Operators intertwining with isometries and Brownian parts of 2-isometries

Linear Algebra and its Applications 509 (2016) 168–190

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Linear Algebra and its Applications www.elsevier.com/locate/laa

Operators intertwining with isometries and Brownian parts of 2-isometries Witold Majdak a,∗ , Mostafa Mbekhta b , Laurian Suciu c a AGH University of Science and Technology, Faculty of Applied Mathematics, al. A. Mickiewicza 30, 30-059 Krakow, Poland b UFR de Mathématiques, Laboratoire Paul Painlevé, UMR CNRS 8524, Université Lille 1, 59655 Villeneuve d‘Ascq Cedex, France c Department of Mathematics and Informatics, “Lucian Blaga” University of Sibiu, Dr. Ion Raţiu 5-7, Sibiu, 550012, Romania

a r t i c l e

i n f o

Article history: Received 6 June 2016 Accepted 19 July 2016 Available online 26 July 2016 Submitted by V. Muller MSC: 47A05 47A15 Keywords: Partial isometry Quasi-isometry Hyponormal operator Invariant subspace

a b s t r a c t For two operators A and T (A ≥ 0) on a Hilbert space H satisfying T ∗ AT = A and the A-regularity condition AT = A1/2 T A1/2 we study the subspace N (A − A2 ) in connection with N (AT − T A), for T belonging to different classes. Our results generalize those due to C. Kubrusly concerning the case when T is a contraction and A = ST is the asymptotic limit of T . Also, the particular case of a 2-isometry in the sense of S. Richter as well as J. Agler and M. Stankus is considered. For such operators, under the same regularity condition we completely describe the reducing Brownian unitary and isometric parts, as well as the invariant Brownian isometric part. Some examples are provided in order to illustrate the limits of the theoretical results. © 2016 Elsevier Inc. All rights reserved.

* Corresponding author. E-mail addresses: [email protected] (W. Majdak), [email protected] (M. Mbekhta), [email protected] (L. Suciu). http://dx.doi.org/10.1016/j.laa.2016.07.014 0024-3795/© 2016 Elsevier Inc. All rights reserved.

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1. Introduction It was shown in [13] that for some A-contractions T , the maximum subspace of H reducing A1/2 T to a quasi-isometry can be related to the subspace N (A − A2 ), which intrinsically depends of A. In the present paper we continue the line of research initiated in [13]. Namely, we deal with the natural question of the role played by this subspace for an A-contraction, by analogy with the subspace N (ST − ST2 ) associated to a contraction T , where ST is the asymptotic limit of T . As is known from [10] (see also [8]), N (ST − ST2 ) is then the maximum invariant subspace for T , which is contained in N (T ST − ST T ). The organization of the paper and the main results are as follows. In Section 2 we set up notation and terminology. In Section 3 we characterize some properties of the subspace N (A − A2 ) in the context of A-contractions (Theorem 3.1) and A-isometries (Theorem 3.6, Proposition 3.7) under the A-regularity condition. Further, we dwell upon the case when the latter condition implies that A = A2 . In particular, if the sequence of norms of the powers for an operator T ∈ B(H) is convergent on each element in H, similarly as in the case of a contraction, we may associate the operator ST with T . Then we investigate when the ST -regularity of T is equivalent to the fact that the equality ST = ST2 holds (Proposition 3.10). In Section 4 we establish a relationship between N (A−A2 ) and the maximum subspace M∗ ⊂ H, which reduces A−1/2 A1/2 T to a normal partial isometry (Theorem 4.1). Further, we examine the case when N (A − A2 ) = M∗ (Proposition 4.6). We close this section by giving an example of a T T ∗ -isometry T with ST = ST2 . As an application of the preceding results, in Section 5 we provide a characterization of a 2-isometry T which is ΔT -regular, where ΔT = T ∗ T − I ≥ 0, in terms of operators appearing in the block matrix form of T (see Proposition 5.1). Next, we describe when such a 2-isometry is reducible to the Brownian isometry or unitary (Theorem 5.4). In this way we in fact obtain the Wold-type decomposition of T consisting of its Brownian unitary and its completely non-Brownian part in place of the unitary and completely non-unitary parts, respectively. Finally, we describe the structure of a pure 2-isometry (Theorem 5.7) and consider the case when it does not have the Brownian isometric part (Corollary 5.9). We close the paper by showing that the Brownian isometric part cannot be defined without the ΔT -regularity condition. 2. Preliminaries In the paper we use the following notation. N, Z and C stand for the sets of all positive integers, integers, and complex numbers, respectively. H denotes a Hilbert space and B(H) is the Banach algebra of all bounded operators on H. By I we mean the identity operator on a considered Hilbert space. Given T ∈ B(H), we write R(T ), N (T ) and T ∗ for the range, the kernel and the adjoint of T , respectively. If T is a positive operator, T 1/2 denotes its square root. We set |T | = (T ∗ T )1/2 for the modulus of T . If M ⊂ H,

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then T M := {T h : h ∈ M}. By M we mean the closure of M in the norm topology  of H. If Mn (n = 0, 1, . . .) are subsets of H, we denote by n≥0 Mn the closure in H of their linear span. A closed subspace M of H is invariant for T if T M ⊂ M. When M is invariant for both T and T ∗ , it is called a reducing subspace for T . For completeness of exposition, recall that an operator T ∈ B(H) is called hyponormal (resp. quasinormal, normal) if T T ∗ ≤ T ∗ T (resp. T T ∗ T = T ∗ T 2 , T ∗ T = T T ∗ ). We say that T is a contraction if T  ≤ 1, a partial isometry if T h = h for every h ∈ R(T ∗ ) = H  N (T ), an isometry if T ∗ T = I, a coisometry if T T ∗ = I, and a unitary operator if T is a surjective isometry. T is said to be a 2-isometry if T ∗2 T 2 −2T ∗ T +I = 0. In the last part of the paper we focus on these operators (for example, see [4,14,18] for more information on the subject), but the relevant background material concerning a more general concept of an m-isometry is to be found in [21] and [1–3]. The interest for this subject is motivated also by the Dirichlet space type model for 2-isometries developed in [21] by exploiting a beautiful extension of the wandering subspace argument for isometries to the class of 2-isometries which leads to a rich function theoretic analysis of such operators. Consider operators A, T ∈ B(H) such that A is nonzero and positive. T is said to be an A-contraction (resp. an A-isometry) if T ∗ AT ≤ A (resp. T ∗ AT = A). Remark that for such a T the space N (A) is invariant (but not necessarily reducing). The reader is referred to [25–29] and [15] for a comprehensive treatment of A-contractions, or to [11] for the more general class of A-power bounded operators. In this class of operators we distinguish some particular members T which will be of our interest in this paper. We now invoke their definitions. Namely, we say that T is a regular A-contraction (or in short A-regular) if AT = A1/2 T A1/2 , and following [17,19] (see also [7]) T is called a quasi-contraction (resp. quasi-isometry) if T is a T ∗ T -contraction (resp. T ∗ T -isometry). Next, recall also (cf. [30]) that if the sequence {T n h}n∈N is convergent for every h ∈ H, then T is a ST -isometry, where the positive operator ST ∈ B(H) is given by ST h, k = limn→∞ T ∗n T n h, k for h, k ∈ H. Clearly, if T  ≤ 1, then ST is defined as the strong limit of the sequence {T ∗n T n }n∈N and called the asymptotic limit of T . The reader is urged to consult [8] for more information on the subject. Recall that for a contraction T the defect operator of T is DT = (I − T ∗ T )1/2 and DT = R(DT ) stands for the defect space of T . 2 We write l+ (L) (resp. lZ2 (L)) for the set of all square-summable sequences in L indexed 2 2 by N (resp. Z), where L is a Hilbert space. For simplicity we put l+ = l+ (C) and 2 2 lZ = lZ (C). If V ∈ B(H) is an isometry and L is a wandering space for V , we denote ∞ n 2 n=0 V L by l+ (V, L). 3. About the subspace N (A − A2 ) We start our considerations by discussing some properties of the subspace N (A − A2 ) in the context of A-contractions, and further A-isometries, under the condition of the A-regularity.

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Theorem 3.1. Let T ∈ B(H) be a regular A-contraction. Then the following assertions hold. (a) N (A − A2 ) is an invariant subspace for A and T , and it reduces A1/2 T to an A-contraction. In addition, T and A1/2 T are simultaneously A-isometries on N (A − A2 ). (b) N (I − A) is an invariant subspace for A and T ∗ , and it reduces A1/2 T , while T ∗ and A1/2 T are contractions on N (I − A). (c) The following conditions are equivalent: (i) N (I − A) is invariant for T , (ii) N (A − A2 ) ⊂ N (AT − T A), (iii) T h = A1/2 T h for every h ∈ N (I − A). In addition, if any of these conditions is satisfied, then T is an isometry on N (I −A) if and only if T is an A-isometry on N (A − A2 ). Proof. (a) Clearly, AN (A − A2 ) ⊂ N (A − A2 ). For h ∈ N (A − A2 ), we have A1/2 h = A3/2 h. By this and AT = A1/2 T A1/2 , we obtain AT h = A2 T h and AT ∗ A1/2 h = A2 T ∗ A1/2 h. Therefore, T h, T ∗ A1/2 h and A1/2 T h belong to N (A − A2 ), so the latter subspace is invariant for T and reduces A1/2 T . Since T ∗ A2 T = T ∗ AT ≤ A on N (A − A2 ), A1/2 T is an A0 -contraction on N (A − A2 ), where A0 = A|N (A−A2 ) . In addition, the last fact shows that A1/2 T is an A0 -isometry on N (A − A2 ) if and only if A1/2 T h = A1/2 h for h ∈ N (A − A2 ). If this is the case, A1/2 T and T are A0 -regular operators on N (A − A2 ) because T is A-regular on H. (b) First notice that N (A − A2 ) = N (I − A) ⊕ N (A),

(3.1)

and N (I − A) reduces A and A1/2 T . As T ∗ AT ≤ A = I on N (I − A), A1/2 T is a contraction on N (I − A). Next, for h ∈ N (I − A) we get AT ∗ h = AT ∗ Ah = T ∗ A2 h = T ∗ h = T ∗ A1/2 h, so T ∗ h ∈ N (I − A). Hence N (I − A) is invariant for T ∗ and T ∗ |N (I−A) is a contraction. (c) If the subspace N (I − A) is invariant for T and h ∈ N (I − A), then AT h = T h = T Ah, so N (I −A) ⊂ N (AT −T A) and by (3.1) it follows that N (A −A2 ) ⊂ N (AT −T A). We proved that (i) implies (ii). Next, from (ii) we obtain T h = T Ah = AT h for h ∈ N (I − A). This gives us the relation T h2 = T ∗ T h, h = T ∗ AT h, h = A1/2 T h2 , which yields (iii). Observe that (iii) entails (i). Indeed, for h ∈ N (I − A), by (iii) and the A-regularity of T , (I − A1/2 )T h2 = 2(T h2 − Re T h, A1/2 T A1/2 h ) = 0.

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As N (I − A) reduces A1/2 T , we deduce that T h = A1/2 T h ∈ N (I − A). Hence, the conditions (i), (ii) and (iii) are equivalent. The last assertion follows immediately from (ii) and (3.1). 2 It is worth noting that an inspection of the above proof of the implication (iii)⇒(i) reveals that (iii) can be replaced by the condition: T h ≤ A1/2 T h for every h ∈ N (I − A). For A-isometries we obtain the ensuing result as completeness of Theorem 3.1. Proposition 3.2. Let T ∈ B(H) be a regular A-isometry such that T ∗ T ≤ An for some integer n ≥ 0. Then the subspace N (I − A) reduces T to an isometry. In addition, if T ∗ T ≤ I, then N (A − A2 ) reduces T and the equality AT = T A holds. Proof. Assume that T ∗ T ≤ An for some integer n ≥ 0. As T is a regular A-isometry, it is also an A2 -isometry. As a consequence, we have (AT − T A)h2 ≤ Ah2 + An+2 h, h − 2Ah2 = An+2 h, h − Ah2

(3.2)

for every h ∈ H. Thus, when n = 0 we get AT = T A on H. If n > 0, then we have An+2 h = A2 h for h ∈ N (A − A2 ), so (3.2) implies AT h = T Ah. This means that N (A − A2 ) ⊂ N (AT − T A). By Theorem 3.1, N (I − A) reduces and T is an isometry on N (I − A). If T ∗ T ≤ I, we have AT ∗ = T ∗ A and by Theorem 3.1 (a) we see that N (A − A2 ) reduces T . 2 Remark 3.3. Let T ∈ B(H) be an A-contraction with an injective A such that A = I. Then T is A-regular if and only if AT = T A. In this case T is a contraction on H = R(A), and the subspace N (A −A2 ) = N (I −A) is strictly contained in H and reducing for T . In addition, if T is an A-isometry, then T is an isometry and AT = T A. In turn, if a unitary operator U ∈ B(H) is a regular A-contraction, where A is not necessarily injective, then AU = U A, hence U and U ∗ are A-isometries. An interesting example in this sense is given below. Example 3.4. Let T ∈ B(H) be a hyponormal operator and T = U |T | be its polar decomposition. Then we have U |T |U ∗ = U T ∗ = U U ∗ |T ∗ | = |T ∗ | ≤ |T |, where the last estimation follows from the Löwner–Heinz inequality (cf. [20]), having in mind that T is hyponormal. Hence U ∗ is a |T |-contraction. Clearly, U ∗ is a |T |-isometry if and only if T is normal. In the latter case U commutes with |T |, so U ∗ is also |T |-regular. In general, U ∗ is |T |-regular as a |T |-contraction if and only if T = Δ(T ), where Δ(T ) = |T |1/2 U ∗ |T |1/2 is the Aluthge transform of T (cf. [5]). Thus, the |T |-regularity of U ∗ is equivalent to the relation U |T | = |T |U .

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Since |T ∗ | ≤ |T |, we get U |T ∗ |U ∗ ≤ U |T |U ∗ = |T ∗ |, so U ∗ is a |T ∗ |-contraction. In this case U ∗ is |T ∗ |-regular if and only if |T ∗ |T ∗ = Δ(T ∗ )|T ∗ |, and in this case U is an isometry on R(T ). So, we conclude that U ∗ is a |T ∗ |-isometry if and only if T is normal. In particular, if T is invertible, then U is unitary, and if U ∗ is either |T |-regular or |T ∗ |-regular, then T is normal. Remark 3.5. Let T be a regular A-isometry with A ≤ 1 such that AT = T A. Then n T is a regular A1/2 -isometry for every n ∈ N (cf. [25, Theorem 2.6]). As a result, we n n n n+1 n+1 have T ∗ A1/2 T = A1/2 and A1/2 T = A1/2 T A1/2 for n ∈ N, and passing with ∗ n to ∞ we get T P T = P and P T = P T P , where P is the orthogonal projection of H onto R(A). So T is a regular P -isometry on H with N (P − P 2 ) = H. On the other hand, since T is A-regular and AT = T A, R(A) = N (I − P ) is not invariant for T (cf. [27, Lemma 1]). Hence P T = T P , and consequently N (P T − T P ) = H. This shows (by Theorem 3.1) that for a regular A-isometry T we have N (A − A2 )  N (AT − T A), in general. Our next result furnishes more information about the subspace N (A − A2 ). Theorem 3.6. Let T ∈ B(H) be an A-isometry with A ≤ 1 and P be the orthogonal projection with R(P ) = R(A). If T ∗ |R(A) is a contraction, then N (A − A2 ) ⊂



{h ∈ H : AT n h = Ah} =

n∈N



N (AT n − P T n A),

(3.3)

n∈N

and A = A2 if and only if AT = AT A. Moreover, if T is A-regular, then N (A − A2 ) =



{h ∈ H : AT n h = A1/2 h}

n∈N

=



N (AT n − P T n A1/2 ) = N (AT − P T A1/2 ),

(3.4)

n∈N

and the right-hand side of the inclusion in (3.3) is equal to the whole space H. Proof. First suppose that T∗ = T ∗ |R(A) is a contraction. Then, for every n ∈ N, so is T∗∗n = P T n |R(A) . For h ∈ H and n ∈ N, we obtain AT n h − P T n Ah2 ≤ AT n h2 + Ah2 − 2Ah2 = AT n h2 − Ah2 ,

(3.5)

using the fact that T ∗n AT n = A. On the other hand, for h ∈ H and n ∈ N, we get P T n Ah ≤ Ah = T ∗n AT n h ≤ AT n h ≤ A1/2 T n h = A1/2 h.

(3.6)

Now, since 0 ≤ A ≤ I, we have A − A2 ≥ 0, and so h ∈ N (A − A2 ) if and only if Ah = A1/2 h. For such h, by (3.6), this equality entails AT n h = Ah for

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every n ∈ N, which by (3.5) implies that AT n h = P T n Ah, and this by (3.6) yields AT n h = Ah. These implications justify the inclusion and the equality of (3.3). Next, if A = A2 , then A = P , so in view of (3.3) we get AT = P T A = AT A. Conversely, if AT = AT A, then A = T ∗ AT = T ∗ AT A = A2 . Now assume that the operator T is A-regular. Since T is an A-isometry there exists an isometry T on R(A) satisfying the relation TA1/2 k = A1/2 T k, which yields Tn A1/2 k = A1/2 T n k for n ∈ N, k ∈ H. By this and the A-regularity of T , for k ∈ H and n ≥ 1, we obtain An/2 Tn A1/2 k = An/2 P T n A1/2 k. Since An/2 is injective on R(A) = R(An/2 ) and An/2 T n A1/2 k = An/2 A1/2 T n k by the A-regularity of T , we get A1/2 T n k = Tn A1/2 k = P T n A1/2 k. If h ∈ N (A − A2 ), then AT n h = A1/2 T n h = P T n A1/2 h (by Theorem 3.1 (a)), which yields N (A − A2 ) ⊂



N (AT n − P T n A1/2 ) ⊂ N (AT − P T A1/2 ).

n∈N

In turn, if k ∈ N (AT − P T A1/2 ), then by Remark 3.5 we have Ak = T ∗ P T A1/2 k = P A1/2 k = A1/2 k, which implies k ∈ N (A − A2 ). Thus, the above inclusions become equalities. We established all identities of (3.4) except for the first one. However, the missing equality follows immediately from the relation A2 h = T ∗n A2 T n h for all h ∈ H and n ∈ N. This relation also shows that AT n h = Ah for all h ∈ H and n ∈ N.  Hence n∈N {h : AT n h = Ah} = H in (3.3), which proves the last assertion of the theorem. 2 The case when T is a contraction on H is now considered. Proposition 3.7. Let T, A ∈ B(H) be contractions such that T is an A-isometry. Then N (A − A2 ) =



{h ∈ H : AT n h = A1/2 h}

n∈N

⊂



{h ∈ H : AT n h = Ah} = N (AT − T A),

(3.7)

n∈N

the subspaces N (A − A2 ) and N (I − A) are invariant for T , and T is an isometry on N (I − A). Proof. Since T is a contraction and an A-isometry, we have Ah = T ∗n AT n h ≤ AT n h for h ∈ H and n ∈ N. So we obtain (AT n h − Ah)2 ≤ AT n h − T n Ah2 ≤ AT n h2 − Ah2 , which gives the last equality in (3.7).

(3.8)

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This equality and the inclusion of (3.3) give that N (A − A2 ) ⊂ N (AT − T A), whence one obtains immediately AT h = T A2 h = A2 T h. This shows that the subspaces N (A − A2 ) and N (I − A) are invariant for T . Clearly, T is an isometry on N (I − A). Finally, the first equality in (3.7) results from the fact that T is A-isometric and N (A −A2 ) is invariant for T . 2 Corollary 3.8. Let T, A ∈ B(H) be contractions such that T is an A-isometry. Then T is A-regular if and only if A and T commute, and this is the case when A = A2 . Remark 3.9. The condition of the A-regularity means that A1/2 (A1/2 T − T A1/2 ) = 0, that is, R(AT −T A) ⊂ N (A). For every A-contraction T , we have N (A) ⊂ N (AT −T A), so R(AT − T A) ⊂ N (AT − T A) ⊂ N (A1/2 (A1/2 T − T A1/2 )) = H, when the A-regularity of T is assumed. The first inclusion above ensures that (AT − T A)2 = 0, but if N (A) = {0}, the second inclusion can be strict (as in Remark 3.5). Thus, the condition A = A2 for a regular A-isometry does not guarantee that AT = T A in general, when T is not a contraction. On the other hand, if AT = T A (and T is not assumed to be a contraction), then A = A2 if and only if AT n h → A1/2 h as n → ∞ for every h ∈ H. This follows from the relations (3.3) and (3.4), because the operator P can be dropped in the last set appearing in (3.3) under the assumption AT = T A. In particular cases the A-regularity of T ensures in itself that A = A2 . For instance, if T is a quasi-isometry (i.e. T ∗2 T 2 = T ∗ T ), then A = T ∗ T and the condition AT = T A simply means that T is quasinormal, or equivalently that T is A-regular. But in this case A = A2 (i.e. T is a partial isometry) if and only if T  ≤ 1. Now let us discuss a case slightly different than this one. For this purpose, we will employ the operator ST invoked in Section 2. Proposition 3.10. Let T ∈ B(H) be such that the sequence {T n h}n∈N is convergent for every h ∈ H. Then ST = ST2 if and only if ST  ≤ 1 and T is ST -regular. In addition, if ST  ≤ 1 and ST T = T ST , then N (I −ST ) is the maximum subspace of H which reduces T to an isometry. Moreover, when T  ≤ 1, the following statements are equivalent: (i) ST = ST2 , (ii) ST T = T ST , (iii) T is ST -regular. Proof. Suppose that ST = ST2 . Then ST  = 1 and, by Theorem 3.6, T is ST -regular. Conversely, assume that ST  ≤ 1 and T is ST -regular. Let V ∈ B(R(ST )) be an 1/2 1/2 isometry such that V ST = ST T on R(ST ). From the proof of Theorem 3.6 we infer that V n h = P |R(ST ) T n h for h ∈ R(ST ) and n ∈ N. Thus h = V n h ≤ T n h for n ∈ N, which implies that ST h, h = h2 for h ∈ R(ST ), and so ST = I on R(ST ). Hence ST2 = ST = P |R(ST ) . Thus we have proved the first assertion, and the equivalence (i)⇔(iii). Note that (ii)⇔(iii) holds by Corollary 3.8.

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Now, if ST  ≤ 1 and ST T = T ST , then clearly N (I − ST ) reduces T to an isometry, and it is the maximum subspace with this property. 2 Remark 3.11. The above proposition can be applied to a m-quasi-isometry T for a fixed integer m ≥ 2, that is, to an operator T which is a T ∗m T m -isometry (see [7,12]). For such a T , ST = T ∗m T m and when T m = 0 we have ST  = 1 if and only if T m is hyponormal (cf. [7, Proposition 3.9]). In addition, ST = ST2 if and only if T m is a partial isometry, but in this case ST T = T ST , in general. In fact, if ST T = T ST , then T m is just quasinormal, which yields ST = ST2 . Consequently, ST T = T ST implies ST = ST2 , and the converse implication is also true when T  ≤ 1 (by the relation (3.7) or Proposition 3.10). Furthermore, when T is a contraction and an isometry on R(T m ), which does not imply ST T = T ST in general, then T ∗ has the completely non-unitary part strongly stable ([7, Theorem 3.11]). Hence ST ∗ = ST2 ∗ , which, by Proposition 3.10, yields ST ∗ T = T ST ∗ . Now let us consider an interesting particular case when the assertions of Proposition 3.10 can be improved. This result complements also [30, Proposition 3.6 and Corollary 3.7]. Theorem 3.12. Let T ∈ B(H) be such that the sequence {T n h}n∈N is convergent for every h ∈ H. Then the subspace N (ST − ST2 ) = N (ST T − T ST ) is invariant for T , and the maximum subspace of H which reduces T to an isometry (or to a unitary operator) is N (I − ST ) = N (I − T ) ∩ N (I − T ∗ ).

(3.9)

Proof. It is known (cf. [30]) that the strong limit of {T n }n∈N is the idempotent operator PT ∈ B(H) with R(PT ) = N (I − T ) and N (PT ) = R(I − T ). Hence ST = PT∗ PT , which in turn entails R(ST ) = R(PT∗ ) = N (I − T ∗ ). But PT has a matrix representation of the form   I S PT = 0 0 with respect to the decomposition H = R(PT )⊕N (PT∗ ). This leads to the representations  ST =

I S∗

S S∗S



 ,

I − ST =

0 −S ∗

−S I − S∗S

 .

So, if h ⊕ k ∈ N (I − ST ) with h ∈ N (I − T ) and k ∈ R(I − T ∗ ), then Sk = 0 and (I − S ∗ S)k − S ∗ h = 0, which yields k = S ∗ h as well as SS ∗ h = Sk = 0. Hence k = S ∗ h = 0 and h = PT∗ h ∈ N (I − T ∗ ). This implies the equality in (3.9) and that N (I −ST ) reduces T to an isometry (in fact to a unitary operator), and it has the desired property of maximality. Next, the subspace N (ST −ST2 ) = N (I −ST ) ⊕N (ST ) is invariant for T and (by (3.9)) one has N (ST − ST2 ) ⊂ N (ST T − T ST ). Conversely, let h ∈ H be such that ST T h =

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T ST h. Then (I − T )ST h = ST (I − T )h = 0 because R(I − T ) = N (ST ). This gives ST h = T ST h = T n ST h for n ≥ 1, so ST h = ST2 h. Hence N (ST T − T ST ) ⊂ N (ST − ST2 ). The proof is complete. 2 Remark 3.13. (a) Some assertions of Theorem 3.1 were obtained for the ST -isometry from Theorem 3.12 without the condition of ST -regularity. But, in general, the assumption on T to be A-regular in Theorem 3.1 cannot be dropped. For instance, every contraction T on H is also a T ∗ T -contraction, but in this case the subspace N (T ∗ T − (T ∗ T )2 ) = N (I − T ∗ T ) ⊕ N (T ) is not invariant for T (even if N (T ) = {0}). (b) Notice that (3.9) cannot be complemented to the equality with N (I − T ), because this subspace reduces T if and only if PT is an orthogonal projection. Hence the equivalent conditions (i)–(iii) of Proposition 3.10 hold when the sequence {T n }n∈N is strongly convergent, even if T is not a contraction. 4. Partial isometries associated to A-isometries In this section we study the properties of the operator A1/2 T on the subspace N (A − A2 ) in the context of A-isometries. We first establish a relationship between this subspace and the maximum subspace M∗ ⊂ H, which reduces M := A−1/2 A1/2 T to a normal partial isometry. Note that M∗ is well-defined owing to [13, Theorem 2.1]. Theorem 4.1. Let T ∈ B(H) be a regular A-isometry. Then the following assertions hold. (i) N (A −A2 ) is the maximum subspace of H which reduces A1/2 T to a partial isometry. In addition, M∗ ⊂ N (A − A2 ) if and only if either M∗ = N (A) or A = 1. (ii) N (I − A) is the maximum subspace of H which reduces A1/2 T to an isometry. If A = 1, then M∗ = H∗ ⊕ N (A),

(4.1)

where H∗ = M∗ ∩ N (I − A) is the maximum subspace of H which reduces A1/2 T to a unitary operator, and also H∗ is the maximum subspace of N (I − A) which is invariant for T ∗ such that T ∗ |H∗ is an isometry. In addition, we have H∗ =



{h ∈ H : AT n h = h = T ∗n Ah}

n∈N

=



{h ∈ N (A − A2 ) : P T n h = h = T ∗n P h}

n∈N

=



{h ∈ N (I − A) : T ∗n h = h},

n∈N

where P ∈ B(H) is the orthogonal projection of H with R(P ) = N (I − A).

(4.2)

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Proof. (i) If h ∈ N (A − A2 ), then we derive from Theorem 3.1 that T h ∈ N (A − A2 ). By this and the regularity of T , we get A1/2 T h = A3/2 T h = A1/2 T Ah = A1/2 T T ∗ AT h, which means that N (A − A2 ) reduces A1/2 T to a partial isometry. Now, if M ⊂ H is another subspace with this property, then the previous relations hold for h ∈ M. This combined with the fact that T is also a regular A2 -isometry lead to the inclusion M ⊂ N (A − A2 ). So N (A − A2 ) is the maximum subspace which reduces A1/2 T to a partial isometry. Obviously, if M∗ = N (A) or A = 1, then M∗ ⊂ N (A − A2 ). Conversely, assume that N (A) = M∗ ⊂ N (A − A2 ). Since T ∗ AT h = Ah = 0 for a nonzero h ∈ M∗  N (A) and A1/2 T is a partial isometry on N (A − A2 ), we infer that A1/2 T |M∗  = 1. As (A1/2 T )|M∗ = A1/2 W , where W is a nonzero partial isometry on M∗ , we get A = 1. (ii) Clearly A1/2 T is an isometry on N (I − A), and it is the maximum subspace of H, which reduces A1/2 T to an isometry. Suppose now that A = 1. Let H∗ be the maximum subspace in H, which reduces 1/2 A T to a unitary operator. Then by the last assertion in (i), we necessarily have H∗ = M∗ ∩ N (I − A). But M∗  H∗ reduces A1/2 T to a normal and completely nonunitary operator. For this reason A1/2 T is also completely non-isometric on M∗  H∗ . So M∗  H∗ = N (A) because N (A) ⊂ M∗ ⊂ N (I − A) ⊕ N (A). Hence, we obtain the decomposition (4.1). Next, by the Nagy–Foiaş–Langer decomposition [8,31] we may write H∗ as H∗ =



{h ∈ H : (A1/2 T )n h = h = (T ∗ A1/2 )n h}

n∈N

=



{h ∈ H : An/2 T n h = h = T ∗n An/2 h},

(4.3)

n∈N

where the second equality (between sets) follows from the relation AT = A1/2 T A1/2 . Now let h ∈ H∗ . Since A is a contraction, we have N (I − A) = N (I − A1/2 ) ⊂ N (A − A2 ), whence we infer An/2 h = Ah for n ∈ N. Therefore, T ∗n An/2 h = T ∗n Ah, and also An/2 T n h = AT n h because N (A − A2 ) is invariant for T (by Theorem 3.1 (a)). This enables us to deduce that AT n h = h = T ∗n Ah, n ∈ N, and, by (4.3), this gives the first set representing H∗ in (4.2). Next, if P is the orthogonal projection of H with R(P ) = N (I − A), then P k = Ak for k ∈ N (A − A2 ), and the above equalities can be equivalently written as P T n h = h = T ∗n P h, n ∈ N. In this way we obtain the second set representing H∗ in (4.2), and it is easy to see that this set is contained in the last set representing H∗ in (4.2). Finally, for h from the latter set and n ∈ N, we have h = An/2 h and T ∗n An/2 h = T ∗n h = h = An/2 h = An/2 T n h, because T n is an An -isometry on H (by the A-regularity of T ). Hence h ∈ H∗ by (4.3). Consequently, all equalities in (4.2) hold. From the last set in (4.2) we also infer that H∗ is the maximum subspace of N (I − A) which is invariant for T ∗ , while T ∗ |H∗ is an isometry. 2

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Corollary 4.2. If T is a regular A-isometry on H, then N (A − A2 ) is the maximum subspace of H which reduces A1/2 T to a quasi-isometry. In fact, A1/2 T is a quasinormal quasi-isometry on N (A − A2 ). The reducing quasi-isometric part of A1/2 T for an A-contraction T with A = 1 was obtained in [13], while Corollary 4.2 appears also as the last assertion of [13, Theorem 2.3]. Corollary 4.3. Let T be a regular A-isometry on H with A = 1. If the subspace N (I −A) is invariant for T , then H∗ = M∗ ∩ N (I − A) is the maximum subspace contained in N (I − A) which reduces T to a unitary operator. Remark 4.4. In certain cases H∗ is even the maximum subspace of H which reduces T to a unitary operator. This surely happens if T is a contractive regular T ∗ T -isometry. Another case is given by Theorem 3.12 when ST  = 1, which means that ST is the orthogonal projection onto N (I − T ). Then N (I − ST ) = N (I − T ) = N (I − T ∗ ) is the corresponding subspace H∗ from (4.1). A different setting will be presented in the next section, where T is not a contraction and A may be an orthogonal projection. Corollary 4.5. Let T ∈ B(H) be an A-isometry such that AT = T A. Then N (A − A2 ), N (I − A) and N (A) reduce the operator T , and T is an isometry on R(A). Moreover, if A = A2 , then H = H∗ ⊕ H1 ⊕ N (A), where H∗ = M∗ ∩ N (I − A) = N (I − ST ∗ ) reduces T to a unitary operator and H1 = N (ST ) reduces T to a shift, where T = T |N (I−A) . Finally, we pass to discussing the case when N (A − A2 ) coincides with M∗ . Proposition 4.6. Let T ∈ B(H) be a regular A-isometry such that A = 1 and N (A) = M∗ . Then the following statements are equivalent: (i) (ii) (iii) (iv) (v)

N (A − A2 ) = M∗ , AT is a normal operator on N (A − A2 ), A1/2 T is a unitary operator on N (I − A), T ∗ is an isometry on N (I − A), T ∗ is an A-isometry on N (I − A).

Moreover, if these conditions are satisfied and the subspace N (I − A) is invariant for T , then N (I − A) reduces T to a unitary operator. The proof is immediate, so we omit it. Remark that in some cases all conditions (i)–(v) hold without assuming that T is A-regular. This, for example, happens when T is a T T ∗ -isometry, where N (I − T ∗ T ) = N (I − T T ∗ ) and this subspace reduces T . Then T  ≤ 1 and ST = T T ∗ , and the T T ∗ -regularity yields N (ST − ST2 ) = N (T T ∗ − (T T ∗ )2 ) = H, i.e. T is a partial isometry. But, as we see below, one can easily find a T T ∗ -isometry T with ST = ST2 .

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Example 4.7. Let {en }n∈Z be the canonical orthonormal basis of lZ2 , and {an }n∈Z be an increasing sequence of reals such that 0 < an < 1, n ∈ Z. Let T be the bilateral weighted shift on lZ2 defined by T en = an en+1 , n ∈ Z. It is known that T is hyponormal (cf. [9, 23]), and T is a T T ∗ -isometry if and only if an−1 = a2n for every n ∈ Z. In addition, T is normal if and only if {an }n∈Z is a constant sequence. So if the sequence {an }n∈Z is not constant and it satisfies the previous recurrence relation, Proposition 3.10 yields ST = ST2 , or equivalently T is not T T ∗ -regular. In fact, the T T ∗ -regularity ensures that T ∗ is quasinormal, and so hyponormal. As T is also hyponormal, it is normal. This contradicts the choice of {an }n∈Z . 5. Applications to 2-isometries Let T ∈ B(H) be a 2-isometry. Then ΔT = T ∗ T −I ≥ 0 (cf. [1]), so T is a ΔT -isometry. Hence N (ΔT ) is an invariant subspace for T which, in fact, is an isometry on N (ΔT ). It is known (cf. [1, Proposition 1.25], [24, Theorem 3.6], and [16, Proposition 1.1]) that for a 2-isometry T , the maximum subspaces in H which reduce T to an isometry and to a unitary operator, respectively, are H0 = H 

n≥0

T n R(ΔT ),

H∞ =



T n H.

(5.1)

n≥1

In addition, it is easy to see that if T is a 2-isometry, then N (ΔT ) = N (ΔT T − T ΔT ),

(5.2)

while N (ΔT − Δ2T ) is invariant for T when T is ΔT -regular. In this case N (I − ΔT ) is invariant for T ∗ , but is not invariant for T when 2 ∈ σp (T ∗ T ), having in view the previous equality and Theorem 3.1 (ii). Recall from [2] that for a 2-isometry T the scalar δ = δT := ΔT 1/2 > 0 is called the covariance of T . Clearly, such a T is also a δ −2 ΔT -isometry, and it is δ −2 ΔT -regular if and only if is ΔT -regular. If this is the case, then the subspace (compare with (5.2)) N (δ −2 ΔT − δ −4 Δ2T ) = N (ΔT T − T ΔT ) ⊕ N (δ 2 I − ΔT ) is invariant for T . Further, N (δ 2 I − ΔT ) is invariant for T ∗ , but not for T when δ 2 + 1 ∈ σp (T ∗ T ). The previous equality shows that the inclusion (3.7) need not hold when T is not a contraction, and the inclusion (3.3) can be strict, in the general context of A-isometries. A description of the structure of a 2-isometry in terms of block operators was given in [1, Theorem 1.26]. Below we characterize those 2-isometries which are ΔT -regular. Proposition 5.1. Let T ∈ B(H) be with δ = δT > 0. Then T is a ΔT -regular 2-isometry if and only if it has the block matrix form

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 T =

V 0

δE W

181

 (5.3)

with respect to the decomposition H = N (ΔT ) ⊕ R(ΔT ), where (i) V is an isometry, (ii) E is an injective contraction with V ∗ E = 0, and (iii) W is an isometry which commutes with E ∗ E. Proof. Assume that T is a ΔT -regular 2-isometry. Since δ > 0, we have ΔT = 0, so ΔT = 0 ⊕ Δ0 with Δ0 = ΔT |R(ΔT ) ≥ 0 and Δ0 an injective operator. As N (ΔT ) is T -invariant, T has a block matrix of the form (5.3) with V being an isometry on N (ΔT ) (which means that (i) holds) and some operators E and W . It is easy to see that W is a regular Δ0 -isometry, and as Δ0 is injective, this just means that Δ0 W = W Δ0 , 1/2 1/2 1/2 which implies that W Δ0 k = Δ0 W k = Δ0 k for k ∈ R(ΔT ). Hence W is an isometry on R(ΔT ). Next, expressing ΔT = T ∗ T − I by using (5.3) and the fact that N (ΔT ) reduces ΔT and W is an isometry, we get the relations V ∗ E = 0 and Δ0 = δ 2 E ∗ E, whence we obtain the remaining assertions of (ii) and (iii). Conversely, if T has the block matrix form (5.3) and (i)–(iii) hold, then it immediately follows that T is a regular ΔT -isometry (see also [30, Lemma 2.1 and Proposition 2.11]). 2 Remark 5.2. The condition δ = δT > 0 in Proposition 5.1 signifies that T is not an isometry. Therefore, N (ΔT ) = {0} for a ΔT -regular 2-isometry T . Remark 5.3. It was proved in [1, Theorem 1.26] that every 2-isometry has a block matrix of the form (5.3), where V is an isometry, V ∗ E = 0, but E is not necessarily injective, W is a Δ0 -isometry, but it need not be a contraction. Hence for a 2-isometry T , which is not ΔT -regular, T ∗ |R(ΔT ) is not a contraction, in general. The same observation remains valid if we consider T being an A-isometry with A = δ −2 ΔT (so with A = 1). This justifies that the hypothesis on T ∗ |R(ΔT ) in Theorem 3.6 cannot be dropped. Let us remain in the framework of 2-isometries. Following [2, Proposition 5.37] recall that a 2-isometry T is called a Brownian isometry of covariance δ = δT > 0 if T has a block matrix of the form (5.3), where W is a unitary operator commuting with E ∗ E. In this case Δ0 = δ 2 E ∗ E, so E is an injective contraction. In particular, if W is a unitary operator which commutes with E ∗ E, and E is an isometry with R(E) = N (V ∗ ), then according to [2, Proposition 5.12] such a T is called a Brownian unitary of covariance δ > 0. Proposition 5.1 makes it evident that Brownian isometries (unitaries) are ΔT -regular 2-isometries, and the inclusion between these classes of operators are strict. These “Brownian parts” can be determined as follows.

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Theorem 5.4. Let T ∈ B(H) be a ΔT -regular 2-isometry of covariance δ > 0 having a block matrix of the form (5.3). Then the following assertions hold. (i) If W is not a shift operator, then the maximum subspace which reduces T to a Brownian isometry of covariance δ is 2 Hb0 := [H0 ⊕ l+ (V, EK∞ )] ⊕ K∞ ,

(5.4)

where K∞ ⊂ R(ΔT ) is the unitary part of W and H0 is as in (5.1). (ii) If E is not a proper contraction, then N (DE ) reduces W and 2 E := [H0 ⊕ l+ (V, EN (DE ))] ⊕ N (DE )

(5.5)

is the maximum subspace in H which reduces T such that the operator E0 = E|R(ΔT | ) in the corresponding matrix (5.3) of T |E is an isometry. E (iii) If W |N (DE ) is not a shift operator, then the maximum subspace in H which reduces T to a Brownian unitary of covariance δ is 2 Hb1 := [H∞ ⊕ l+ (V, ER0 )] ⊕ R0 ,

(5.6)

where H∞ is as in (5.1) and R0 = K∞ ∩ N (DE ). Proof. (i) Assume that W is not a shift. Then K∞ = {0} and, in view the structure (5.1) of K∞ , we see that this subspace reduces E ∗ E and E ∗ EW = W E ∗ E. Now let E = J|E| be the polar decomposition of E. Since E is injective and V ∗ E = 0 (by Proposition 5.1 (ii)), the operator J is an isometry from R(ΔT ) = R(E ∗ E) onto R(E) ⊂ N (V ∗ ). In fact R(E) = N (V1∗ ), where V1 = V |N (ΔT )H0 . As K∞ reduces E ∗ E, we have R(E) = J|E|K∞ ⊕ J|E|(R(E ∗ )  K∞ ) = EK∞ ⊕ E(R(E ∗ )  K∞ ) in N (V ∗ ) because R(E ∗ ) = R(ΔT ). So EK∞ is a wandering subspace for V . Let us set 2 N := l+ (V, EK∞ ). Then the subspace H1 := N ⊕ K∞ reduces T . Indeed, it is immediate from (5.3) that T H1 ⊂ H1 . Clearly, N reduces V because V ∗ E = 0, and also E∗N ⊂



E ∗ V n EK∞ ⊂ E ∗ EK∞ = K∞ .

n≥0

From these facts and the matrix form (5.3) of T ∗ we derive that T ∗ H1 ⊂ H1 . Now, it is clear that the subspace Hb0 (defined in (5.4)) reduces T to a Brownian isometry of covariance δ because ΔT |Hb0  = δ 2 E ∗ E|K∞  = δ 2 , taking into account that E = 1 in the corresponding matrix (5.3) of any ΔT -regular 2-isometry with a positive covariance. We now prove that Hb0 is the largest subspace with this property. Indeed, if M ⊂ H is another such subspace, then M reduces ΔT and M1 = R(ΔT ) ∩ M reduces W

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183

to a unitary operator (T |M being a Brownian isometry). Therefore, M1 ⊂ K∞ . On the other hand, N (ΔT ) ∩ M = (H0 ∩ M) ⊕ M0 , where H0 is as in (5.1) and M0 = (N (ΔT )  H0 ) ∩ M, but M0 does not contain any other subspace reducing T to an isometry. Thus T |M0 = V |M0 is a shift. Taking into account the above inclusion and the  form (5.3) of T |M we infer that M0 = n≥0 V n R(E1 ) with E1 = E|M1 . Finally, we deduce that M = (H0 ∩ M) ⊕ M0 ⊕ M1 is contained in the subspace Hb0 , which proves the maximality property of this subspace. Let us proceed to the proof of (ii) and (iii). Assume that the operator E in (5.3) is not proper, i.e. N (DE ) = {0}, where DE is the defect operator. Since DE commutes with W , N (DE ) reduces W and E maps isometrically N (DE ) into N (V ∗ ). Then it is easy to see (as above) that E (defined in (5.5)) reduces T , therefore R(ΔT |E ) = N (DE ) and the restriction of E to this subspace is an isometry. In addition, if W is not a shift operator on N (DE ), then R0 := K∞ ∩ N (DE ) = {0} and this subspace reduces W to a unitary operator. Also, we deduce from (5.3) that Hb1 (defined in (5.6)) reduces T to a Brownian unitary of covariance δ because E maps isometrically R0 onto ER0 = N (V  ∗ ), where V  = V |Hb1 R0 . In both cases, similar arguments as above show that E and Hb1 have the required maximality properties. 2 Remark 5.5. The meaning of the subspace E is the following: the 2-isometry T |E has a block matrix of the form (5.3) with all corresponding operators V, E and W being isometries, and clearly, in this case ΔT |E = δ 2 I. Such a 2-isometry is ΔT -regular but, as Proposition 5.1 shows, not every regular ΔT -isometry has this form. Remark also that δ −2 ΔT |E is an orthogonal projection in B(E), so H = N (δ 2 ΔT −Δ2T ) if and only if H = E. Hence the 2-isometries T for which δ −2 ΔT are orthogonal projections are exactly those with all isometric operators in their block matrices (5.3). Among them there are, of course, the Brownian unitaries. It is worth noting that Hb1 ⊂ Hb0 ∩ E and this inclusion is strict even in the case when E = H. In the context of a 2-isometry we have E ⊂ N (δ 2 ΔT − Δ2T ), where the inclusion is strict, in general. Also, we see that the subspace R0 from the assertion (iii) of Theorem 5.4 is just the corresponding subspace H∗ from Theorem 4.1 for T as a regular δ −2 ΔT -isometry. Clearly, R0 ⊂ N (δ 2 I − ΔT ) = N (DE ) with a strict inclusion again, in general. For T as in the above theorem, when the operators T |Hb1 and T |Hb0 are non-zero they are called the Brownian unitary part and the Brownian isometric part of T , respectively. Theorem 5.4 enables us in fact to get the Wold-type decomposition of T , where the Brownian unitary part appears in place of the unitary part from the classical Wold decomposition of an isometry, while the completely non-Brownian unitary part can be refined as below. Corollary 5.6. Let T ∈ B(H) be a ΔT -regular 2-isometry of covariance δ > 0. Then T can be represented as the direct sum

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T = T0 ⊕ T 1 ⊕ T 2

on

H = Hb1 ⊕ (Hb0  Hb1 ) ⊕ (H  Hb0 )

(5.7)

such that T0 is a Brownian unitary, T1 is a Brownian isometry without Brownian unitary part, and T2 is a ΔT2 -regular 2-isometry without Brownian isometric part. Also, each of these parts of T is of covariance δ when it acts on a non-zero subspace of H. Moreover, such a representation of T is uniquely determined by the above properties. According to [1, Definition 1.8] a 2-isometry T is called pure if it has no non-zero isometric direct summand. For a ΔT -regular 2-isometry T , this means that V in (5.3) is a pure isometry. Note that the Brownian isometry T1 in (5.7) is pure if and only if the (reducing) isometric part of T reduces to the unitary part of T . On the other hand, the 2-isometry T2 is pure, being completely non-Brownian isometric. The block matrix form of T2 will be given in Corollary 5.9 below. Now let us describe the structure of such a pure 2-isometry with the help of reducing subspaces introduced above. Theorem 5.7. Let T ∈ B(H) be a pure ΔT -regular 2-isometry of covariance δ > 0, having the block matrix (5.3). Let K∞ be the unitary part of W and set R0 = K∞ ∩ N (DE ), 2 R1 = K∞ ∩ R(DE ), R2 = R(ΔT )  K∞ , Ej = E|Rj and Nj = l+ (V, R(Ej )) for 2 j = 0, 1, 2. Then N (ΔT ) = j=0 Nj and T has the block matrix ⎛

S2 ⎜0 ⎜ ⎜0 T =⎜ ⎜0 ⎜ ⎝0 0

0 S1 0 0 0 0

0 0 S0 0 0 0

0 0 δE0 U0 0 0

0 δE1 0 0 U1 0

⎞ δE2 0 ⎟ ⎟ 0 ⎟ ⎟ 0 ⎟ ⎟ 0 ⎠ S

⎡

⎤ N2 ⎢ N1 ⎥ ⎢ ⎥ ⎢ N0 ⎥ ⎢ ⎥, ⎢ R0 ⎥ ⎢ ⎥ ⎣ R1 ⎦ R2

(5.8)

where S and Sj are shifts, U0 and U1 are unitary operators, E0 is an isometry, E1 is a proper contraction, and E2 is a contraction of type E0 ⊕ E1 such that E ∗ E commutes with U0 , U1 and S on the corresponding subspaces. In addition, Hb1 = N0 ⊕ R0

and

Hb0 = N1 ⊕ N0 ⊕ R0 ⊕ R1

are, respectively, the pure Brownian unitary and the pure Brownian isometric parts of T in H. Proof. First note that ΔT = 0 and E = 0 in (5.3) with T being non-isometric and E injective. Since K∞ is the reducing unitary part of W , the subspace R2 is reducing for W , and also R0 and R1 = K∞  R0 are such, as we have seen in the above proof. But each Rj reduces the injective contraction E ∗ E, therefore Rj = E ∗ ERj and R(ΔT ) = E ∗ ER(ΔT ) = R0 ⊕ R1 ⊕ R2 . By using the polar decomposition E = J|E| of E, where J is an isometry, and putting Ej = E|Rj we obtain

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R(E) = ER(ΔT ) =

2 

JRj =

2 

j=0

185

R(Ej ) ⊂ N (V ∗ ).

j=0

So, each subspace R(Ej ) induces a “shift” subspace for V into N (ΔT ), namely Nj := 2 l+ (V, R(Ej )), which reduces V . Since, in view of the representation (5.3) of T , we have R(Ej ) ⊂ R(E) ⊂ T R(ΔT ), owing to (5.1) and the fact that H0 = {0} (in our hypothesis) it follows that Nj ⊂



T n R(ΔT ) = H  H0 = H.

n≥0

Now the subspace N (V ∗ )  R(E) also generates a “shift” subspace for V (as above) denoted by N , which is invariant for T . Since N (V ∗ )  R(E) ⊂ N (E ∗ ), we see by (5.3) that N is invariant, too, for T ∗ . Thus, N reduces T to an isometry, hence N = {0} because T is pure as a 2-isometry. We conclude that T |N (ΔT ) is a shift with the wandering subspace N (V ∗ ) = R(E), and from the definitions before of Nj we obtain

2 N (ΔT ) = l+ (V, R(E)) =

2 

Nj .

j=0

Further, it is clear that the subspaces Hb1 = N0 ⊕ R0 and Hb0 = N1 ⊕ N0 ⊕ R0 ⊕ R1 are, respectively, the pure Brownian unitary and the pure Brownian isometric parts of T in H. Having this in mind, we immediately derive from (5.3) and (5.7) that T has the 2 2 block matrix form (5.8) with respect to the decomposition H = j=0 N2−j ⊕ j=0 Rj . All operators in this matrix are zero except for those on the main diagonal, being unitary or shifts, and those lying on the antidiagonal on the right-hand side, which are injective δ-multiples of contractions, where δ = δT > 0. Clearly, E0 is an isometry from R0 into N0 and E1 is a proper contraction from R1 = K∞ ∩ R(I − E ∗ E) into N1 . Also, since R2 = (R2 ∩ N (DE )) ⊕ (R2 ∩ DE ) and these two summands are invariant subspaces for E ∗ E, it follows that E2 = E2 ⊕ E2 with respect to this decomposition, where E2 is an isometry and E2 is a proper contraction. 2 Remark 5.8. Observe also from the matrix representation (5.8) that Hb := N2 ⊕ N1 ⊕ N0 ⊕ R0 ⊕ R1 is the maximum invariant subspace for T on which T is a Brownian isometry. Corollary 5.9. Let T ∈ B(H) be a ΔT -regular 2-isometry of covariance δ > 0. Then T does not have the Brownian isometric part if and only if T has the block matrix form  T =

S 0

δE S

 (5.9)

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with respect to the decomposition H = M ⊕ M⊥ , where S and S  are shifts, E is an injective contraction such that R(E) = N (S ∗ ) and E ∗ ES  = S  E ∗ E. In addition, if (5.9) holds, then δ −2 ΔT is an orthogonal projection if and only if E is an isometry. Note that the ΔT -regular 2-isometries without Brownian isometric part have specific block matrices of the form (5.9), where S, S  are shift operators and E ∗ E is not only a Toeplitz operator for S  but (being a commutant of S  ) is even S  -analytic (cf. [22]). On the other hand, E is closely related by its range to S. These operators show that the class of ΔT -regular 2-isometries strictly contains the Brownian isometries. Corollary 5.10. If T ∈ B(H) is a pure ΔT -regular 2-isometry of covariance δ > 0 and δ −2 ΔT is an orthogonal projection, then the Brownian unitary and isometric parts of T coincide. The structure of a 2-isometry T obtained above, in which the Brownian unitary and isometric parts of T can be concretely described, is essentially based on the ΔT -regularity of T which imposes that T ∗ |R(ΔT ) is a coisometry commuting with Δ0 . As we now show, the Brownian unitary part of a 2-isometry may be constructed under other more general conditions. Theorem 5.11. Let T ∈ B(H) be a 2-isometry of covariance δ > 0, having the block matrix of the form (5.3) with V being an isometry and E, W being contractions. Then the reducing Brownian unitary part of T is the subspace Hb1 from (5.6), and T |Hb1 is of covariance δ when K∞ ∩ N (DE ) = {0}. Moreover, δ −2 ΔT is an orthogonal projection if and only if the operators E and W are isometries. Proof. Since T is a ΔT -isometry with ΔT = 0, T has always a block matrix of the form (5.3) with V being an isometry on N (ΔT ), E ∈ B(R(ΔT ), N (ΔT )) such that V ∗ E = 0 (because ΔT ≥ 0), and W being a Δ0 -isometry (i.e. W ∗ Δ0 W = Δ0 ), where Δ0 = ΔT |R(ΔT ) = δ 2 E ∗ E + W ∗ W − I. By our assumption, W and E are contractions, so E is an injective operator because Δ0 is also injective. Next, note that W is also a δ −2 Δ0 -isometry, so then we deduce from Proposition 3.7 that N (δ 2 I −Δ0 ) is invariant for W , while W is an isometry on this subspace. Therefore, N (δ 2 I − Δ0 ) ⊂ N (I − SW ), SW being the asymptotic limit of the contraction W . In 2 turn, since δ 2 I − Δ0 = δ 2 (I − E ∗ E) + DW ≥ δ 2 (I − E ∗ E), we infer that N (δ 2 I − Δ0 ) ⊂ ∗ N (I−E E). A combination of these two inclusions and the relationship between Δ0 , E ∗ E 2 yields and DW N (δ 2 I − ΔT ) = N (I − E ∗ E) ∩ N (I − SW ) = N (I − E ∗ E) ∩ N (I − W ∗ W ).

(5.10)

They imply that δ −2 ΔT is an orthogonal projection (i.e. N (δ 2 I − ΔT ) = R(ΔT )) if and only if W and E are isometries. This gives us the second assertion of the theorem.

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Looking again at W as a δ −2 Δ0 -isometry, we appeal one more time to Proposition 3.7 to get N (δ 2 I−Δ0 ) ⊂ N (Δ0 W −W Δ0 ). So, if K∞ is the unitary part of W in R(ΔT ), then for h ∈ K∞ ∩N (DE ), by (5.9) and the previous inclusion, we have Δ0 W h = W Δ0 h. If we take into account that W h ∈ K∞ and E ∗ Eh = h, we have E ∗ EW h = W E ∗ Eh = W h. Therefore, W h ∈ N (DE ) and thus K∞ ∩ N (DE ) is invariant for W . This can be directly obtained from (5.10), having in view that N (δ 2 I − ΔT ) is invariant for W (as evidenced above). On the other hand, since W is a Δ0 -isometry, it follows for h ∈ K∞ that W ∗ (δ 2 E ∗ E + W ∗ W − I)W h = (δ 2 E ∗ E + W ∗ W − I)h, that is, W ∗ E ∗ EW h = E ∗ Eh because W h ∈ K∞ . Also, as W ∗ h ∈ K∞ and W W ∗ h = h, we can substitute W ∗ h for h in the previous equality to obtain E ∗ EW ∗ h = W ∗ E ∗ Eh = W ∗ h for h ∈ K∞ ∩ N (DE ), which means that W ∗ h ∈ K∞ ∩ N (DE ). Therefore, K∞ ∩ N (DE ) reduces W to a unitary operator, and it trivially reduces E ∗ E. Hence the operator E maps isometrically K∞ ∩ N (DE ) onto a wandering subspace in N (V ∗ ) for V . As a result, the subspace Hb1 can be defined as in (5.6) and it reduces T to a Brownian unitary operator. In addition, if K∞ ∩ N (DE ) = {0}, then T |Hb1 is of covariance δ = ΔT |Hb1  because ΔT |Hb1 = ΔT |Hb1 = Δ0 |K∞ ∩N (DE ) = δ 2 E ∗ E|K∞ ∩N (DE ) = δ 2 I. In this case, when K∞ ∩ N (DE ) = {0} the subspace Hb1 is the largest reducing T to a Brownian unitary. Indeed, let M ⊂ H be another such subspace. Then M reduces ΔT and ΔT |M = R(ΔT ) ∩ M =: M1 . Using the block matrix of T , we see that M1 reduces W to a unitary operator because T |M is a Brownian unitary, and also E|M1 needs to be an isometry, that is to say, E ∗ E|M1 = I. So M1 ⊂ K∞ ∩ N (DE ). Next, N (ΔT |M ) =  N (ΔT ) ∩ M = (H∞ ∩ M) ⊕ M0 , where H∞ is as in (5.1) and M0 := n≥0 V n E1 M1 with E1 = E|M1 , because T |M is Brownian unitary of covariance δ (see (5.17) and (5.18) in [2]). Hence M = (H∞ ∩M) ⊕M0 ⊕M1 ⊂ Hb1 , which proves the maximality property of Hb1 . 2 Corollary 5.12. Let T be as in Theorem 5.11 such that E is an isometry. Then the reducing Brownian isometric part of T in H is the subspace Hb0 = (H0  H∞ ) ⊕ Hb1 , and T |Hb0 is of covariance δ > 0 when T ∗ |R(ΔT ) has non-zero unitary part. As Theorem 5.11 shows, the Brownian unitary part of a 2-isometry T can be defined under more general conditions than the ΔT -regularity. But even in a more particular setting than in Theorem 5.11 (with W  = 1, and so E ≤ 1), it may not be welldefined (see Example 5.13). This makes the class of ΔT -regular 2-isometries particularly interesting in itself. Regarding the structure of the Brownian isometric part of a 2-isometry T (being the form (5.3)), this part needs to be “supported” on the unitary part K∞ of W in

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R(ΔT ), which imposes that K∞ is reducing for E ∗ E. Owing to this fact, we construct a 2-isometry T (not being ΔT -regular) with an isometry W and K∞ = {0} such that K∞ does not reduce E ∗ E. 2 Example 5.13. Let S be the forward shift on l+ and U be the bilateral shift (i.e. the 2 2 2 minimal unitary extension of S) on lZ . Consider the isometry W =  K = lZ ⊕ l + .  U ⊕ S on 2I J on K, where Then K∞ = lZ2 with our above notation. Define the operator A = J ∗ 2I 2 2 ∗ 2 2 denotes J : l+ → lZ the natural injection. Clearly J is the projection of lZ onto l+ . It ∗ is also plain that A ≥ 0, A is injective (even invertible), and W AW = A because U J = JS. 2  = l+  be the natural embedding of K into K,  E  = JA  1/2 , and Let K (K), J : K → K  We can write E  = δE, where δ = E  = A1/2  and V be the forward shift on K. 2 ∗ ∗  E  = A, and from the above matrix E is an injective contraction. Then δ E E = E of A it follows that the subspace K∞ (i.e. the unitary part of W ) does not reduce E ∗ E. Equivalently, we have E ∗ EW = W E ∗ E because J ∗ U = SJ ∗ (U ∗ is not an extension of S ∗ ), but W ∗ E ∗ EW = E ∗ E.   V δE  ⊕ K. Clearly, R(E) = R(A) = Now consider the operator T = on G = K 0 W  and since V , W are isometries, it is easy to verify (or see [1, K = N (V ∗ ) into K,

Theorem 1.26]) that T is a 2-isometry on G which is not ΔT -regular. Notice that T is  Also, the subspace EK∞ = {0} pure as a 2-isometry on G because V is a shift on K. 2  generates the subspace L = l+ (V, EK∞ ) in K, but the subspace L ⊕ K∞ ⊂ G does not reduce T . In fact, L ⊕ K∞ is invariant for T , but not for T ∗ because E ∗ EK∞  K∞ as we already observed. So, in this case the Brownian isometric part of T cannot be defined. Finally, we claim that the Brownian unitary part of T equals to {0}. To see this, we first note that R(ΔT ) = K and ΔT |K = δ 2 E ∗ E, so ΔT  = δ 2 = A. But one can easily convince himself that 1, 2 and 3 are the eigenvalues of A, hence we get δ 2 = A = 3. Next, we need to determine explicitly the subspace K∞ ∩ N (I − E ∗ E) of K. For this purpose, observe that N (I − E ∗ E) = N (3I − A). Representing A as the 3 × 3 operator   2I1 0 I1 2 2 2 0 2I2 0 matrix on K = l+ ⊕ (lZ2  l+ ) ⊕ l+ , where I1 and I2 are the identity I1 0 2I1 2 2 2 operators on l+ and lZ2  l+ , respectively, we see that N (3I − A) = {x ⊕ 0 ⊕ x : x ∈ l+ }, so K∞ ∩ N (3I − A) = {0}. Thus, by Theorem 5.11, the Brownian unitary part of T is {0}, and so our claim is established. Remark 5.14. It is known (cf. [1, Proposition 1.5] or [6, Proposition 2.3]) that for a 2-isometry T , the generalized Toeplitz operator ΔT of T can be considered as an asymptotic limit of the sequence { n1 T ∗n T n }n∈N in the sense that ΔT h, h = limn→∞ n1 T n h2 for h ∈ H. Hence N (ΔT ) = {h ∈ H : √1n T n h → 0 as n → ∞}, so if the equality N (ΔT ) = H holds, then in the standard terminology T is said to be of class C0· as a 2-isometry, i.e. T  = 1. When N (ΔT ) = {0} one says that T is of class C1· as

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a 2-isometry, and this means that { √1n T n h}n∈N does not converge to zero for any h ∈ H \ {0}. But the corresponding operator W in the block matrix (5.3) of T does not have this property, because W is not a 2-isometry, in general. However, if W  ≤ 1, then W ∗ ST W = ST , and as W ∗ δ −2 Δ0 W = δ −2 Δ0 , we have Δ0 ≤ δ 2 ST , and so W is of class C1· as a contraction on R(Δ0 ) ⊂ R(ST ). In this case, the block matrix (5.3) of T can be compared to the canonical triangulation of a contraction with respect to C0· and C1· parts (cf. [31]), or more general of an operator T with {T n h}n∈N convergent for every h ∈ H (cf. [30, Proposition 2.5]), which we referred to in Section 2. Acknowledgements The first author was supported by the Polish Ministry of Science and Higher Education (AGH local grant). The second author was supported in part by the Labex CEMPI (ANR-11-LABX-0007-01). The third author was supported by the project financed from Lucian Blaga University of Sibiu research grants LBUS-IRG-2016-02. The authors thank the referee for valuable comments that helped to improve the paper. References [1] J. Agler, M. Stankus, m-isometric transformations of Hilbert spaces, Integral Equations Operator Theory 21 (4) (1995) 383–429. [2] J. Agler, M. Stankus, m-isometric transformations of Hilbert spaces II, Integral Equations Operator Theory 23 (10) (1995) 1–48. [3] J. Agler, M. Stankus, m-isometric transformations of Hilbert spaces III, Integral Equations Operator Theory 24 (1996) 379–421. [4] A. Aleman, The Multiplication Operator on Hilbert Spaces of Analytic Functions, Habilitationsschrift, Fern Universität, Hagen, 1993. [5] A. Aluthge, On p-hyponormal operators for 0 < p < 1, Integral Equations Operator Theory 13 (1990) 307–315. [6] T. Bermúdez, I. Marrero, A. Martinón, On the orbit of an m-isometry, Integral Equations Operator Theory 64 (2009) 487–494. [7] G. Cassier, L. Suciu, Mapping theorems and similarity to contractions for classes of A-contractions, in: Hot Topics in Operator Theory, in: Theta Ser. Adv. Math., 2008, pp. 39–58. [8] C.S. Kubrusly, An Introduction to Models and Decompositions in Operator Theory, Birkhäuser, Boston, 1997. [9] C.S. Kubrusly, Hilbert Space Operators: A Problem Solving Approach, Birkhäuser, Boston, 2003. [10] C.S. Kubrusly, P.C.M. Vieira, D.O. Pinto, A decomposition for a class of contractions, Adv. Math. Sci. Appl. 6 (1996) 523–530. [11] W. Majdak, N.-A. Secelean, L. Suciu, Ergodic properties of operators in some semi-Hilbertian spaces, Linear Multilinear Algebra 61 (2) (2012) 139–159. [12] M. Mbekhta, L. Suciu, Classes of operators similar to partial isometries, Integral Equations Operator Theory 63 (4) (2009) 571–590. [13] M. Mbekhta, L. Suciu, Quasi-isometries associated to A-contractions, Linear Algebra Appl. 459 (2014) 430–453. [14] S. McCullough, Subbrownian operators, J. Operator Theory 22 (1989) 291–305. [15] M.S. Moslehian, S.M.S. Nabavi Sales, H. Najafi, On the binary relation ≤u on self-adjoint Hilbert space operators, C. R. Math. Acad. Sci. Paris 350 (7–8) (2012) 407–410. [16] A. Olofsson, A von Neumann–Wold decomposition of two-isometries, Acta Sci. Math. (Szeged) 70 (2004) 715–726. [17] S.M. Patel, A note on quasi-isometries, Glas. Mat. 35 (55) (2000) 307–312. [18] S.M. Patel, 2-isometric operators, Glas. Mat. 37 (57) (2002) 141–145.

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