MAPPING THEOREMS AND SIMILARITY TO CONTRACTIONS FOR

MAPPING THEOREMS AND SIMILARITY TO CONTRACTIONS FOR CLASSES OF A-CONTRACTIONS GILLES CASSIER AND LAURIAN SUCIU

In the present paper we derive a functional calculus for A-contractions in the case when T is polynomially bounded and we give some mapping theorems, with applications to some operators similar to contractions, and in the ergodic theory. A bounded linear operator T on a complex Hilbert space is an A-contraction if T ∗ AT ≤ A for a positive operator A. We will also consider the special cases when A = (T ∗ T )α , with α ∈ (0, 1], respectively A = T ∗n T n , which are in fact power bounded, respectively similar to contractions and other related operators. Abstract.

1. Introduction Let H be a complex separable Hilbert space, and B(H) the Banach algebra of all bounded linear operators on H, where I = IH is the identity operator. The adjoint operator of T ∈ B(H) is denoted by T ∗ , and R(T ), N (T ) stand for the range and the null space of T , respectively. An operator T on H is a contraction if T ∗ T ≤ I , or equivalently ||T || ≤ 1. Also, T is a hyponormal operator if T T ∗ ≤ T ∗ T . All invariant (reducing) subspaces in H for subsets of B(H) will be supposed to be closed. The largest subspace in H which reduces an operator T to a unitary operator is called the unitary part in H of T . If such a part reduces to {0}, then T is called a completely non unitary operator. The class of contractions is one of the most studied and understood class of operators on H, and the investigations concerning other operators in B(H) have a starting point the theory of contractions. We refer 2000 Mathematics Subject Classication. Primary 47A15, 47A20, 47A63; Secondary 47B20. Key words and phrases. A-contraction, A-isometry, functional calculus, n-quasicontractions, similarity to contraction, Aluthge transform. 1

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GILLES CASSIER AND LAURIAN SUCIU

to a class of operators which generalize the contractions in dierent ways, where many questions naturally arise. More precisely, if A ∈ B(H) is a xed positive operator (i.e. hAh, hi ≥ 0 for any h ∈ H ), then an operator T ∈ B(H) is called an A-contraction on H if it satises the inequality (1.1)

T ∗ AT ≤ A.

If the equality in (1.1) occurs, then T is called an A-isometry on H. Such operators appear in dierent papers as [C1], [C2], [CMS], [D2], [S1-4], [K], and others. Clearly an ordinary contraction T is an I contraction, and a T ∗ T -contraction. On the one hand, we observe that if T is an A-contraction for some positive invertible operator A, then A1/2 T A−1/2 is a contraction. On the other hand, if T = B −1 CB where C is a contraction and B is an invertible operator, then T is a Acontraction for A = B ∗ B . Consequently, an operator T on H is similar to a contraction if and only if T is an A-contraction for some positive invertible operator A. It is clear from (1.1) that the null space N (A) is an invariant subspace for T (and trivially, for A), but N (A) is not reducing for T , in general. N (A) reduces T if AT = T A. For instance, if T is a quasinormal contraction (that is T ∗ T 2 = T T ∗ T ) then T and T ∗ are T ∗ T -contractions and T ∗ T commutes with T and T ∗ . Let T be an A-contraction. Applying Douglas criterion [D1], we see that there exists a unique contraction Tb ∈ B(R(A)) satisfying A1/2 T = TbA1/2 , where A1/2 is the square root of A. We call Tb the Douglas contraction associated to T . Since {T ∗n AT n ; n ≥ 1} is a bounded decreasing sequence of positive operators, it converges strongly to an operator AT ∈ B(H). When A = I , that is, if T is a contraction, then we denote by ST the strong limit of {T ∗n T n ; n ≥ 1}. Clearly, ST = 0 if and only if T n h → 0 (n → ∞) for any h ∈ H, and this means that T is strongly stable on H. If Tb is the Douglas contraction on R(A) associated to the A-contraction T , then STb will be the strong limit in B(R(A)) of the sequence {Tb∗n Tbn ; n ≥ 1}. Since A1/2 T n h = Tbn A1/2 h and T ∗n A1/2 k = A1/2 Tb∗n k for h ∈ H, k ∈ R(A) and n ≥ 1, one has (1.2)

AT h = A1/2 STb A1/2 h (h ∈ H).

This gives AT ≤ A because STb ≤ I . We also have STb = Tb∗ STb Tb and AT = T ∗ AT T . In order to get more important facts on A-contractions, we have introduced the class of regular A-contractions [S1-2]. We say that an

MAPPING THEOREMS AND SIMILARITY TO CONTRACTIONS

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A-contraction T is regular (shortly, T is A-regular ) if it satises the condition AT = A1/2 T A1/2 .

(1.3)

Equivalently, this means that A1/2 and A1/2 T commute. Recall that R(A) = R(A1/2 ) and that A1/2 is injective on R(A). It is clear that if T is A-regular such that N (A) reduces T , then A1/2 T = T A1/2 , or equivalently AT = T A. Consequently, for a regular A-contraction T we have AT = T A if and only if N (A) reduces T . We will see here that the class of A-regular contractions is stable for the functional calculus. We say also that an A-contraction T is ergodic (shortly T is Aergodic ) if n−1

lim n

n−1

1 X 1/2 j 1 X ∗j 1/2 A T h = lim T A h = A1/2 P h, n n n j=0 j=0

where P is the orthogonal projection onto N (A − AT ) (see [S2, S4]). It was proved [S2] that if T is A-ergodic then AN (A−AT ) ⊂ N (A−AT ). When A is injective one has N (A − AT ) = N (I − T ). Also, if T is power bounded (T ∈ P W B(H)) that is supn∈N ||T n || < ∞, then T is A-ergodic if and only if AN (A − AT ) ⊂ N (A − AT ) (see [S5]). For an operator T we consider his polar decomposition T = U |T | where U is the partial isometry verifying N (T ) = N (U ), and |T | = (T ∗ T )1/2 . We associate to T his α-Aluthge transform given by ∆α (T ) = |T |α U |T |1−α , for α ∈ (0, 1] [O]. In the case α = 1, ∆1 (T ) = |T |U is known as Duggal's transform of T that we will note Te [FJKP]. In the sequel we will see the relations between these transformations and T as a (T ∗ T )α -contraction. Also, according to [P1] we say that T ∈ B(H) is n-quasihyponormal, if ||T n+1 h|| ≥ ||T ∗ T n h|| for all h ∈ H. The aim of the paper is to give a functional calculus, to derive some mapping theorems for A-contractions and to continue the study from [S1-4] for some particular classes of A-contractions. Also, some applications will be obtained related to ρ-contractions [CF], [CS], [CZ], [NF].

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2. Mapping theorems for A-contractions An operator T ∈ B(H) is polynomially bounded, (T ∈ P B(H)) if there exists a constant M ≥ 1 such that (2.1)

||p(T )|| ≤ M sup |p(z)| |z|≤1

for all complex polynomial p. Recall that a polynomially bounded operator T is said to be absolutely continuous (T ∈ P Babs (H)) if and only if all its scalar spectral measures are absolutely continuous with respect to the Haar measure on T. If T is polynomially bounded and absolutely continuous [NF] the Nagy-Foias-Mlak functional calculus for T can be constructed, i.e. there exists a uniquely determined unital continuous homomorphism f 7→ f (T ) from H ∞ with the weak-∗ topology into B(H) with the weak topology. As usually, H ∞ denote the Hardy algebra of bounded, analytic functions, dened on the open unit disc D, endowed with ||f ||∞ = sup|z| n. In particular, a quasicontraction (quasi-isometry) is a n-quasicontraction (n-quasi-isometry) for any n ≥ 1.

Remark 3.2. Any n-quasicontraction T is a power bounded operator

because one has

sup ||T m || ≤ max ||T j ||. m≥0

1≤j≤n

Hence any n-quasicontraction T has its spectrum in the closed unit disc, but ||T || ≥ 1 in general. In fact, if T n 6= 0 and T is a n-quasi-isometry then ||T || ≥ 1. For example, any idempotent operator T which is not an orthogonal projection has ||T || > 1.

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From now on, for an operator T ∈ B(H) we consider the matrix representation µ ¶ C S (3.1) T = 0 0 with respect to the decomposition H = R(T ) ⊕ N (T ∗ ), where C = T |R(T ) and S ∈ B(N (T ∗ ), R(T )). We characterize this matrix representation in the case of n-quasicontractions.

Theorem 3.3. An operator T ∈ B(H) is a n-quasicontraction with

n ≥ 1 if and only if in the matrix (3.1) C is a (n − 1)-quasicontraction. Moreover, T is a n-quasi-isometry if and only if C is a (n − 1)-quasiisometry.

Proof. Suppose that T is a n-quasicontraction with n ≥ 1, that is T is a contraction on R(T n ). Since R(T ) is an invariant subspace for T , T has a matrix representation of the form (3.1) on H = R(T ) ⊕ N (T ∗ ), and C = T |R(T ) is a contraction on R(T n ). As R(T n ) = R(C n−1 ) it follows that C is a contraction on R(C n−1 ), hence C is a (n − 1)quasicontraction on R(T ). In the case when T is n-quasi-isometry, the operator C is an isometry on R(C n−1 ) and therefore C is a (n − 1)-quasi-isometry on R(T ). Conversely, let T be of the form (3.1) with C a (n−1)-quasicontraction on R(T ). For j ≥ 1 one has µ ¶ C ∗j C j C ∗j C j−1 S ∗j j T T = . S ∗ C ∗(j−1) C j S ∗ C ∗(j−1) C j−1 S Since C0 = C|R(C n−1 ) is a contraction, for h ∈ R(T ) and k ∈ N (T ∗ ) we get h(T ∗n T n − T ∗(n+1) T n+1 )(h ⊕ k), h ⊕ ki =

hC ∗n (I − C ∗ C)C n h + C ∗n (I − C ∗ C)C n−1 Sk, hi+ hS ∗ C ∗(n−1) (I − C ∗ C)C n h + S ∗ C ∗(n−1) (I − C ∗ C)C n−1 Sk, ki = ||DC0 C n h||2 + 2RehDC0 C n h, DC0 C n−1 Ski + ||DC0 C n−1 Sk||2 ≥ ||DC0 C n h||2 − 2||DC0 C n h|| ||DC0 C n−1 Sk|| + ||DC0 C n−1 Sk||2 ≥ 0, where DC0 is the defect operator of C0 . This yields that

T ∗(n+1) T n+1 ≤ T ∗n T n , that is T is a n-quasicontraction on H.


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In the case when C is a (n − 1)-quasi-isometry on R(T ) the operator C0 will be an isometry, so DC0 = 0 and it follows that T is a n-quasiisometry on H. This ends the proof. 2

Corollary 3.4. An operator T ∈ B(H) is a quasicontraction if and

only if C in (3.1) is a contraction. In particular, T is a quasi-isometry if and only if C is an isometry.

Remark 3.5. A natural problem arises regarding the powers of a n-

quasicontraction T . Clearly, T j is a quasicontraction for any j ≥ n, and T j is a (n−j+1)-quasicontraction if 2 ≤ j ≤ n−1 because (n−j+1)j > n in this case. In particular, T n−1 is a 2-quasicontraction, T n−2 is a 3-quasicontraction (if n > 3), ... , T 2 is a (n − 1)-quasicontraction. Furthermore, if n ≥ 2 is the smallest integer such that T is a nquasicontraction, then the smallest integer q ≥ 1 such that T j is a q -quasicontraction for 2 ≤ j ≤ n − 1 is q = [ nj ] + 1, where [ nj ] denote the integer part of nj . This one occurs because it needs to be qj ≥ n. Similar conclusions can be formulated for the powers of a n-quasi-isometry. Also, it is easy to see that if T is a n-quasicontraction with n ≥ 2, then T n is quasi-isometry if and only if T is n-quasi-isometry. Using the matrix (3.1) one can see when ||T || ≤ 1.

Proposition 3.6. Let T be a n-quasicontraction, then ||T || ≤ 1 if and

only if CC ∗ + SS ∗ ≤ I on R(T ). Also, T is a partial isometry if and only if CC ∗ + SS ∗ = I . Proof. From (3.1) we get ∗

TT =

µ

¶ CC ∗ + SS ∗ 0 0 0

and so ||T ||2 = ||CC ∗ + SS ∗ || = ||(CC ∗ + SS ∗ )1/2 ||2 . Hence ||T || ≤ 1 if and only if ||(CC ∗ + SS ∗ )1/2 || ≤ 1, or equivalently, CC ∗ + SS ∗ ≤ I . Since T partial isometry means T T ∗ an orthogonal projection, we infer that this happens if and only if CC ∗ + SS ∗ is idempotent, or equivalent CC ∗ + SS ∗ = I . 2

Corollary 3.7. Let T be a quasicontraction with T |R(T ) a coisometry. Then ||T || ≤ 1 if and only if S = 0 in the matrix (3.1) of T .

In particular, from this corollary it follows the well-known fact that an idempotent T is an orthogonal projection if and only if ||T || = 1.

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Corollary 3.8. A quasi-isometry T is a partial isometry if and only

if S is a partial isometry and C ∗ S = 0, where C and S are as in the matrix ( 3.1) of T . Furthermore, S = 0 if and only if C is unitary. Proof. Let T be a quasi-isometry. Then C = T |R(T ) is an isometry, therefore CC ∗ is an orthogonal projection, and by Proposition 3.6 we have that SS ∗ = I − CC ∗ is also an orthogonal projection. Hence S is a partial isometry. In addition, since CC ∗ + SS ∗ = I we have CC ∗ C +SS ∗ C = C , or equivalently SS ∗ C = 0, which implies C ∗ S = 0. Conversely, we assume that S is a partial isometry and C ∗ S = 0. Then clearly one has (CC ∗ + SS ∗ )C = C and (CC ∗ + SS ∗ )S = S , and since T (h0 , h1 ) = Ch0 + Sh1 for h0 ∈ R(T ) and h1 ∈ N (T ∗ ) it follows that (CC ∗ + SS ∗ )T h = T h for any h ∈ H. Consequently CC ∗ + SS ∗ = I on R(T ), which gives by Proposition 3.6 that T is a partial isometry. 2 Recall that M. Patel proved in [P2] that a quasi-isometry T is a contraction if and only if S is a contraction and C ∗ S = 0. Thus, the previous corollary completes this result. In the sequel we see when ||T n || = 1 for a n-quasi-isometry, by using the matrix representation (3.1).

Proposition 3.9. Let T be a n-quasi-isometry with n ≥ 1 such that T n 6= 0. The following statements are equivalent: (i) ||T n || = 1; (ii) T n is hyponormal; (iii) C n C ∗n + C n−1 SS ∗ C ∗(n−1) ≤ I. Moreover, if these conditions are satised then ||C n || = ||C n−1 || = 1, S ∗ C n = 0, C j is hyponormal for j ≥ n − 1, and T is a (n − 1)quasicontraction. Proof. Since T is a n-quasi-isometry with the matrix representation (3.1), we have µ n ¶ µ n ∗n ¶ C C n−1 S C C + C n−1 SS ∗ C ∗(n−1) 0 n n ∗n T = , T T = . 0 0 0 0 Hence ||T n || = 1 if and only if C n C ∗n + C n−1 SS ∗ C ∗(n−1) ≤ I , which gives (i) ⇐⇒ (iii). The equivalence (i) ⇐⇒ (ii) follows from [P1,P2] because T n is a quasi-isometry. Suppose now the condition (iii) satised. This implies ||C n || ≤ 1, and since T n 6= 0, or equivalently T n+1 6= 0 (T being a n-quasiisometry), one has C n 6= 0. But C n is a quasi-isometry because C is a n-quasi-isometry, hence ||C n || = 1. Also, we infer from (iii) that

C ∗(2n−1) C n C ∗n C 2n−1 +C ∗(2n−1) C n−1 SS ∗ C ∗(n−1) C 2n−1 ≤ C ∗(2n−1) C 2n−1 .


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Since C is an isometry on R(C n−1 ), this inequality is equivalent to

C ∗(n−1) C n−1 + C ∗n SS ∗ C n ≤ C ∗(n−1) C n−1 , which gives S ∗ C n = 0. Next, if ||T n || = 1 one has for h ∈ H,

||T n h|| = ||T 2n−1 h|| ≤ ||T n || ||T n−1 h|| = ||T n−1 h||. Thus, T is a (n − 1)-quasicontraction, and having in view the matrix of T n−1 on H = R(T ) ⊕ N (T ∗ ) it follows from Theorem 3.3 that ||C n−1 || ≤ 1. Also, C n−1 is a quasi-isometry (C being a (n − 1)-quasiisometry) and C n−1 6= 0 because T n 6= 0. Hence ||C n−1 || = 1 and C n−1 is hyponormal by Theorem 2.2 [P1]. Analougously, since C is a m-quasi-isometry for any m ≥ n − 1 and ||C m || = ||C m−1 || = 1, it follows that C m is hyponormal. This ends the proof. 2 Next we see when ||T || = 1 for a n-quasi-isometry. First we give the following remark.

Remark 3.10. Let T be a n-quasi-isometry with T n 6= 0, so that

V = T |R(T n ) is an isometry. Then, since R(T n ) is an invariant subspace for T , it follows that T has a matrix representation with respect to the decomposition H = R(T n ) ⊕ N (T ∗n ) of the form µ ¶ V R (3.2) T = , 0 Q

where R ∈ B(N (T ∗n ), R(T n )) and Q ∈ B(N (T ∗n )). Moreover, Qn = 0 because T n is a quasi-isometry and so T n need to have a matrix of the form (3.1) with respect to the above decomposition of H. Clearly, in the case n = 1, that is T is a quasi-isometry, one has Q = 0 in (3.2). Also, the converse assertion is true, namely any operator T having a matrix of the form (3.2) with V an isometry and Qn = 0 is a n-quasi-isometry. We use now the representation (3.2) in the following result.

Theorem 3.11. Let T be a n-quasi-isometry on H with T n 6= 0, T

having the matrix representation (3.2). The following statements are equivalent : (i) ||T || = 1; (ii) T is n-quasihyponormal; (iii) R∗ R + Q∗ Q ≤ I and V ∗ R = 0. Moreover, if ||T || = 1 then the unitary part of T in H is R = T j j≥1 R(T ), and the completely nonunitary part of T is ∗-strongly stable.

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Proof. (i) ⇐⇒ (iii). Using the matrix (3.2) of T we get µ ¶ µ ¶ V V ∗ + RR∗ RQ∗ I V ∗R ∗ ∗ TT = , T T = . QR∗ QQ∗ R ∗ V R ∗ R + Q∗ Q Thus, if ||T || = 1 then T ∗ T ≤ I and T T ∗ ≤ I , which provide that R∗ R + Q∗ Q ≤ I and, respectively V V ∗ + RR∗ ≤ I . Also, the last inequality implies V ∗ (V V ∗ +RR∗ )V ≤ V ∗ V , or equivalently I +V ∗ RR∗ V ≤ I (V being an isometry on R(T )), that is V ∗ R = 0. Conversely, assuming that R∗ R + Q∗ Q ≤ I and V ∗ R = 0, we infer from the matrix of T ∗ T in this case that T ∗ T ≤ I , and so ||T || ≤ 1. Since T n 6= 0 and T is a n-quasi-isometry it follows that ||T || = 1. (i) ⇐⇒ (ii). Suppose that ||T || = 1. Then for h ∈ H we have

||T n h − T ∗ T n+1 h||2 = ||T n h||2 + ||T ∗ T n+1 h||2 − 2RehT n h, T ∗ T n+1 hi ≤ 2||T n h||2 − 2||T n+1 h|| = 0, hence we obtain T n = T ∗ T n+1 . Next, for h ∈ H we get

||T ∗ T n h||2 = ||T ∗2 T n+1 h||2 ≤ ||T ||2 ||T n+1 h||2 = ||T n+1 h||2 . Thus we have ||T ∗ T n h|| ≤ ||T n+1 h|| for every h ∈ H, which means that T is n-quasihyponormal. Conversely, if T is n-quasihyponormal then T is normaloid [DK2], and since the spectral radius of T is equal to 1 (by Remark 3.1, because T n 6= 0) it follows that ||T || = 1. Assume ||T || = 1. As T n = T ∗ T n+1 we deduce that T ∗ R(T j+1 ) ⊂ T R(T j ) for any j ≥ n. Thus, if Rn = j≥n R(T j ) we infer

T ∗ Rn ⊂ T ∗ R(T j+1 ) ⊂ R(T j ) for j ≥ n, whence T ∗ Rn ⊂ Rn that is Rn is invariant for T ∗ . Also, since R(T j ) is invariant for T , j ≥ n, it follows that Rn is invariant for T , and so Rn is reducing for T . But we infer from the relation T n = T ∗ T n+1 that

T T ∗ T n+1 = T n+1 = T ∗ T T n+1 which yields that T T ∗ |Rn = T ∗ T |Rn , that is T is unitary on Rn . Hence Rn ⊂ Hu , Hu being the unitary part of T in H. On the other hand, we have Hu = T j Hu ⊂ R(T j ) for j ≥ 1, and so we deduce that T Hu ⊂ j≥1 R(T j ) = R and nally that

R ⊂ Rn ⊂ Hu ⊂ R, that is Hu = R. The fact that the completely nonunitary part of T is ∗-strongly stable (that is T ∗ is strongly stable on H ª Hu ) follows from Corollary 1 in [DK1]. 2


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Corollary 3.12. If T is a n-quasi-isometry with ||T || = 1 then N (T ) ⊂ N (T ∗n ).

Proof. One uses the relation T ∗n = T ∗(n+1) T .

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4. Similarity to contractions We remarked that n-quasicontractions are power bounded, but it is possible to show the following

Theorem 4.1. A n-quasicontraction is similar to a contraction, for any n ≥ 1.

Proof. First we prove that a quasicontraction T is similar to a contraction (the case n = 1). Using Paulsen's criterion [Pal] it is sucient to prove that T is a completely polynomially bounded operator. Thus, for every matrix of polynomials (pij ) we have for T of the form (3.1) µ ¶ pij (C) pf ij (C)S , 0 pij (0)I having in view the powers of T m for m ≥ 1, where for a polynomial p we denote by pe the polynomial

p(z) − p(0) . z Lm But (pij (T )) ∈ B(Hm ) for some P m ≥ 1, where Hm = 1 H. Let m 2 2 0 1 h = ⊕m h ∈ H with ||h|| = ||h || ≤ 1 . If h = h m i i i=1 i i ⊕ hi ∈ i=1 0 R(T ) ⊕ N (T ∗ ) for 1 ≤ i ≤ m then h = h0 ⊕ h1 where h0 = ⊕m i=1 hi , 1 h1 = ⊕m i=1 hi . Thus, from the above matrix of pij (T ) we obtain pe(z) =

||(pij (T ))h||2 =

m m X X || pij (T )hi ||2 = i=1

j=1

m m m X X X 0 1 2 [|| (pij (C)hi + pf pij (0)h1i ||2 ] ≤ ij (C)Shi )|| + || i=1

j=1

j=1

m m m m m X X X X X 1 2 0 2 || pij (0)h1i ||2 = pf 2 (|| pij (C)hi || + || ij (C)Shi || ) + i=1

j=1

j=1

i=1

j=1

2(||(pij (C))h0 ||2 + ||f pij (C)Sh1 ||2 ) + ||(pij (0))h1 ||2 ≤ 2(||(pij (C))||2 + ||f pij (C)||2 ||S||2 ) + ||(pij (0))||2 ≤ 3||(pij )||2 + 2||S||2 ||(f pij )||2 .

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Here we used that ||hi || ≤ 1 for i = 0, 1 and the fact that C being a contraction it satises the stronger version of von Neumann inequality relative to the matrix (pij ). But we also have

||(f pij )|| = sup ||(f pij (λ))|| = sup ||[pij (λ)] − [pij (0)]|| |λ|=1

|λ|=1

≤ 2||(pij )||. Then from the above inequality we infer that p ||(pij (T ))|| ≤ 3 + 8||S||2 ||(pij )||, which yields that T is a completely polynomially bounded operator. Consequently, applying Paulsen's criterion [Pal] we see that any quasicontraction T is similar to a contraction. Now we suppose that any (n − 1)-quasicontraction (n ≥ 2) is similar to a contraction. Let T be a n-quasicontraction of the form (3.1). Then the operator C being a (n − 1)-quasicontraction, C will be similar to a contraction, hence there exists a contraction B on R(T ) and a positive invertible operator A in B(R(T )) such that AC = BA. If we dene µ ¶ µ ¶ A 0 B AS e= A , Te = 0 I 0 0

e with respect to the decomposition H = R(T ) ⊕ N (T ∗ ), then clearly A is a positive invertible operator in B(H) and Te is a quasicontraction e = TeA e. Thus T is similar to Te and hence T is on H, and also AT similar to a contraction, using the above remark (from the case n = 1). We conclude (by induction) that any n-quasicontraction is similar to a contraction, and this ends the proof. 2 Observe that a n-quasicontraction is a A-contraction for some positive invertible A (by using Theorem (4.1) and a remark from the introduction). Moreover, the argument from the previous proof (in the case n = 1) can be used to prove that an operator T ∈ B(H) is similar to a quasicontraction if T |R(T ) is similar to a contraction (as an operator in

B(R(T ))). But by Paulsen's criterion one can see that TR(T ) is similar to a contraction if T is such. Thus one infers from previous theorem the following

Corollary 4.2. An operator T is similar to a contraction if and only

if T |R(T ) is similar to a contraction.

Also, we deduce from this corollary the following more general result in the context of A-contractions.


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Proposition 4.3. If T is an A-contraction on H with R(A) a closed

subspace such that N (A) = N (T ) then T is similar to a contraction. Proof. Clearly, T has a matrix of the form µ ¶ T0 0 T = , T1 0

with respect to the decomposition H = R(T ∗ )⊕N (T ) = R(A)⊕N (A). 1/2 1/2 If A0 = A|R(A) then A0 is invertible in B(R(A)) and A0 T0 = TbA0 where Tb is the Douglas contraction associated to the A-contraction T . 1/2 Since A0 is invertible, it follows that T0 is similar with Tb and so T0∗ is similar to the contraction Tb∗ . But T0∗ = T ∗ |R(T ∗ ) , and by Corollary 4.2 one obtains that T ∗ , and consequently T , is similar to a contraction. 2 We remark that any quasicontraction T satises the condition N (T ) = N (A) where A = T ∗ T , but R(A) is not closed, in general. In the regular case we can obtain the following

Corollary 4.4. If T is a regular A-contraction on H with N (A) ⊂ N (T n ) for some n ≥ 1 then T ∗ is a n-quasicontraction.

Proof. The regularity condition AT = A1/2 T A1/2 implies A1/2 TbA1/2 h = A1/2 T A1/2 h for h ∈ R(A). Using the matrix representation of T with respect to the decomposition H = R(A) ⊕ N (A) and putting 1/2 1/2 1/2 1/2 T0∗ = T ∗ |R(A) we infer that A0 TbA0 h = A0 T0 A0 h for h ∈ R(A), 1/2 1/2 and since A1/2 is injective on R(A) it follows that TbA0 = T0 A0 . This gives T0 = Tb that is T0 is a contraction. As T ∗ = T ∗ | and 0

R(A)

R(T ∗n ) ⊂ R(A), T ∗ is also a contraction on R(T ∗n ), hence T ∗ is a n-quasicontraction. 2 In some cases this corollary shows that T is also a n-quasicontraction. For instance, any non-injective regular A-contraction T with A injective is necessarily a contraction. In fact, in this case the condition AT = A1/2 T A1/2 implies T A1/2 h = A1/2 T h = TbA1/2 h for h ∈ H, and so T = Tb on H = R(A), that is T is a contraction with N (T ) 6= {0} = N (A).

Corollary 4.5. If T is a regular A-isometry on H with N (A) = N (T n )

for some n ≥ 1 then T ∗ is a n-quasicontraction with T ∗ |R(T ∗n ) a coisometry. Proof. By Corollary 4.4, T ∗ is a n-quasicontraction. In addition, since T ∗ AT = A it follows that T0 = Tb (from the previous proof) is an isometry on R(A) = R(T ∗n ), hence T ∗ |R(T ∗n ) = T0∗ is a coisometry. 2

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Corollary 4.6. Let T be a n-quasicontraction on H such that T |T n | =

|T n |T . Then T ∗ is also a n-quasicontraction, and when T is a n-quasiisometry, T ∗ is a coisometry on R(T ∗n ). Proof. The condition T |T n | = |T n |T means that T is a regular T T n -contraction, and as N (T n ) = N (T ∗n T n ) we can apply the last two corollaries to obtain the desired facts. 2 ∗n

Remark 4.7. If T is a n-quasicontraction satisfying T |T n | = |T n |T

then clearly T n is quasinormal, and when T is a n-quasi-isometry one has T ∗n T n = T ∗2n T 2n = (T ∗n T n )2 , hence T n is a partial isometry.

Remark 4.8. If T is as in Corollary 4.6 then T ∗ as a T n T ∗n -contraction

is not regular, in general. Although, since Tb = T |R(T n ) is a contraction, T ∗ is a lifting of the contraction Tb∗ which assures that T ∗ is a P -

contraction, P being the orthogonal projection on R(T n ) = R(T n T ∗n ). Also, T as a regular T n T ∗n -contraction is a P∗ -contraction, where P∗ is the orthogonal projection on R(T ∗n ) = R(T ∗n T n ). Finally, from Theorem 4.1 we can derive even a more general result. We denote by σa (T ) the approximate point spectrum of T .

Theorem 4.9. Let T ∈ B(H) such that σ(T ) ⊆ D. One has

(i) If T is a b(T )∗ b(T )-contraction for some b ∈ B0 then T is similar to a contraction. (ii) If T ∈ P Babs (H) with σa (T ) ∩ D 6= D and u is an inner function such that T is a u(T )∗ u(T )-isometry, with u(T ) 6= 0, then T is similar to a quasi-isometry. Proof. (i) Considering T a b(T )∗ b(T )-contraction, of course with b(T ) 6= 0, we obtain by Theorem 2.1 that b(T ) is a quasicontraction, and Theorem 4.1 gives that b(T ) is similar to a contraction. Now using a theorem of Mascioni [M] (which asserts that if σ(T ) ⊆ D and b(T ) is a contraction, for a nite Blaschke product, then T is similar to a contraction), we infer that T is similar to a contraction. (ii) Now if T is u(T )∗ u(T )-isometry we obtain by Theorem 2.3 that u(T ) is a quasi-isometry. With respect to the orthogonal decomposition H = R(T ) ⊕ N (T ∗ ) one has µ ¶ µ ¶ T1 R u(T1 ) R(u) T = , u(T ) = 0 0 0 0 where we can easy see that u(T1 ) is a quasi-isometry, being the restriction of u(T ) to an invariant subspace. Since u(σa (T ) ∩ D) = σa (u(T )) ∩ D, u(D) = D and using the hypothesis we deduce that


19

there exists an open disc D(α, r) ⊆ D − (σa (u(T )) ∩ D). We get that β−u(z) there is β ∈ D(α, r), β 6= 0 such that uβ (z) = 1−βu(z) is a inner function. Thus uβ (T1 ) is an injective quasi-isometry with closed range, and by [P2] this means that uβ (T1 ) is similar with an isometry. Finally by Corollary 3.13 [C1] we obtain that T1 is similar to an isometry V on R(T ) by an invertible operator A1 ∈ B(R(T )). Taking µ ¶ µ ¶ A1 0 V A1 R A= , W = 0 I 0 0 it is easy to see that AT = W A. As A is invertible it follows that T is similar to a quasi-isometry. 2 5. Applications for some classes of operators We will see that other characterizations of quasicontractions can be expressed in terms of Duggal and Aluthge transforms. Recall that if T = U |T | is the polar decomposition of T ∈ B(H) where U is the unique partial isometry with N (U ) = N (T ), then the operator Te = |T |U is called the Duggal transform of T (see [FJKP]). Clearly, ||Te|| ≤ ||T || in general. One has the following

Proposition 5.1. An operator T ∈ B(H) is a quasicontraction if and only if Te is a contraction.

Proof. Let T be a quasicontraction, that is T ∗2 T 2 ≤ T ∗ T . Then the Douglas contraction Tb associated to T on R(T ∗ T ) = R(T ∗ ) satises Tb|T |h = |T |T h, or equivalently Te|T |h = Tb|T |h, for h ∈ R(T ∗ ). Hence Te|R(T ∗ ) = Tb, and since Te = 0 on N (T ) = N (U ) it follows that Te is a contraction on H. The converse also holds because if Te is a contraction then T ∗2 T 2 = |T |Te∗ Te|T | ≤ |T |2 = T ∗ T, that is T is a quasicontraction.

2

Proposition 5.2. An operator T ∈ B(H) is a quasi-isometry if and

only if Te is a partial isometry with N (Te) = N (T ).

Proof. Clearly, N (T ) ⊂ N (Te), so that H ª N (Te) ⊂ H ª N (T ) = R(T ∗ ). Thus, if T is a quasi-isometry, that is Te|R(T ∗ ) = Tb is an isometry on R(T ∗ ) (Tb being as in the previous proof), then Te is also an isometry on H ª N (Te) which means that Te is a partial isometry. In this case one has N (Te) = N (T ) as in the proof of Theorem 2.2 from [P1].

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Conversely, if Te is a partial isometry with N (Te) = N (T ) then R(Te∗ ) = R(T ∗ ) and Tb = Te|R(T ∗ ) is an isometry, hence T is a quasiisometry. 2 In fact, the Proposition 5.2 is just Theorem 2.1 [P1] reformulated. We obtained this result by using that Te|R(T ∗ ) = Tb. The Proposition 5.1 gives the largest class of operators for which the Duggal transform is a contraction. We can now nd the largest class of operators for which the Duggal transform is a ρ-contraction (ρ > 0) in the sense of Sz. Nagy-Foias [NF] (see also [CF], [CS], [CZ]). Recall that in [FJKP] was proved that if T is a ρ-contraction then Te is also a ρ-contraction. Thus, our class of operators needs to contain the class of ρ-contractions. In the sequel we say that T ∈ B(H) is a quasi ρ-contraction if T |R(T ) is a ρ-contraction (T ∈ Cρ (H)). Obviously ρ-contractions areµquasi ρ¶U S contractions. It was proved in [CZ] that operators of the form 0 0 with U a unitary operator which acts on a nite dimensional space, is a ρ-contraction for some ρ ≥ 1 if and only if S = 0. But such an operator is always a quasi ρ-contraction for any ρ ≥ 1. Thus there exist quasi ρ-contractions which are not ρ-contractions. We have the following

Theorem 5.3. An operator T ∈ B(H) is a quasi ρ-contraction if and only if Te is a ρ-contraction on H.

Proof. Let T = U |T | be the polar decomposition of T . Since R(T ) = R(U ), with respect to the decomposition H = R(T ) ⊕ N (T ∗ ), T and U have the form µ ¶ µ ¶ T1 R U1 U2 (5.1) , U= . 0 0 0 0 Suppose rstly that T is a quasi ρ-contraction, that is T1 is a ρcontraction. As Te = |T |U , it follows immediately that T U = U Te and also U ∗ T U = Te, because U is an isometry on R(T ∗ ) which contains R(Te). Therefore we have ¶ ¶ µ ¶µ µ ∗ ¶µ U1 T1 U1 U1∗ T1 U2 T1 R U1 U2 U1 0 e = T = U2∗ T1 U1 U2∗ T1 U2 0 0 0 0 U2∗ 0 ¶ ¶µ µ ∗ ¶µ T1 0 U1 U2 U1 0 = U ∗ T 0 U, = 0 0 0 0 U2∗ 0


21

where T 0 is a ρ-contraction on H, R(T ) being a reducing subspace for T 0 . This is equivalent (see [C3]) to ρ(Te) = ρ(U ∗ T 0 U ) ≤ ρ(T 0 ) where ρ(T ) = inf{ρ ∈ R∗+ : T ∈ Cρ } and ρ(T 0 ) is similarly dened. We have ρ(T 0 ) ≤ ρ, T 0 being a ρ-contraction, and since ρ(Te) ≤ ρ(T 0 ) it follows that Te is a ρ-contraction. Conversely, we suppose that Te is a ρ-contraction on H. As N (T ) = N (Te) is invariant for Te and R(T ∗ ) is also invariant for Te because Te = |T |U , we infer that R(T ∗ ) reduces Te and Te0 = Te|R(T ∗ ) is a ρcontraction. Now if U0 = U |R(T ∗ ) then U0 is unitary from R(T ∗ ) onto R(T ) and we have U0 Te0 = T1 U0 . Hence T1 is unitarily equivalent to Te0 , and consequently T1 is a ρ-contraction. Finally, T is a quasi ρcontraction, and the proof is nished. 2 Next using [CS] and [CZ] we can give two applications of quasi ρcontractions

Proposition 5.4. Let f ∈ A(D) be a non constant function such that

||f || ≤ 1. If T is a quasi ρ-contraction for ρ > 0, then f (T ) is a quasi ρ0 -contraction, where (

(5.2)

0

ρ =

(0)| 1 + (ρ − 1) 1−|f if ρ ≤ 1; 1+|f (0)| 1+|f (0)| 1 + (ρ − 1) 1−|f (0)| if ρ ≥ 1.

Proof. By Corollary 4.2 a quasi ρ-contraction is similar to a contraction, thus f (T ) is well dened for f ∈ A(D). The conclusion follows directly from [CS] using the matrix representation of T and f (T ) with respect to the orthogonal decomposition, H = R(T ) ⊕ N (T ∗ ), namely µ ¶ µ ¶ T1 R f (T1 ) R(f ) T = , f (T ) = 0 0 0 0 where T1 is a ρ-contraction. Hence the corresponding result of [CS], f (T1 ) is a ρ0 -contraction with ρ0 given by (5.2). Thus f (T ) is a quasi ρ0 -contraction, and this ends the proof. 2 Notice that if f ∈ A(D) with f (0) = 0 then ρ = ρ0 in the previous proposition. Recall that the numerical radius of T ∈ B(H) is dened by

w(T ) = sup |hT h, hi|. |h|=1

Also, w(T ) ≤ 1 if and only if T is a 2-contraction ([B]). Remember that every ρ-contraction is similar to a contraction [NF, p. 92].

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Now, as a consequence from [CZ] we can give a necessary and sucient condition for a quasi 2-contractions, to have the numerical radius less or equal to 1, which is related to Proposition 3.6.

Proposition 5.5. Let T be a quasi 2-contraction. Then T has the

numerical radius less or equal to 1, if and only if T1 is a 2-contraction and |α|2 RR∗ ≤ 4(I − Re(αT1 )) (α ∈ D) where T1 , R being as in (5.1). Proof. It is obtained from Theorem 1 [CZ] as an immediate consequence. 2 Next, we recall that if T = U |T | is as above, then the operator ∆α (T ) = |T |α U |T |1−α , 0 < α ≤ 1 is called the α-Aluthge transform of T (see [A], [O]). In general, ||∆α (T )|| ≤ ||T || and we can see now when ∆α (T ) is a contraction. We remark that for α = 1 we have ∆1 (T ) = Te, the Duggal transform of T .

Proposition 5.6. For T ∈ B(H) we have that ∆α (T ) is a contraction if and only if T is a (T ∗ T )α -contraction, 0 < α ≤ 1.

Proof. The cases α = 1 was treated before, so we consider 0 < α < 1. Let T be a |T |2α -contraction, that is T ∗ |T |2α T ≤ |T |2α . Then there exists a contraction Tb on R(|T |2α ) = R(T ∗ ) such that Tb|T |α = |T |α T , or equivalently ∆α (T )|T |α = Tb|T |α . Therefore ∆α (T )|R(T ∗ ) = Tb, hence ∆α (T ) is a contraction because ∆α (T ) = 0 on N (T ). Conversely, if ∆α (T ) is a contraction then we have

T ∗ |T |2α T = |T |U ∗ |T |2α U |T | = |T |α ∆α (T )∗ ∆α (T )|T |α ≤ |T |2α , and consequently, T is a (T ∗ T )α -contraction.

2

We have the following properties of the operators described above

Proposition 5.7. If T is a (T ∗ T )α -contraction with 0 < α < 1 then T is a power bounded operator.

Proof. One has T ∗ |T |2α T ≤ |T |2α . Hence |||T |α T n h|| ≤ |||T |α h|| for n ∈ N and h ∈ H. But we have

|||T |T n h|| ≤ |||T |1−α || |||T |α T n h|| ≤ |||T |1−α || |||T |α || ||h||, which means

||T n+1 h|| ≤ |||T |1−α || |||T |α || ||h|| (h ∈ H).


23

Therefore ||T n || ≤ |||T |1−α || |||T |α ||, n ≥ 1, that is T is a power bounded. 2 In some cases one can see more than in the previous proposition. We give rstly the following

Proposition 5.8. Let T ∈ B(H) an operator injective with dense

range. Then T and ∆α (T ) are quasi-similar (0 < α ≤ 1).

Proof. By the denition of α-Aluthge transforms one obtains

∆α (T )|T |α = |T |α U |T |1−α |T |α = |T |α U |T | = |T |α T with |T |α injective, hence a quasi-anity. Also, we have

U |T |1−α ∆α (T ) = T U |T |1−α . As T is with dense range, it follows that R(U ) = R(T ) = H = R(T ∗ ) = R(U ∗ ), therefore U is unitary and hence U |T |1−α is a quasi-anity. In conclusion T and ∆α (T ) are quasi-similar operators. 2 We remark that in this case T has a non trivial hyperinvariant subspace if and only if ∆α (T ) have a non trivial hyperinvariant subspace. Recall that a subspace of H is hyperinvariant for an operator T if it is invariant for all operators that commute with T . In the injective case we infer the following result.

Corollary 5.9. Let T be a (T ∗ T )α -contraction injective. Then T |R(T ) is quasi-similar to a contraction.

Proof. It follows immediately from the previous proposition observing that in this case U |T |1−α is a quasi-anity from H to R(T ) and by Proposition 5.6, ∆α (T ) is a contraction. 2 Concerning the closed range case we obtain the following result.

Corollary 5.10. Let T be a (T ∗ T )α -contraction with closed range. Then T is similar to a contraction.

Proof. Clearly, U |T |1−α R(T ∗ ) ⊂ R(T ), and it is easy to see that U |T |1−α is an invertible operator from R(T ∗ ) to R(T ). From the proof of Proposition 5.8 we have that ∆α (T )|R(T ∗ ) is similar to T |R(T ) by the operator U |T |1−α |R(T ∗ ) , and the Proposition 5.6 gives that ∆α(T ) is a contraction. Hence T |R(T ) is similar to a contraction, and using now the Corollary 4.2 one obtains that T is similar to a contraction. 2 Acknowledgements. The authors wish to thank the referee for the short delay and also for his remarks which have contributed to improve the presentation of the original version.

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