposition 2.4. If T ∈ B(H)
and m ∈ N, then the following are equivalent: For the reader's convenience we recall the classification of all composition operators induced by self-maps of directed graphs having only one vertex with degree greater than one and all other vertices having degrees equal to one. It turns out that there are only two possible cases (see [8, Theorem 3.2.1]). The first case is related to the directed trees with one branching vertex and the second one is connected with the composition operators on directed graphs with one circuit (see Fig. 1 below). From now on we focus on the latter class of operators. Let us now recall the definition and fundamental facts regarding composition operators. By a discrete measure space we mean a measure space (X, A , μ), where X is a countably infinite set, A is the σ-algebra of all subsets of X and μ is a positive measure on A such that μ(x) := μ({x}) ∈ (0, ∞) for all x ∈ X. Suppose now that (X, A , μ) is a measure space and a mapping φ : if this is the case, then C φ 2 = h φ L ∞ (μ) (see [24,Theorem 1]). If n ∈ N, then by φ n we denote the n-fold composition of φ with itself; φ 0 is the identity mapping. It is easily seen that if φ is nonsingular and n ∈ Z + , then φ n is also nonsingular, so h φ n makes sense; in particular h φ 0 ≡ 1 and Combining this, Proposition 2.3 and [20, Lemma 2.1] we get the following fact.
The next result, which is a direct consequence of [20, Lemma 2.3(ii)], Lemma 2.1 and (2.5), provides a characterization of bounded m-isometric composition operators on discrete measure spaces.

Proposition 2.5.
If m ∈ N, (X, A , μ) is a discrete measure space and C φ ∈ B(L 2 (μ)), then the following are equivalent: x κ . . Figure 1. The graph connected with a composition operator on a directed graph with one circuit Observe that, if card(φ −1 (x)) = 0 for some x ∈ X, then C φ is not an m-isometry, as condition (ii) from Proposition 2.5 is not satisfied.
Let us gather the following assumptions.
are two disjoint systems of distinct points of X. Assume that (X, A , μ) is a discrete measure space and a self-map φ of X is defined by (2.8) An operator C φ defined on a discrete measure space (X, A , μ) satisfying (2.8) is called a composition operator on a directed graph with one circuit.
For κ ∈ N, denote by Φ 1 : Z → Z and Φ 2 : Z → {1, . . . , κ} the functions uniquely determined by the following formula It follows from the definition of functions Φ 1 and Φ 2 that It is easily seen that the Radon-Nikodym derivative h φ and Observe also that (cf. [8, (3.4 Using (2.11) and (2.13) we can prove that Now we collect some necessary facts needed in the proof of Theorem 2.10. First recall that (cf., [20, p. 524 ]) if m ∈ N and j ∈ J [1,m] , then The following result is used in the proof of Lemma 2.7.
Now we prove a technical lemma which is vital for later results.
Proof. First observe that by our assumptions h φ p (x r ) ∈ R + for all p ∈ J [1,m] and r ∈ J [1,κ] , and for r ∈ J [1,κ] , The latter, combined with the fact that This completes the proof.
This is a consequence of (2.4), (2.11) and the fact that C k φ = C φ k for any k ∈ N. The below proposition is the main tool used in the proof of Theorem 2.10.  [1,η] . Applying Lemma 2.1 we deduce that m γ i = 0 for i ∈ J [1,η] . Hence, by Corollary 2.2, there exists a system (2.21) Using Lemma 2.7 we get This combined with the fact that a i ≥ 0 for i ∈ J [1,η] , (2.21) and Lemma 2.1 proves the required equivalence and completes the proof. The next theorem states that if C φ is a bounded m-isometric composition operator on a directed graph with one circuit, then measure on branches form, in fact, polynomials of degree at most m − 2.
The following theorem, which is the main result of this section, establishes a characterization of m-isometric composition operators on directed graphs with one circuit. Theorem 2.11. Suppose (2.8) holds, m is an integer such that m ≥ 2 and C φ ∈ B(L 2 (μ)). Then the following conditions are equivalent: μ(x i,lκ+Φ2(p+r) ) = 0, r ∈ J [1,κ] . Our next result is a consequence of Proposition 2.9 and Theorem 2.11. Corollary 2.12. Suppose that (2.8) holds with κ = 1, C φ ∈ B(L 2 (μ)) and m is an integer such that m ≥ 2. Then C φ is an m-isometry if and only if μ(x i,j+1 ) is a polynomial in j of degree at most m − 2 for all i ∈ J [1,η] . Moreover, if there exists i 0 ∈ J [1,η] Note that, under the assumptions of Corollary 2.12, if C φ ∈ B(L 2 (μ)) is m-isometric, then for every t ∈ R + , C φ ∈ B(L 2 (μ t )) is also m-isometric, where μ t is the measure on A uniquely determined by Below we show that bounded composition operators on directed graphs with one circuit separate the class of m-isometric operators for m ≥ 2. Example 2.13. Suppose (2.8) holds with κ = 1, η ∈ J [1,∞] and m ≥ 2 is an integer. Let q be a real polynomial of degree m − 2 such that all coefficients of q are nonnegative, and set μ(x 1 ) = 1 and Observe that by (2.10), [1,η] and j ∈ N.
The result below provides a sufficient condition for a bounded composition operator on a directed graph with one circuit to be an m-isometry.
This and an induction argument on j implies that 0 (2.24) and consequently Combined with (2.25), (2.26) and Proposition 2.5, this completes the proof.
Since bounded 2-isometric operators are completely hyperexpansive and completely hyperexpansive operators are 2-expansive, the following corollary is an immediate consequence of Proposition 2.14.

Corollary 2.15. If (2.8) holds and C φ ∈ B(L 2 (μ)) then C φ is completely hyperexpansive if and only if it is 2-isometric.
Concerning Proposition 2.14, one may ask if m ≥ 2 and C φ ∈ B(L 2 (μ)) are such that (−1) m B m (C φ ) ≥ 0 implies B m (C φ ) = 0. The following example shows that this is not true. Example 2.16. Assume (2.8) holds with κ = 1 and η = 1 and suppose that m ≥ 2 is an integer. Let p be a polynomial of degree m − 1 such that p(j) > 0 for all j ∈ N. Set μ(x 1,j ) = p(j) for j ∈ N and μ(x 1 ) = 1. Then, by Corollary 2.12 the operator C φ is strictly (m + 1)-isometric, in particular, it is not m-isometric. It follows from Proposition 2.3 that (−1) m B m (C φ ) ≥ 0.

m-isometric completion problem
In this section we discuss some results related to the m-isometric completion problem. The counterparts of it for unilateral weighted shifts appeared in e.g. [1,22]. In [ (m+2)-isometric unilateral weighted shift with weight sequence starting with {a n } m n=1 (see [1,Proposition 2.7]). In this section we focus also on the misometric completion problem for composition operators on directed graphs with one circuit.
For future reference we need a technical lemma. Proof. We use induction on l. The case l = 0 is obvious. Assume that lemma holds for a fixed unspecified l ∈ N and let {b n } l+1 n=0 ⊆ (0, ∞). By the induction hypothesis there exists a polynomial v in one indeterminate x with real coefficients of degree l + 1 such that v(n) = b n for n ∈ J [0,l] , v(l . It is easily seen that w s is a polynomial in one indeterminate x with real coefficients of degree l + 2 and for any s ≥ 1, w s (n) = b n for n ∈ J [0,l+1] and w s (n) > 0 for n ∈ J [l+2,∞] . Since the function ψ : [1, ∞) s → w s (l + 2) ∈ R + is continuous and lim s→∞ ψ(s) = +∞, the proof is completed. [1,k] .
Recall that an operator S ∈ B( 2 ) is called a unilateral weighted shift with is a bounded sequence. It is a matter of a straightforward verification to see that these operators are bounded and injective. The reader should be aware that our notation for unilateral weighted shifts is different from the conventional one, i.e., Se n = λ n e n+1 . Let C be a class of operators. For a classical weighted shift, the completion problem within the class C entails determining whether or not a given initial finite sequence of positive weights may be extended to the sequence of weights of an injective, bounded unilateral weighted shift which belongs to the class C; such a shift is called a C class completion of the initial weight sequence. Now we provide the solution of the completion problem within the class of m-isometries for the classical weighted shift (cf. [1,Proposition 2.7]). Since a unilateral weighted shift with weights {λ n } ∞ n=1 ⊆ C \ {0} is unitarily equivalent to the shift with weight sequence {|λ n |} ∞ n=1 ⊆ (0, ∞), the following result can be generalized to unilateral weighted shifts with non-zero weights.
Since w m (i) > 0 for i = 0, . . . , m − 1, the sequence {λ n } m−1 n=1 given by the formula is well-defined and consists of positive real numbers. This and the fact that w m (0) = 1 imply that Combined with Proposition 3.2(ii) and w m (m) = 0, this implies that there is no solution of the m-isometric completion problem for {λ n } m−1 n=1 . If m is even, then it is enough to consider We leave the details to the reader. Now we concentrate on the completion problem for m-isometric composition operators on directed graphs with one circuit. Proposition 3.4. Suppose m, κ ∈ Z are such that κ > m ≥ 2. Let η, X, A and φ be as in (2.8), M ∈ (0, ∞) and {w i } η i=1 be a system of polynomials of degree at most m − 2 such that w i (j) > 0 for all i ∈ J [1,η] and j ∈ N and Then there exists a measure μ on A such that μ(x i,j ) = w i (j) for i ∈ J [1,η] , j ∈ N and the associated C φ ∈ B(L 2 (μ)) is an m-isometry. Moreover, there exists t 0 ∈ R such that all measures μ t having the above properties can be Proof. Define μ(x i,j ) = w i (j) for i ∈ J [1,η] and j ∈ N. In view of (2.4), (2.10) and Theorem 2.11 any solution [μ(x 1 ), . . . , μ(x κ )] T of (2.22) fulfills our requirements. Observe that (2.22) can be written in the following matrix form ⎡ μ(x i,lκ+Φ2(p+r) ), r ∈ J [1,κ] .
Since the matrix in (3.2), call it A, is circulant, it follows from [19, Proposition 1.1] that the rank of A is equal to κ−1 because the associated polynomial of A is (x − 1) m and the degree of the greatest common divisor of 1 − x κ and (x − 1) m is equal to 1. We claim that the rank of the augmented matrix of (3.2) is equal to κ−1. To prove this assume that μ(x r ) = 1 for every r ∈ J [1,κ] . It follows from (2.4) and (2.11) that C φ is bounded. Thus, by Remark 2.8, the assumptions of Proposition 2.9 are satisfied. Hence (2.20) holds with μ(x r ) = 1 for every IEOT r ∈ J [1,κ] . Now (2.19) implies that the sum of the rows of the augmented matrix of (3.2) is equal to 0. Thus the range of the augmented matrix is at most κ − 1. Combined with the fact that the rank of A is equal to κ − 1, this proves our claim.
By the Rouché-Capelli theorem there exists a solution of (3.2) and the solutions of the system (3.2) form a one-dimensional affine subspace of R κ . Note that the vector v = [t, . . . , t] for t ∈ R satisfies the equality Av = 0. The proof is complete.
By using similar reasoning we prove the following useful characterization of 2-isometric and 3-isometric composition operators on directed graphs with one circuit, which provides an explicit description of the measure μ on X.
Theorem 3.5. Suppose (2.8) holds with κ > 1. Then the following are equivalent: where w κ,t c,d is a polynomial of degree at most 2 given by Moreover, C φ is 2-isometric if and only if (ii) holds with a system of polynomials {w i } η i=1 of degree equal to 0 and with d i = 0 for all i ∈ J [1,η] . Proof. Let us first assume that κ ≥ 4. In view of Proposition 3.4 and its proof it is enough to show that a vector [μ(x i )] κ i=1 given by (3.3) is a solution of (3.2). This can be proved by using the fact that 3 w κ,t c,d = 0. It is a matter of direct verification that our result is true for κ = 2, 3.
Using the above tools we are ready to prove some results regarding the m-isometric completion problem for composition operators on directed graphs with one circuit. One can think of many ways to state the m-isometric completion problem in the case of these composition operators, thus we limit our considerations to only a selected number of possibilities.
First we begin with the simplest situation, namely, when the circuit has only one element. In this situation we can prove a result, similar in its nature, to the analogous completion problem for unilateral weighted shifts. Proposition 3.6. Let m ∈ N and assume that {a n } m n=1 ⊆ (0, ∞). Then there exist a discrete measure space (X, A , μ) and a self-map φ : X → X satisfying (2.8) with κ = 1 and η = 1 such that μ(x 1,n ) = a n for n ∈ J [1,m] and C φ is an (m + 2)-isometry. Moreover, if m = 1, then (X, A , μ) and φ : X → X can be chosen so that C φ is a 2-isometry. Proof. If m ≥ 2, then we use Lemma 3.1 for {a n } m n=1 and Corollary 2.12. If m = 1, then the result follows directly from Corollary 2.12.
Let us now prove another result regarding the 2-isometric completion problem in which we assume that measure of elements located in the circuit is given a priori. (i) there exist a discrete measure space (X, A , μ) and a self-map φ : X → X satisfying (2.8) such that μ(x n ) = a n for n ∈ J [1,κ] and Proof. (i)⇒(ii). This implication is a straightforward application of The- is a constant sequence and ( a) 1 . Define X and A as in (2.8). Let μ be a discrete measure on A such that [1,η] and j ∈ N, (3.5) where Theorem 3.5 implies that C φ is a 2-isometry.
Let us note that, if η > 1 in Proposition 3.7, then, if exists, the 2isometric operator that solves the completion problem is not unique. Indeed, let (X, A , μ) be as in Proposition 3.7(i). For t ∈ (0, 1) define a measure ν t : A → (0, ∞) in the following way: otherwise.
It is an easy observation that C φ satisfies Proposition 3.7(i) with ν t in place of μ.
The following important result is a direct consequence of Theorem 2.10 and Proposition 4.2. In what follows we investigate whether the Cauchy dual operator of a composition operator on a directed graph with on circuit is subnormal. The notion of the Cauchy dual operator was introduced by Shimorin in [26] on the occasion of a study of the Wold-type decomposition and the wandering subspace property. Recently, these topics have been very popular and authors established interesting results (see e.g. [2,3,6]). In particular, there is still an open problem of determining a characterization of the subnormality of the Cauchy dual operator of a 2-isometry (for a preliminary version of this problem, see [15]). Anand et al. provided two sufficient conditions under which 2-isometric operator satisfies the above property, namely, the kernel condition (see [3,Theroem 3.3]) and Δ T -regularity (see [3,Theorem 4.5]). In what follows, we also characterize when composition operators on directed graphs with one circuit satisfy these properties.
Let us recall that for a left-invertible operator T ∈ B(H) the Cauchy dual operator T of T is given by T = T (T * T ) −1 . It is well-known (see [27]) that, if C φ ∈ B(L 2 (μ)) is a composition operator, then and consequently, if moreover C φ is left-invertible, then C φ is a weighted composition operator with symbol φ and weight w φ := 1 h φ •φ , i.e., In what follows we use the following notation. If (X, A , μ) is a discrete measure space and w : X → (0, ∞) is a function, then by μ w and {ŵ n } ∞ n=0 we denote a discrete measure on A and a sequence of functions uniquely determined by μ w (x) = |w(x)| 2 μ(x), x ∈ X (4.7)