Taylor spectrum approach to Brownian-type operators with quasinormal entry

In this paper, we introduce operators that are represented by upper triangular $2\times 2$ block matrices whose entries satisfy some algebraic constraints. We call them Brownian-type operators of class $\mathcal Q,$ briefly operators of class $\mathcal Q.$ These operators emerged from the study of Brownian isometries performed by Agler and Stankus via detailed analysis of the time shift operator of the modified Brownian motion process. It turns out that the class $\mathcal Q$ is closely related to the Cauchy dual subnormality problem which asks whether the Cauchy dual of a completely hyperexpansive operator is subnormal. Since the class $\mathcal Q$ is closed under the operation of taking the Cauchy dual, the problem itself becomes a part of a more general question of investigating subnormality in this class. This issue, along with the analysis of nonstandard moment problems, covers a large part of the paper. Using the Taylor spectrum technique culminates in a full characterization of subnormal operators of class $\mathcal Q.$ As a consequence, we solve the Cauchy dual subnormality problem for expansive operators of class $\mathcal Q$ in the affirmative, showing that the original problem can surprisingly be extended to a class of operators that are far from being completely hyperexpansive. The Taylor spectrum approach turns out to be fruitful enough to allow us to characterize other classes of operators including $m$-isometries. We also study linear operator pencils associated with operators of class $\mathcal Q$ proving that the corresponding regions of subnormality are closed intervals with explicitly described endpoints.


Introduction
Given two complex Hilbert spaces H and K, we denote by B(H, K) the Banach space of all bounded linear operators from H to K. The kernel, the range, the adjoint and the modulus of an operator T ∈ B(H, K) are denoted by N (T ), R(T ), T * and |T |, respectively. We regard B(H) := B(H, H) as a C * -algebra. The identity operator on H is denoted by I H , or simply by I if no ambiguity arises. Recall that an operator T ∈ B(H) is said to be quasinormal if T T * T = T * T T, or equivalently if T |T | = |T |T. We say that T is subnormal if there exist a complex Hilbert space K and a normal operator N ∈ B(K) such that H ⊆ K (an isometric embedding) and Sh = N h for all h ∈ H. It is well known that quasinormal operators are subnormal (see [17,Proposition II.1.7]). We refer the reader to [17] for more information on these classes of operators.
Let T ∈ B(H). We say that T is a 2-isometry if T * 2 T 2 − 2T * T + I = 0. We call T a Brownian isometry if T is a 2-isometry such that △ T △ T * △ T = 0, where △ T = T * T −I. If △ T 0 and △ T T = △ 1/2 T , we say that T is △ T -regular. By a quasi-Brownian isometry we mean a △ T -regular 2-isometry. It is well known that any 2-isometry is left-invertible 1 and △ T 0 ([40, Lemma 1]). The notion of a 2isometry was invented by Agler in [2], while the notion of a Brownian isometry was introduced by Agler and Stankus in [3]. The class of 2-isometric operators emerged from the study of the time shift operator of the modified Brownian motion process from one side [3], and from the investigation of invariant subspaces of the Dirichlet shift from the other [40]. The class of △ T -regular 2-isometries were investigated in [36,10] and in [5,6] under the name of quasi-Brownian isometries.
Given a left-invertible operator T ∈ B(H), we set T ′ = T (T * T ) −1 . Following [42], we call T ′ the Cauchy dual operator of T. Recall that if T is left-invertible, then so is T ′ and T = (T ′ ) ′ . Athavale noticed that the Cauchy dual operator of a completely hyperexpansive injective unilateral weighted shift is a subnormal contraction (see [9,Proposition 6] with t = 1), but not conversely (see [9,Remark 4]). The Cauchy dual subnormality problem asks whether the Cauchy dual operator of 1 In this paper, left-invertibility and invertibility of an operator T ∈ B(H) refer to the algebra B(H). a completely hyperexpansive operator (see Section 9 for the definition) is a subnormal contraction (see [15,Question 2.11]). As shown in [5], the answer is in the negative even for 2-isometries, that is, there are 2-isometries whose Cauchy dual operators are not subnormal (recall that each 2-isometry is completely hyperexpansive and that the Cauchy dual operator of a completely hyperexpansive operator is always a contraction). However, as proved in [5,Theorem 4.5], the Cauchy dual operator T ′ of a quasi-Brownian isometry T is a subnormal contraction (see also [10,Theorem 3.4] for a recent generalization of this result to the case of completely hyperexpansive △ T -regular operators). This leads to the question of why this phenomenon can happen. We will try to answer it by regarding quasi-Brownian isometries as elements of a larger class of operators which is closed under the operation of taking the Cauchy dual (note that the class of quasi-Brownian isometries is not closed under this operation). As a consequence, in the larger class of operators, the Cauchy dual subnormality problem becomes a part of the more general question of finding necessary and sufficient conditions for subnormality.
Let us recall that non-isometric Brownian and quasi-Brownian isometries have upper triangular 2 × 2 block matrix representations with entries satisfying some algebraic constraints (see the remark just after Definition 1.1). For the purposes of our paper explained in the above discussion, we introduce a wider class of operators consisting of the so-called Brown-type operators.
Definition 1.1. We say that T ∈ B(H) is a Brownian-type operator if it has the block matrix form with respect to a nontrivial 2 orthogonal decomposition H = H 1 ⊕ H 2 , where the operators V ∈ B(H 1 ), E ∈ B(H 2 , H 1 ) and Q ∈ B(H 2 ) satisfy the following conditions: V is an isometry, i.e., V * V = I, Moreover, if Q is quasinormal, we call T a Brownian-type operator of class Q and write T = V E 0 Q ∈ Q H1,H2 ; to simplify the terminology, we say that T is an operator of class Q. By analogy, if Q is isometric (resp. unitary, normal, etc.), then T is called an operator of class I (resp. U, N , etc.). If K is a complex Hilbert space and H = K ⊕ K (understood as an external orthogonal sum), then we abbreviate Q K,K to Q K .
In Definition 1.1, we have decided to exclude the case when one of the summands H 1 or H 2 is absent because otherwise the operator T is quasinormal. Moreover, by (2) and (3), the hypothesis that E = 0 excludes the case when H 1 is finite dimensional. Notice also that by the square root theorem [43,Theorem 2.4.4], the equality (4) is equivalent to Q|E| = |E|Q. One can deduce from [3, Proposition 5.37 and Theorem 5.48] (resp., [36,Proposition 5.1]) that a non-isometric operator T ∈ B(H) is a Brownian isometry (resp., a quasi-Brownian isometry) if and only if T is of class U (resp., of class I) (to avoid injectivity of E postulated in [3,Proposition 5.37] and [36,Proposition 5.1], consult [5,Theorem 4.1]). This means that Brownian isometries are quasi-Brownian isometries. In view of [5,Example 4.4], the converse implication is not true in general.
It is worth pointing out that upper triangular 2 × 2 block matrices appear in different parts of operator theory and functional analysis on the occasion of investigating variety of topics; for example, the hyperinvariant subspace problem [22,34,32,33], the Halmos similarity problem for polynomially bounded operators [24,38], the task of finding models for the time shift operator for modified Brownian motion process [3], the question of characterizing invertibility of upper triangular 2 × 2 block matrices [29], the task of searching for a model theory for 2-hyponormal operators [20], the problem of determining a complete set of unitary invariants for the class of Cowen-Douglas operators realized as upper triangular 2 × 2 block matrices [31], and many others.
We state now the main result of this paper which characterizes subnormality of operators of class Q in terms of the Taylor spectrum σ(|Q|, |E|) of the pair (|Q|, |E|). The spectral region for subnormality of operators of class Q is described by Theorem 1.2(iii) and illustrated in Figure 1. We refer the reader to Section 2 for the necessary definitions and notations. H2 . Let P ∈ B(H 2 ) be the orthogonal projection of H 2 onto M := R(|E|). Then the operators |Q|, |E| and P commute, M reduces |Q| and |E|, and the following conditions are equivalent: The proof of Theorem 1.2 is fairly long and it occupies most of Sections 3, 4 and 5. The theorem itself has many applications spread over Sections 5, 6 and 10. In particular, we show that contractions of class Q are subnormal (see Corollary 5.2), we solve the Cauchy dual subnormality problem for expansive operators of class Q in the affirmative (see Corollary 6.2) and, what is more important, we completely characterize subnormality of the Cauchy-duals of left-invertible operators of class Q (see Theorem 6.1). The study of linear operator pencils associated with operators of class Q provides a useful test of the applicability of the main theorem (see Theorems 10.1 and 10.2).
The Taylor spectrum approach developed in this paper for the purpose of investigating subnormality turns out to be efficient when studying other collections of operators of class Q including m-contractions, m-isometries, etc. (see Section 9). In fact, it appears to be effective even in providing explicit formulas for the norm of operators of class Q (see (17) in Section 3) and for the right endpoints of the intervals of subnormality of linear operator pencils associated with operators of class Q (see (75) and (79) in Section 10). The Taylor spectrum technique is also applied to characterize quasi-Brownian and Brownian isometries of class Q in Sections 7 and 8, respectively. Unexpectedly, the Brownian case is essentially more complicated. The reader has to be aware of the fact that quasi-Brownian (and so Brownian) isometries are always of class Q, however relative to properly selected orthogonal decompositions (of the underlying Hilbert spaces), which are not necessarily easy to be determined in concrete cases.
We conclude Introduction by pointing out that the overwhelming majority of the characterizations of selected subclasses of the class Q that appear in this paper consist in finding for a given subclass a minimal universal subset of the Euclidean plane having the property that an operator T = V E 0 Q ∈ Q H1,H2 belongs to the subclass if and only if the Taylor spectrum σ(|Q|, |E|) of the pair (|Q|, |E|) is contained in the aforementioned subset. The universality of this subset lies in the fact that it does not depend on the choice of the orthogonal decomposition H 1 ⊕ H 2 of the underlying Hilbert space H relative to which a given operator T ∈ B(H) is of class Q, i.e., T has the block matrix form (1) with V, E and Q satisfying (2)- (5). What is more interesting, there may exist different orthogonal decompositions of H relative to which the given operator T is of class Q and the Taylor spectra σ(|Q|, |E|) of the corresponding pairs (|Q|, |E|) are significantly different (see Example 7.3). It turns out that the class of Brownian isometries is the only subclass of Q considered in this paper which cannot be characterized by the Taylor spectrum σ(|Q|, |E|) of the pair (|Q|, |E|) (see Remark 8.5).

Prerequisites
In this section we fix notation and terminology and give necessary facts. Let Z, R and C stand for the sets of integers, real numbers and complex numbers, respectively. Denote by N the set of positive integers. Set Given a set X, we write χ ∆ for the characteristic function of a subset ∆ of X. The σ-algebra of all Borel subsets of a topological space X is denoted by B(X). For x ∈ X, δ x stands for the Borel probability measure on R supported on {x}.
Let H be a complex Hilbert space. We call an operator T ∈ B(H) a contraction (resp., an expansion) if T h h for all h ∈ H (resp., T h h for all h ∈ H), or equivalently if T * T I (resp., T * T I). The contractivity of T can also be characterized by requiring that T 1 (however T 1 does not characterize expansivity of T ). Obviously, T is an isometry if and only T is simultaneously a contraction and an expansion. We write σ(T ) for the spectrum of T . If G is a regular Borel spectral measure on a topological Hausdorff space X, then supp G denotes the closed support of G, i.e., X \ supp G is the largest open subset ∆ of X such that G(∆) = 0. Recall that if T ∈ B(H) is a selfadjoint operator and G is the spectral measure of T, then σ(T ) = supp G. The following elementary fact will be frequently used in this paper.
We refer the reader to [13,Chapter 6] for more details on spectral theory of Hilbert space operators. A pair (T 1 , T 2 ) of commuting operators T 1 , T 2 ∈ B(H) is said to be a spherical contraction (resp., spherical expansion) if T * 1 T 1 +T * 2 T 2 I (resp., T * 1 T 1 +T * 2 T 2 I). If (T 1 , T 2 ) is simultaneously spherical contraction and spherical expansion, that is is called a spherical isometry (see [8]). For a pair (T 1 , T 2 ) of commuting operators T 1 , T 2 ∈ B(H), we denote by σ(T 1 , T 2 ) the Taylor spectrum of (T 1 , T 2 ), and by r(T 1 , T 2 ) the geometric spectral radius of (T 1 , T 2 ), that is, The reader is referred to [48,50,19,37,16] for the definitions and the basic properties of the Taylor spectrum and the geometric spectral radius (of commuting n-tuples of operators). In particular, the Taylor spectrum σ(T 1 , T 2 ) is a nonempty compact subset of C 2 whenever H = {0}. Moreover, it has the following projection property (see [48,Lemma 3.1]; see also [19,Theorem 4.9]): where π 1 , π 2 : C 2 → C are defined by π 1 (z 1 , z 2 ) = z 1 and π 2 (z 1 , z 2 ) = z 2 for (z 1 , z 2 ) ∈ C 2 . The following fact follows directly from the projection property of the Taylor spectrum.
Note that under the assumptions of (8), σ(T 1 , However, the first equation may not hold even for positive operators (see (56) in Example 6.4). For a given pair (T 1 , T 2 ) of commuting selfadjoint operators T 1 , T 2 ∈ B(H), there exists a unique Borel spectral measure G : B(R 2 ) → B(H), called the joint spectral measure of (T 1 , T 2 ), such that where as usual C[x 1 , x 2 ] stands for the ring of polynomials in indeterminates x 1 , x 2 with complex coefficients (similar notations are used throughout the paper with no further explanation). The joint spectral measure G is the product of the spectral measures of T 1 and T 2 (see [13, Theorem 6.5.1]). As shown below, in this particular case, the Taylor spectrum σ(T 1 , T 2 ) coincides with the closed support of the joint spectral measure G; this yields the spectral mapping theorem for continuous functions. 3 Theorem 2.1. Suppose that T 1 , T 2 ∈ B(H) are commuting selfadjoint operators with the joint spectral measure G. Then the following assertions are valid: (i) σ(T 1 , T 2 ) = supp G; moreover, if T 1 , T 2 are positive, then σ(T 1 , T 2 ) ⊆ R 2 + , (ii) for any continuous function ψ : Proof. First observe that by (7) we have so if additionally T 1 and T 2 are positive, then σ(T 1 , T 2 ) ⊆ R 2 + . (i) First note that the Taylor spectrum σ(T 1 , T 2 ) coincides with the left spectrum of (T 1 , T 2 ) (see [19,Proposition 7.2]). It is a routine matter to show that the left spectrum of (T 1 , T 2 ) coincides with the approximate point spectrum of (T 1 , T 2 ) (this is true for an arbitrary pair of commuting Hilbert space operators). Hence, for (λ 1 , λ 2 ) ∈ R 2 , (λ 1 , λ 2 ) / ∈ σ(T 1 , T 2 ) if and only if there exists c ∈ (0, ∞) such that or equivalently, by [13,Theorem 6.5.3], if and only if (λ 1 , λ 2 ) / ∈ supp G. Combined with (10), this proves (i).
Hence G • ψ −1 is the joint spectral measure of the pair ψ(T 1 , T 2 ). This yields (To get ( * ) adapt the proof of [46,Lemma 3.2].) This completes the proof. 3 Note that Theorem 2.1 remains true for commuting normal operators with C in place of R.
As a consequence of Theorem 2.1, we obtain the following.
If T 1 , T 2 ∈ B(H) are commuting and selfadjoint operators, then For this, note that σ(T 1 , where p(s, t) = s · t. Hence, applying Theorem 2.1(ii) gives the former equivalence in (11). The latter is a matter of routine verification.
The following lemma is surely folklore. For self-containedness we sketch its proof (the reader can easily formulate a version for commuting normal operators).
We now describe the Taylor spectrum of an orthogonal sum of pairs of commuting selfadjoint operators. Proposition 2.3. Suppose that for every n ∈ N, (T 1,n , T 2,n ) is a pair of commuting selfadjoint operators on a nonzero complex Hilbert space H n . For j = 1, 2, let T j = ∞ n=1 T j,n . Then (T 1 , T 2 ) is a pair of commuting selfadjoint operators such that Proof. Set T = (T 1 , T 2 ) and T n = (T 1,n , T 2,n ) for n ∈ N. Denote by G T and G T n the joint spectral measures of T and T n , respectively. Let G Tj and G Tj,n be the spectral measures of T j and T j,n , respectively, where j = 1, 2 and n ∈ N. It is clear that This implies that Combined with the uniqueness of joint spectral measures, this yields In view of Theorem 2.1(i), it suffices to show that For this, take (s, t) ∈ R 2 . If (s, t) / ∈ supp G T , then there exists an open set ∆ in R 2 such that (s, t) ∈ ∆ and G T (∆) = 0. Therefore by (15), G T n (∆) = 0 for all n ∈ N, which implies that (s, t) / ∈ supp G T n for all n ∈ N. As a consequence, ∞ n=1 supp G T n ⊆ supp G T , which shows that the right side of (16) is contained in the left side. In turn, if (s, t) / ∈ ∞ n=1 supp G T n , then there exists an open set ∆ in R 2 such that (s, t) ∈ ∆ and ∆ ∩ ∞ n=1 supp G T n = ∅. Hence, G T n (∆) = 0 for all n ∈ N, which together with (15) implies that G T (∆) = 0. As a consequence, (s, t) / ∈ supp G T . This completes the proof.
Proof. Since R 2 is separable metric space, so is Γ. Hence, there exists a sequence {(x 1,n , x 2,n )} ∞ n=1 ⊆ Γ which is dense in Γ. The proof is completed by applying Proposition 2.3 to H n = C, T 1,n = x 1,n I C and T 2,n = x 2,n I C and by observing that according to (8), σ(T 1,n , T 2,n ) = {(x 1,n , x 2,n )} for all n ∈ N.
Remark 2.5. A closer inspection of the proof reveals that Proposition 2.3 remains valid for families (of arbitrary cardinality) of pairs of commuting normal operators. As a consequence, Corollary 2.4 remains true if selfadjoint operators are replaced by normal operators and R by C. What is more, using only the definition of the Taylor spectrum, one can show that (14) holds (certainly without the closure) for any finite number of pairs of commuting operators (cf. [18]). ♦

Fundamental properties of operators of class Q
In this section we prove some basic properties of operators of class Q that are needed in this paper. We begin by showing that the operators of class Q form a huge class which can be parameterized in a sense by arbitrary pairs of commuting positive operators. (i) if T = V E 0 Q ∈ Q H1,H2 , then |Q| and |E| are commuting positive opera- Proof. (i) That |Q| and |E| commute follows from (4) and the square root theorem. Let E = U |E| be the polar decomposition of E. Then U maps R(|E|) unitarily onto R(E). Since by (3), By the uniqueness of the square root, we deduce that |E| = B. It is easily seen that T = V E 0 A ∈ Q H1,H2 . The "moreover" part is a direct consequence of (i) and (ii). This completes the proof.
The theorem below is crucial for our further investigations because the overwhelming majority of results of this paper are stated in terms of the Taylor spectrum of the pair (|Q|, |E|). Theorem 3.3. Suppose that Γ is an arbitrary nonempty compact subset of R 2 + and H 2 is a separable infinite dimensional complex Hilbert space. Then there exists a nonzero complex Hilbert space Proof. It follows from Corollary 2.4 that there exists a pair (A, B) of commuting positive operators A, B ∈ B(H 2 ) such that σ(A, B) = Γ. Applying Corollary 3.2 completes the proof.
As shown below the norm of an operator of class Q can be expressed in terms of the geometric spectral radius of the pair (|Q|, |E|).
Proof. Let G be the joint spectral measure of the pair (|Q|, |E|) and let G-ess sup ϕ stand for the essential supremum of a Borel function ϕ : R 2 + → R + with respect to the measure G. It follows from Definition 1.1 that Combined with the hypothesis that the spaces H 1 and H 2 are nonzero, this implies that where ( * ) follows from Theorem 2.1(i) and the continuity of the function (s, t) → (s 2 + t 2 ) 1/2 on R 2 + . This completes the proof.
Remark 3.5. It follows from Proposition 3.4 that if T ∈ B(H) is of class Q and T > 1, then the geometric spectral radius r(|Q|, |E|) does not depend on the choice of an orthogonal decomposition H = H 1 ⊕ H 2 of H relative to which T has a block matrix representation (1) with entries V, E and Q satisfying the conditions (2)-(5). We refer the reader to Example 7.3 for a detailed discussion of the question of the existence of different orthogonal decompositions of the underlying Hilbert space H relative to which a given operator T ∈ B(H) is of class Q. ♦ Next we characterize contractive, isometric and expansive operators of class Q.
For self-containedness, we state the following result whose straightforward proof is left to the reader.
Proposition 3.7. The class Q is closed under the operation of taking orthogonal sums, i.e., if {T ι } ι∈J is a uniformly bounded family of operators of class Q, then ι∈J T ι is an operator of class Q.
The following lemma provides a sufficient condition for the product of two quasinormal operators to be quasinormal.
Lemma 3.8. Suppose that Q 1 , Q 2 ∈ B(H) are commuting quasinormal operators such that Q 1 commutes with Q * 2 Q 2 and Q 2 commutes with Q * 1 Q 1 . Then Q 1 Q 2 is quasinormal. Moreover, any positive integer power of a quasinormal operator is quasinormal.
Proof. We leave the simple algebraic proof of the first part to the reader. The "moreover" part follows from the first part by applying the formula which is valid for any quasinormal operator Q.
Our next goal is to give a sufficient condition for the product of two operators of class Q to be of class Q.
Clearly, V 1 V 2 is an isometry, while by Lemma 3.8, It turns out that the operation of taking positive integer powers is inner in the class Q. The class Q is also closed under the operation of taking the Cauchy dual. Furthermore, we discuss the questions of when an operator of class Q is △ T -regular and when it satisfies the kernel condition introduced recently in [5].
(ii) T * n T n = I 0 0 Ωn ∈ Q H1,H2 for any n ∈ Z + , where (iv) if T is left-invertible, then T ′ ∈ Q H1,H2 and Proof. (i) Using induction, one can verify that By induction, (24) implies (20). Clearly for any n ∈ Z + , V n is an isometry and, by Lemma 3.8, Q n is a quasinormal operator. Since V is an isometry, we infer from (3) and (20) that V * n E n = 0 for any n ∈ Z + . Employing (24), we see that Using induction and (4), we deduce that Q commutes with E * n E n for all n ∈ Z + . This implies that T n is of class Q for any n ∈ Z + .
(iii) It is clear that T is left-invertible if and only if T * T is invertible, which by (ii) with n = 1 is equivalent to the invertibility of Ω 1 . The remaining statement in (iii) is a direct consequence of Theorem 2.1(i) and Lemma 2.2.
(iv) It is a routine matter to show that (23) holds and then to verify that T ′ is of class Q.
(v) The "only if" part is obvious. To prove the "if" part, notice that by (ii), Since T is an expansion, we see that Ω 1 − I 0 and Knowing that Q commutes with Ω 1 and using the square root theorem, we deduce that Q commutes with (Ω 1 − I) 1/2 , and consequently by (27), To prove the "if" part, suppose that If h 1 ⊕ h 2 ∈ N (T * ), then in view of (28) and (29), we have Hence by (ii) with n = 1 and (28), , which justifies the "if" part. The "only if" part goes by reversing the above argument. This completes the proof.

Moment theoretic necessities
In this section we prove a series of lemmata concerning Hamburger and Stieltjes moment problems needed in subsequent sections of this paper. We state some of them in a more general context, namely for the multi-dimensional moment problems, because the proofs are essentially the same.
Below we use the standard multi-index notation, that is, where |µ| denotes the total variation measure of µ. We write supp µ for the closed support of a finite positive Borel measure µ on R d (the support exists because such µ is automatically regular, see [41,Theorem 2.18]). We say that a multi-sequence If such µ is unique, then Suppose that µ 1 and µ 2 are compactly supported complex Borel measures on R d such that Then Let f be a continuous complex function on R d vanishing at infinity. By the Stone- Since we deduce from (32) that R d f dµ = 0. Applying [41, Theorems 6.19 and 2.18] yields µ = 0, or equivalently, µ 1 = µ 2 .
As a consequence, we have Moreover, the mapping C ∋ ϑ −→ ρ + ϑµ is an injection. ♦ Proof. Since |x α | R α for all α ∈ Z d + and x ∈ supp |µ|, we get Lemma 4.4. Let d ∈ N, µ be a compactly supported complex Borel measure on R d and γ α = R d x α dµ(x) for α ∈ Z d + . Then the following conditions are equivalent: We state now the following fact which we need in the proof of Lemma 4.6. It can be proved by induction on the degree of the polynomial in question.
where △ : C Z+ → C Z+ is the linear transformation given by (△γ) n = γ n+1 − γ n for n ∈ Z + and γ ∈ C Z+ ,p ∈ C Z+ is given byp n = p(n) for n ∈ Z + and p (m) (0) stands for the mth derivative of p at 0.
As shown below, a nonconstant polynomial perturbation of a Hamburger moment sequence is never a Hamburger moment sequence. (i) the sequence {γ n + p(n)} ∞ n=0 is a Hamburger moment sequence, (ii) p is a constant polynomial and µ({1}) + p(0) 0. Moreover, if (ii) holds, then µ+p(0)δ 1 is a compactly supported representing measure of {γ n + p(n)} ∞ n=0 . Proof. Without loss of generality we may assume that the polynomial p is nonzero, that is k := deg p 0.
The following is an immediate consequence of Lemma 4.6 applied to γ n = 0 and µ = 0.
n=0 is a Stieltjes moment sequence, (iii) p is a constant polynomial and p(0) 0.
Denote by a the leading coefficient of p. The above inequality implies that which contradicts the fact that a = 0. Therefore, p is a constant polynomial. ♦ For the sake of completeness, we provide a proof of the following lemma which will be used in subsequent parts of this paper. Proof. Since sup x∈supp G |ϕ(x)| < ∞, Σ ϕdG ∈ B(H). To prove the "only if" part, assume that Σ ϕdG 0.
n=0 is a Stieltjes moment sequence for every h ∈ H.
Before concluding this section, we recall the celebrated criterion for subnormality of bounded operators essentially due to Lambert (see [35]; see also [45,Proposition 2.3]).

An operator T ∈ B(H) is subnormal if and only if for every
The following general characterization of subnormal operators fits nicely into the scope of the present investigations. It will be used to provide the second proof of Corollary 5.2.
Theorem 4.11. Suppose that ϕ n : X → R, n ∈ Z + , are continuous functions on a topological Hausdorff space X of the form where each µ x is a compactly supported complex Borel measure on R + . Furthermore, assume that T ∈ B(H) is an operator for which there exists a regular spectral measure G : B(X) → B(H) with compact support such that Then T is subnormal if and only if µ x is a positive measure for every x ∈ supp G.
Proof. By (35) and (37), the operator T is subnormal if and only if the sequence { X ϕ n (x) G(dx)h, h } ∞ n=0 is a Stieltjes moment sequence for every h ∈ H. By Lemma 4.10, the latter holds if and only if {ϕ n (x)} ∞ n=0 is a Stieltjes moment sequence for every x ∈ supp G, which in view of (36) and Lemma 4.4 is equivalent to the fact that µ x is a positive measure for every x ∈ supp G.

Proof of the main result and some consequences
Before proving Theorem 1.2, which is the main result of this paper, we make the following useful observation being a direct consequence of (8) and (10).
Proof of Theorem 1.2. (i)⇔(iii) In view of Proposition 3.1(i), (|Q|, |E|) is a pair of commuting positive operators. Let G be the joint spectral measure of (|Q|, |E|). Then, by Theorem 2.1(i) and [41,Theorem 2.18], the measure G is compactly supported and regular. It follows from (9) and (21) that where ϕ n : R 2 + → R + is the continuous function defined by Notice that by Proposition 3.10(ii) and (35), the operator T is subnormal if and only if { Ω n h, h } ∞ n=0 is a Stieltjes moment sequence for every h ∈ H 2 . Hence in view of (39) and Lemma 4.10, T is subnormal if and only if supp G ⊆ Ξ, where Ξ is the set of all points (s, t) ∈ R 2 + for which {ϕ n (s, t)} ∞ n=0 is a Stieltjes moment sequence. Therefore, according to Theorem 2.1(i), to get the equivalence (i)⇔(iii), it is enough to show that Ξ =D + ∪ R + × {0} . For this purpose, take (s, t) ∈ R 2 + and consider two cases. Case 1. s = 1. Then by (40), we have ϕ n (s, t) = 1+nt 2 . Applying Lemma 4.7 to p(x) = 1+t 2 x, we see that (1, t) ∈ Ξ if and only if t = 0. Case 2. s = 1. Then by (40) we have This implies that where µ s,t : B(R + ) → R is the signed measure of the form Using Lemma 4.4, we conclude that (s, t) ∈ Ξ if and only if the measure µ s,t is positive, or equivalently if and only if 0 If t = 0, then (44) holds. If t = 0, then (44) holds if and only if (s, t) ∈D + . Thus (s, t) ∈ Ξ if and only if (s, t) ∈ D + ∪ R + × {0} \ {(1, 0)}. Summarizing Cases 1 and 2, we conclude that Ξ =D + ∪ R + × {0} , which gives the desired equivalence (i)⇔(iii).
(v)⇔(vi) This equivalence can be proved in the same way as the equivalence (ii)⇔(iii) of Proposition 3.6.
The "moreover" part is a direct consequence of (iii) and (10). This completes the proof.
In the rest of this section we record some consequences of Theorem 1.2. We begin with the following corollary which is immediate from Theorem 1.2(v).
The next corollary follows from Proposition 3.6 and Theorem 1.2 (recall that by Proposition 3.4 the contractions of class Q are of norm 1).
Corollary 5.2. Any contraction of class Q is subnormal.
As shown below, Corollary 5.2 can also be deduced from Theorem 4.11.
The following is a variant of Corollary 5.3 with essentially the same proof. H2 , where Q = αU, α ∈ C and U ∈ B(H 2 ) is an isometry. If E = 0, then the following conditions are equivalent: (i) T is subnormal, (ii) |α| 2 + E 2 1, (iii) T is a contraction.

A solution to the Cauchy dual subnormality problem in the class Q
We begin by providing a complete answer to the question of when the Cauchy dual of an operator of class Q is subnormal.
We now show that within the class Q the Cauchy dual subnormality problem has an affirmative solution. What is more surprising is that we can solve it affirmatively even if complete hyperexpansivity is replaced by expansivity. For a more detailed discussion of this question, see Proposition 9.6 and Example 9.7. The solution is given in Corollary 6.2 below which is a direct consequence of Proposition 3.6 and Theorem 6.1. Another way of obtaining Corollary 6.2 is to apply Proposition 3.10(iv), Corollary 5.2 and the well-known and easy to prove fact that the Cauchy dual of an expansive operator is a contraction.
Corollary 6.2. The Cauchy dual of an expansive operator of class Q is a subnormal contraction.
Below we recapture the affirmative solution to the Cauchy dual subnormality problem for quasi-Brownian isometries. Proof. Let T ∈ B(H) be a quasi-Brownian isometry. If T is an isometry, then T ′ = T is subnormal. If T is not an isometry, then by [36,Proposition 5.1], T has the block matrix form (1) with entries satisfying the conditions (2), (3) and (4), Q being an isometry. Since each isometry is quasinormal, we deduce that T is an operator of class Q and Q * Q + E * E I. Combined with Proposition 3.6 and Corollary 6.2, this implies that T ′ is a subnormal contraction, which completes the proof.
Regarding Corollaries 5.2 and 6.2, it is worth pointing out that there are subnormal operators of class Q that are not contractive, and non-expansive left-invertible operators of class Q whose Cauchy dual operators are subnormal. This can be deduced from Theorems 1.2(iii) and 6.1 and Propositions 3.6 and 3.10(iii) via an abstract non-explicit procedure given in Theorem 3.3. Explicit instances are given in Example 6.4 below which will be continued in Sections 9 and 10 under different circumstances.
Example 6.4. Our goal in this example is to show that 1 • for any θ ∈ (1, ∞), there exists a subnormal operator T of class Q such that T = θ, (21) and (22)).
For this purpose, let K be an infinite dimensional complex Hilbert space and τ, η be complex numbers such that η = 0. Take a non-unitary isometry V ∈ B(K) and a quasinormal operatorQ ∈ B(R(V )). Define the operators Q τ , E η ∈ B(K) by where P ∈ B(K) is the orthogonal projection of K onto N (V * ). Then the operator Q τ is quasinormal. It is easily seen that T τ,η := V Eη 0 Qτ ∈ Q K (see Definition 1.1). The operators |Q τ | and |E η | can be represented relative to the orthogonal decomposition K = N (V * ) ⊕ R(V ) as follows: Since N (V * ) = {0}, we infer from (8) and Remark 2.5 that According to (6), (54) and (55), the following chain of equivalences holds Combined with (55), Theorem 1.2(ii) implies that T τ,η is subnormal if and only if (|τ |, |η|) ∈D + .
Since by (54), we deduce from Proposition 3.10(iii) that T τ,η is left-invertible if and only if |Q| is invertible.

Quasi-Brownian isometries of class Q
In this section we provide a few characterizations of quasi-Brownian isometries of class Q. Given an isometry V ∈ B(H), we say that ; recall that H 1 reduces V to a unitary operator and H 2 reduces V to a unilateral shift of multiplicity dim N (V * ) (see [47,Theorem 1.1] for more details). It is clear that (62) (ii)⇔(iii) This equivalence is a straightforward consequence of (4) and Proposition 3.10(ii).
(iii)⇒(v) Since Q is quasinormal, |Q| 2 − I commutes with Q and so H i := N (|Q| 2 − I) reduces Q to an isometry. Set H si = H 2 ⊖ H i = R(|Q| 2 − I). Clearly, H 2 = H i ⊕ H si and H si reduces Q. Since |Q| 2 − I commutes with |E|, we see that H i , and consequently H si , reduces |E|. Notice that which implies that |Q| 2 + |E| 2 is the identity operator on H si . This shows that Q| Hsi , |E| Hsi is a spherical isometry.
(v)⇒(iii) This implication is a matter of routine verification. We now prove the "moreover" part. Let H i = H u ⊕ H s be the von Neumann-Wold decomposition of H i for Q| Hi . Since H u and H s reduce Q| Hi and H i reduces Q, we deduce that H u and H s reduce Q, the operator Q| Hu is unitary and the operator Q| Hs is a unilateral shift (of finite of infinite multiplicity). Because H u = ∞ n=0 Q n (H i ), |E|(H i ) ⊆ H i and Q commutes with |E|, we see that which implies that H u reduces |E|. Since H si also reduces |E|, we conclude that H s reduces |E|. This completes the proof.
Below we show that there are operators of class Q with injective E, which are not 2-isometries (the case when E = 0 is obvious due to the fact that quasinormal 2-isometries are isometric; see [25, Theorem 1 in §2.6.2] and [30, Theorem 3.4]).
, where E is an isometry. Then the following conditions are equivalent: (i) T is a 2-isometry, (ii) Q = 0 ⊕ U, where U ∈ B(H 2 ⊖ N (Q)) is an isometry.
Proof. In view of the equivalence of (ii) and (iii) in Theorem 7.1, T is a 2-isometry if and only if (Q * Q) 2 = Q * Q. Hence, by [28,Problem 127], T is a 2-isometry if and only if Q is a partial isometry. Since Q is quasinormal, we infer from [28,Problem 204] that Q is a partial isometry if and only if Q = 0 ⊕ U, where U ∈ B(H 2 ⊖ N (Q)) is an isometry.
Taking any quasinormal operator Q which is not of the form as in the condition (ii) of Corollary 7.2 (e.g., when Q / ∈ {0, 1}), we get an operator of class Q which is not a 2-isometry.
The key role which plays the Taylor spectrum σ(|Q|, |E|) in the present paper raises the question of the existence of different orthogonal decompositions of the underlying Hilbert space H relative to which a given operator T ∈ B(H) is of class Q, i.e., T has the block matrix form (1) with V, E and Q satisfying (2)-(5). This question is discussed in the following example.

Brownian isometries of class Q
The aim of this section is to give a deeper insight into the structure of Brownian isometries of class Q. We begin by proving two preparatory lemmata which are of some independent interest. Proof. That the operators |Q|, |E| and |Q * | commute can be deduced from (4) and (5) via the square root theorem (cf. Proposition 3.1(i)). Hence, by the equivalence (ii)⇔(iii) of Theorem 7.1, it suffices to show that △ T △ T * △ T = 0 if and only if (|Q * | 2 − I)(|Q| 2 + |E| 2 − I) 2 = 0. It is a routine matter to verify that where Ω 1 = |Q| 2 + |E| 2 . As a consequence, we get the desired equivalence. Proof. Suppose T is a Brownian isometry. Then by Lemma 8.1, we have where A s := |E| Hs and P ∈ B(H s ) is the orthogonal projection of H s onto N (Q * s ) with Q s := Q| Hs (by the moreover part of Theorem 7.1, H s reduces Q and |E|). Because (P A s ) * = P A s , we infer from (69) that P A s = 0. As a consequence, we see that R(A s ) ⊆ N (Q * s ) ⊥ . Since Q s commutes with A s , so does Q * s and consequently A s (N (Q * s )) ⊆ N (Q * s ). Putting all of this together, we see that A s (N (Q * s )) = {0}. Therefore, because Q commutes with |E|, we deduce that |E|Q n N (Q * s ) = {0} for all n ∈ Z + . Since by (62), H s = ∞ n=0 Q n N (Q * s ), we conclude that |E| Hs = 0. To prove the converse implication, assume that |E| Hs = 0. It follows from Theorem 7.1 that H 2 = H u ⊕ H s ⊕ H si , the spaces H u , H s and H si reduce both Q and |E|, Q| Hu is unitary, Q| Hs is a unilateral shift and Q| Hsi , |E| Hsi is a spherical isometry. Now, straightforward calculations show that the condition (ii) of Lemma 8.1 holds. Hence by this lemma, T is a Brownian isometry. This completes the proof.
We are now ready to characterize Brownian isometries of class Q.  As shown below, the class of Brownian isometries is the only subclass of Q considered in this paper which cannot be characterized by the Taylor spectrum σ(|Q|, |E|) of the pair (|Q|, |E|).
We conclude this remark by reexamining [5,Example 4.4]. Let V ∈ B(H 1 ), E ∈ B(H 2 , H 1 ) and Q ∈ B(H 2 ) be isometric operators such that Q is not unitary and V * E = 0. As shown in [5,Example 4.4], the operator T defined by (1) is a quasi-Brownian isometry (obviously of class Q) which is not a Brownian isometry. Clearly, the first inclusion in (70) holds. Hence by the above discussion the second one does not hold. The latter also follows directly from the equality σ(|Q|, |E|, |Q * |) = ∈ QH 1,H2 relative to an orthogonal decompositionH 1 ⊕H 2 , whereQ is a unitary operator. As a consequence, σ(|Q|, |Ẽ|) = {(1, 1)}. This means that Brownian isometries cannot be characterized by the Taylor spectrum σ(|Q|, |E|). ♦ 9. m-isometries and related operators of class Q In this section we characterize m-contractions, m-isometries and m-expansions of class Q by using the Taylor spectrum approach.
Given an integer m 1 and an operator T ∈ B(H), we write Recall that an operator • completely hyperexpansive if T is m-expansive for all m 1. The above-mentioned concepts can be attributed to many authors, such as Agler [1] (m-contractivity), Richter [40] (2-expansivity), Aleman [4] (complete hyperexpansivity for special operators), Agler [2] (m-isometricity) and Athavale [9] (mexpansivity and complete hyperexpansivity). It is well-known that a 2-isometry is m-isometric for every integer m where Ω j are as in (21). In view of Proposition 3.10(ii), we have Let G be the joint spectral measure of (|Q|, |E|). It follows from (39) and (40) that   Hence, we getψ In turn if s = 1, we can argue as follows: Putting all this together we see thatψ m = ψ m for all m ∈ N. Combined with (9), (72) and (73), this completes the proof.
We are now in a position to characterize m-contractivity, m-isometricity and m-expansivity of operators of class Q. The spectral regions for m-contractivity and m-expansivity of operators of class Q are illustrated in Figures 2 and 3 (for the case m = 1, see Proposition 3.6).
Theorem 9.2. Assume that T = V E 0 Q ∈ Q H1,H2 and m 2 is an integer. Then the following assertions hold: (i) T is m-contractive if and only if Proof. Since the proofs of (i) and (ii) are similar, we justify only (i). Let G be the joint spectral measure of the pair (|Q|, |E|). Observe that by (9) Using (71), we verify that Combined with (74), this yields (i). Finally, (iii) can be deduced from (i) and (ii). This completes the proof.
Corollary 9.3. Assume that T = V E 0 Q ∈ Q H1,H2 and m 2 is an integer. Then the following assertions hold: (i) if m is odd (resp., even), then T is m-contractive if and only if T is 3-contractive (resp., 2-contractive), (ii) if m is odd (resp., even), then T is m-expansive if and only if T is expansive (resp., 2-expansive), (iii) T is m-isometric if and only if T is 2-isometric, (iv) T is completely hypercontractive if and only if T is contractive, (v) T is completely hyperexpansive if and only if T is 2-expansive.
Proof. Use Theorem 9.2 and additionally Proposition 3.6 in the cases (ii), (iv) and (v).
The example below illustrates Theorem 9.2. We refer the reader to Figure 4 describing spectral regions for the above-mentioned subclasses of the class Q (except for Brownian isometries, cf. Remark 8.5). ♦ A recent result due to Badea and Suciu (see [10,Theorem 3.4]), which states that a △ T -regular 2-expansive operator T is completely hyperexpansive if and only if its Cauchy dual T ′ is subnormal, solves in the affirmative the Cauchy dual subnormality problem in the class of △ T -regular 2-expansions (see [5,Theorem 4.5] for an earlier solution of this problem in the class of △ T -regular 2-isometries). It is well known and easy to prove that the relation T ←→ T ′ is a one-to-one correspondence between expansive operators and left-invertible contractions. When restricted to operators of class Q, this correspondence becomes a bijection between expansions and left-invertible subnormal contractions (see Corollary 6.2). In view of Proposition 3.10(v), expansions T of class Q are always △ T -regular. This suggests that there may exist △ T -regular operators outside of the class of completely hyperexpansive ones for which the Cauchy dual subnormality problem has an affirmative solution. This is really the case as shown in Example 9.7 below which is based on Proposition 9.6. The proposition itself is a direct consequence of Propositions 3.6 and 3.10(v), Corollary 6.2 and Theorem 9.2(ii).
To have a concrete example of an operator satisfying the assumptions of Proposition 9.6, we revisit Example 6.4 again.

Linear operator pencils built over the class Q
In this section we study linear operator pencils that are associated with operators of class Q. By a linear operator pencil (see [26]) we mean a mapping where A, B ∈ B(H). Given T = V E 0 Q ∈ Q H1,H2 , we define the linear operator pencil T † (λ) by Clearly, T † (λ) ∈ Q H1,H2 for every λ ∈ C. Observe that T † (λ) can be regarded as the perturbation of the quasinormal operator V 0 0 Q by the nilpotent operator λ 0 E 0 0 . It is worth pointing out that the operators V 0 0 Q and 0 E Using β † (T ) we can describe the set S † (T ) explicitly. Proof. First observe that T † (0) is a quasinormal operator and thus by [17, Proposition II.1.7], T † (0) is subnormal, which yields (i). In view of (i) and Theorems 2.1(iii) and 1.2(ii), we have S † (T ) = {0} ∪ α ∈ (0, ∞) : s 2 + α 2 t 2 1, ∀(s, t) ∈ σ ♯ (|Q|, |E|) .
The "moreover" part follows from (i) and (ii). This completes the proof.
There is another possibility of associating a linear operator pencil with an operator of class Q. Namely, given T = V E 0 Q ∈ Q H1,H2 , we define the pencil T † (·) by and the corresponding set S † (T ) by S † (T ) = α ∈ R + : T † (α) is subnormal .
Example 10.4 (Example 6.4 continued). Let T τ,η be as in Example 6.4. We begin by showing that for every b ∈ R + , there exist τ ∈ C and η ∈ C \ {0} such that S † (T τ,η ) = [0, b]. Indeed, it follows from (55) that σ ♯ (|Q τ |, |E η |) = {(|τ |, |η|)}. Assume additionally that |τ | 1. Combined with (75), this gives First, suppose that b = 0. Then by considering the case |τ | = 1 we infer from the moreover part of Theorem 10.1 that S † (T τ,η ) = [0, b]. Let now b > 0. Then by taking into account the case |τ | < 1 we deduce from (82) and Theorem 10.1 that S † (T τ,η ) = 0, β † (T τ,η ) . This together with (82) shows that there exists η ∈ C\{0} such that β † (T τ,η ) = b. Similarly, using Theorem 10.2, one can show that for every b ∈ R + ∪ {∞}, there exist parameters τ and η such that S † (T τ,η ) = [0, b] ∩ R + . We leave the details to the reader. ♦ We conclude this paper by commenting the contents of this section. In view of Theorem 9.2, the technique of using the Taylor spectrum developed here can also be applied to describe the sets of the form α ∈ R + : T † (α) is in C and α ∈ R + : T † (α) is in C , where C is one of the classes of operators appearing in Section 9 including mcontractions, m-expansions, etc. As the number of cases to be considered is large (in particular depends on the parity of m) and each of them requires separate treatment, we decided not to include details in this paper.