On the Structure of Conditionally Positive Definite Algebraic Operators

Recently, the authors have introduced and intensively studied a class of bounded Hilbert space operators called conditionally positive definite. Its origins go back to the harmonic analysis on ∗\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$*$$\end{document}-semigroups, namely to the concept of conditional positive definiteness. Our main aim here is to give a complete description of algebraic conditionally positive definite operators on inner product spaces; we do not assume that the operators under consideration are bounded.


Introduction
The concepts of positive and conditional positive definiteness have their origins in stochastic processes [9,22,24,26,27]. The first concept entered operator theory soon after on the occasion of studying isometries in a group sense and, more generally, subnormal operators in a semigroup sense. One of the main results in this area states that a bounded Hilbert space operator T is subnormal if and only if the sequence { T n h 2 } ∞ n=0 is positive definite on the additive semigroup {0, 1, 2, . . .} for all vectors h (see [1,16,23]). The conditional positive definiteness in a semigroup sense appeared in operator theory for the first time in relation to subnormal contractions (see [28]). Later it appeared in the context of complete hyperexpansivity and complete hypercontractivity of finite order [5,11,12]. In a recent paper [20] we have provided the fundamentals of the theory of operators called conditionally positive definite (CPD for brevity), that is, bounded Hilbert space operators T for which the sequence { T n h 2 } ∞ n=0 is conditionally positive definite on the additive semigroup {0, 1, 2, . . .} for all vectors h. This class of operators contains subnormal operators [14,17], 2-and 3-isometries [2][3][4], complete hypercontractions of order 2 [11,18], certain algebraic operators which are neither subnormal nor m-isometric, and much more. Furthermore, the class of CPD weighted shift operators has been characterized in [21]. The aim of the present paper is to provide a complete description of the structure of (not necessarily bounded) CPD algebraic operators on inner product spaces. Before stating the main result, we recall the necessary concepts.
All Let M be an inner product space. Following [20] we say that an operator T ∈ L(M ) is CPD if We write B(M ) for the algebra of all bounded linear operators on M .
The following theorem is the main result of this paper. It provides a complete description of the structure of CPD algebraic operators on inner product spaces. Theorem 1.1 Let M be an inner product space and T ∈ L(M ). Then the following conditions are equivalent: Moreover, if (ii) holds, then for all j, In particular, if T ∈ B(M ), then each of the spaces X j,1 , X j,2 , Y j and Z j is closed.
The structure of bounded CPD algebraic operators on a Hilbert space with spectral radius not exceeding 1 is given below. Comparing with Theorem 1.1, the decomposition (1.2) below does not contain the component that corresponds to the space k j=1 X j,1 X j,2 appearing in (1.1). Theorem 1.2 Let H be a Hilbert space and T ∈ B(H). Then the following conditions are equivalent: (i) T is a CPD algebraic operator with spectral radius r (T ) 1, (ii) T has an orthogonal decomposition (some of the summands may be absent) relative to an orthogonal decomposition H = H 1 ⊕ · · · ⊕ H n , where 1 Some components of the form k j=1 X j,1 X j,2 , l j=1 Y j and m j=1 Z j may be absent; if the first component is nonzero, then both spaces X j,1 and X j,2 must be nonzero (a similar rule applies to the other two components). The symbols and ⊕ are reserved to denote the algebraic direct sum and the orthogonal direct sum of vector spaces, respectively. if N j = 0, then N 2 j = 0 and |z j | = 1, (1.5) |z j | 1 for j = 1, . . . , n, (iii) T has an orthogonal decomposition T = A ⊕ V , where A is an algebraic normal contraction and V is an algebraic 3-isometry (again some of the summands may be absent).
Moreover, if (i) holds and r (T ) < 1, then T is a diagonal operator, that is, the nilpotent summands in the decomposition (1.2) are absent.
Another consequence of Theorem 1.1 is the following result, which is closely related to Theorem 1.

Preliminaries
Denote by R and C the fields of real and complex numbers, respectively. Set R + = {t ∈ R : t 0} and T = {z ∈ C : |z| = 1}. Write Z + and N for the sets of nonnegative and positive integers, respectively. Denote by B(Ω) the σ -algebra of Borel subsets of a topological space Ω. For z ∈ K, where K ∈ {R, C}, the symbol δ z stands for the Borel probability measure on K concentrated on {z}. The closed support of a complex Borel measure ρ on K is denoted by supp(ρ). As usual, the symbol C[X ] stands for the ring of polynomials in indeterminate X with complex coefficients. The derivative of p ∈ C[X ] is denoted by p . If p ∈ C[X ] is a polynomial of degree at least 1, then there is a unique polynomial q ∈ C[X ] such that In what follows, the rational function p(x)− p(0) x is identified with the polynomial q. The following result of classical nature will be used in the proof of Lemma 3.8. As a consequence, we have Suppose now that w m = 1 for some m ∈ N. By our assumption on w, m must be greater than or equal to 3. It is easily seen that Re (b), Re (wb), . . . , Re (w m−1 b) are the accumulation points of the sequence {Re (w n b)} ∞ n=0 . Hence, we have and so w =w. As |w| = 1, this yields w = ±1, which is a contradiction.
Let γ = {γ n } ∞ n=0 be a sequence of real numbers. We say that γ is positive definite (PD for brevity) if for all finite sequences λ 0 , . . . , λ k ∈ C, 3) holds for all finite sequences λ 0 , . . . , λ k ∈ C such that k j=0 λ j = 0, then we call γ conditionally positive definite (CPD for brevity). Using this terminology, one can rephrase the notion of a CPD operator as follows: an operator T ∈ L(M ) on an inner product space M is CPD if and only if the sequence { T n h 2 } ∞ n=0 is CPD for every h ∈ M . The following fact is a consequence of [20, Proposition 2.2.9].
If lim sup n→∞ |γ n | 1/n < ∞, then γ is C P D i f and only i f 2 γ is PD. (2.4) The CPD sequences of exponential growth have the following integral representation.
n=0 be a sequence of real numbers. Then the following conditions are equivalent: where Q n is the polynomial given by (2.6) The polynomials Q n defined in (2.6) have the following property: where 2 Q (·) (x) denotes the action of the transformation 2 on the sequence n=0 be a sequence of real numbers. We say that γ n is a polynomial in n of degree k if there exists a polynomial p ∈ C[X ] of degree k such that γ n = p(n) for all n ∈ Z + . By the Fundamental Theorem of Algebra such p is unique and its coefficients are real. We say that γ is a Stieltjes moment sequence if there exists a positive Borel measure μ on R + , called a representing measure of γ , such that It is clear that any Stieltjes moment sequence is PD. The celebrated Stieltjes moment theorem states that γ is a Stieltjes moment sequence if and only if the sequences γ and {γ n+1 } ∞ n=0 are PD (see [7, Theorem 6.2.5]). The following result is a counterpart of the Stieltjes moment theorem for CPD sequences.
Let (b, c, ν) be the representing triplet of γ . Then the following conditions are equivalent: This combined with [20, Proposition 2.2.9] completes the proof.
The following fact is a basic characterization of algebraic operators that will be needed in this paper (see e.g., [13,Section 6]).

Lemma 2.4 Let M be a vector space and T ∈ L(M ). Then the following conditions are equivalent:
(i) T is algebraic, (ii) there exist positive integers ι 1 , . . . , ι n , distinct complex numbers w 1 , . . . , w n and nonzero vector subspaces for all finite sequences of integers 1 j 1 < · · · < j s n. In particular, if M is an inner product space and T ∈ B(M ), then each space M j is closed.

Preparatory Lemmas
The proof of the main result of this paper will be preceded by a series of lemmas.
n=0 be a sequence of complex numbers such that

1)
where ρ is a complex Borel measure on C with finite supp(ρ). Then the following two statements are equivalent: Consider the polynomial p ∈ C[X ] given by Substituting this polynomial into (3.2), we deduce from (3.3) that This implies that We claim that supp(ρ) ⊆ R. Suppose, to the contrary, that there exists Since supp(ρ) is finite, this contradicts z 0 ∈ supp(ρ).
(ii)⇒(i) This implication is easily seen to be true.
It turns out that the implication (i)⇒(ii) of Lemma 3.1 is no longer true if supp(ρ) is infinite, even if it is a compact subset of C. Note that the converse implication is always true.
n=0 be the sequence given by It is clear that γ is a Stieltjes moment sequence with a representing measure δ 0 and consequently it is PD. Define the Borel probability measure ρ on C by where χ Δ stands for the characteristic function of Δ. Then a standard measure-theoretic argument gives It is a matter of routine to verify that supp(ρ) = T.
Clearly lim sup n→∞ |γ n | 1/n < ∞. Since we deduce that It follows from (3.7) that where ρ is the complex Borel measure on C defined by If z 1z2 = 1, then, in view of (3.9), we have which together with (3.10) implies that the sequence γ is CPD. Therefore, we can assume that z 1z2 = 1. We will consider three cases. Case 1. z 1z2 / ∈ R. Observe that z 2z1 / ∈ R and z 1z2 = z 2z1 . Combined with (3.9) and (3.10), this implies that γ is CPD if and only if h 1 ⊥ h 2 .
We also need an extension of Lemma 3.3.

Lemma 3.5
Let M 1 and M 2 be vector subspaces of an inner product space M , N j ∈ L(M j ) for j = 1, 2, and z 1 , z 2 be distinct complex numbers. Suppose that the sequence is CPD for all h 1 ∈ N (N 1 ) and h 2 ∈ N (N 2 ). Then the following assertions hold: Proof (i) Fix h 1 ∈ N (N 1 ) and h 2 ∈ N (N 2 ). It follows from Newton's binomial formula that This implies that for all λ 1 , λ 2 ∈ C, Applying Lemma 3.3 to the vector spaces spanned by h 1 and h 2 , we conclude that (ii) Note that |z 1 |, |z 2 | 1 implies that z 1z2 = 1. Indeed, otherwise |z 1 ||z 2 | = 1, so |z 1 | = 1, and thus z 1z1 = 1 = z 1z2 and finally z 1 = z 2 , which is a contradiction. Therefore, (ii) follows from (i).
is CPD for all h 1 ∈ M 1 and h 2 ∈ M 2 . Then M 1 ⊥ M 2 .
By passing to the limits in (3.33) when n is even, we getb = 2ReΞ 1 . The same procedure applied to odd n's leads tob = −2ReΞ 1 . As a consequence,b = ReΞ 1 = 0. Combined with (3.33), this shows that Re h 1 , h 2 = (−1) n Re h 1 , h 2 for all n ∈ Z + . Hence, Re h 1 , h 2 = 0, which implies that h 1 and h 2 are orthogonal. This completes the proof.

Remark 3.9
It is worth pointing out that one of the direct consequences of Lemma 3.8 is that, under its assumptions, there always exists an operator T ∈ L(M 1 + M 2 ) for which the spaces M 1 and M 2 are invariant and T | M j = z j I M j + N j for j = 1, 2. In fact, the following identity holds: A similar observation applies to Lemma 3.7. ♦ X j,1 , X j,2 , Y j and Z j is closed. We can assume that all these spaces are non-zero (similar arguments can be applied to other cases). Define the polynomial p ∈ C[X ] by It follows from Theorem 1.1(ii) that p(T ) = 0. Applying the spectral mapping theorem (see e.g., [6, Lemma 53.3]), we deduce that where σ (T ) stands for the spectrum of T . Since r (T ) 1, we deduce that |x j,1 | 1, |x j,2 | 1 and |y j | 1 for all j (recall that the modulus of each z j is equal to 1). However x j,1 x j,2 = 1, so x j,1 = x j,2 , which is a contradiction. Therefore, the decomposition (1.1) reduces to This shows that (ii) holds. The "moreover" part is a direct consequence of (4.2).
(ii)⇒(iii) Set Clearly, A is an algebraic normal contraction. It follows from [8, Theorem 2.2] that z j I H j + N j is a 3-isometry for each j such that N j = 0. This implies that V is a 3-isometry. Using (1.2), we see that (iii) is valid. (iii)⇒(i) It follows from [11,Proposition 2.7] (see also [20,Proposition 4.3.1]) that V is CPD. As a consequence, T is a CPD algebraic operator. Clearly r (A) 1. It follows from [2, Lemma 1.21] that r (V ) 1. As a consequence, r (T ) 1. This completes the proof.
Proof of Proposition 1.3 According to Theorem 1.1, the operator S takes the form described in part (ii) of this theorem with T = S and M = K, where each of the spaces X j,1 , X j,2 , Y j , Z j and X j,1 X j,2 is closed (cf. (2.8)). As a consequence, each map X j,1 X j,2 x 1 x 2 −→ (x 1 , x 2 ) ∈ X j,1 ⊕ X j,2 is a topological and linear isomorphism. Now, it is matter of routine to verify that the operator S is similar to the operator T defined by T = k j=1 x j,1 I X j,1 ⊕ x j,2 I X j,2 ⊕ l j=1 y j I Y j ⊕ m j=1 (z j I Z j + N j ).
That the operator T takes the form T = A⊕V , where A is an algebraic normal operator and V is an algebraic 3-isometry, can be justified as in the proof of the implication (ii)⇒(iii) of Theorem 1.2.
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