On dilation and commuting liftings of n -tuples of commuting Hilbert space contractions

. The n -tuples of commuting Hilbert space contractions are considered. We give a model of a commuting lifting of one contraction and investigate conditions under which a commuting lifting theorem holds for an n -tuple. A series of such liftings leads to an isometric dilation of the n -tuple. The method is tested on some class of triples motivated by Parrotts example. It provides also a new proof of the fact that a positive deﬁnite n -tuple has an isometric dilation.


Introduction
By the dilation theory, started by Szökefalvi-Nagy, every contraction has a unitary dilation. The dilation provides a simpler proof of the von-Neumann inequality. It is a bit striking that the result does not extend to n-tuples of (commuting) operators. Precisely, a single contraction on a Hilbert space has the minimal, unique up to isomorphism, unitary dilation (Nagy [15,16]), a pair of commuting contractions has a minimal unitary dilation but it is not necessarily unique (Andô [1]), while for n ≥ 3 a unitary dilation may not exist (Parrott [11], Varopoulos [17]). More precisely, the dilation does exist but it may fail to commute. Parrott gave the first example of a triple of commuting contractions not admitting a (commuting) unitary dilation. Varopoulos showed that for any n ≥ 3 the von Neumann inequality may not be satisfied and hence the dilation may not exist. On the other hand, an example of four contractions not admitting a unitary dilation and satisfying von-Neumann inequality can be found in [5]. [122]

Zbigniew Burdak and Wiesław Grygierzec
The dilation theory is well developed, to mention for example results on row contractions. We refer the interested reader for example to [12,Chap. 4,5]. On the other hand, the theory still remains of interest as indicated by the following list of recent results [2,3,4,6,7,9,13,14].
The paper is devoted to the problem of existence of an isometric (equivalently unitary) dilation of three or more contractions. We approach the problem via commuting liftings. We give an equivalent condition for a triple to satisfy commuting lifting theorem (i.e. there is a commuting triple consisting of an isometric dilation of one operator and liftings of the two remaining). The result may be used for n-tuples. The condition is not simple enough to be treated as a solution of the problem. However, we show some examples in which it works well. In particular, we apply it to some class of triples motivated by the Parrott's example. Moreover, we show that a positive definite n-tuple has a unitary dilation. Such a result is known, but our proof gives a direct construction of the dilation.

Preliminaries
If the projection in condition (iii) may be canceled (i.e. T α = U α | H ) the dilation is called an extension. A dilation is called isometric or unitary if operators in U are of the respective type. Since an n-tuple of isometries admits a unitary extension, an arbitrary n-tuple of commuting contractions admits a unitary dilation if and only if it admits an isometric dilation.
Note that a dilation is assumed to be a commuting n-tuple. If n-tuple fails to admit a unitary dilation it is due to the commutativity requirement. As we mentioned, a contraction T ∈ B(H) admits a unique isometric dilation where uniqueness is up to the unitary equivalence. For the model of such dilation recall that D T = √ I − T * T and D T = ran(D T ) are called the defect operator and the defect space of a contraction T , respectively. Let D ⊂ C be the unit disk with boundary T and let L 2 (T) and H 2 (T) denote the space of scalar valued, square integrable functions and the Hardy space, respectively. The space of square integrable functions valued in a separable Hilbert space H with the inner product induced by the norm f = ( T f (z) 2 H dm(z)) 1/2 (m -normalized Lebesgue measure) is unitarily equivalent to L 2 (T) ⊗ H and is denoted with this symbol. Similarly, H 2 (T) ⊗ H denotes the space of analytic, square integrable, H-valued functions. Set P n ∈ B(H 2 (T)) for the projection onto the subspace Cz n and P n ⊗ I for the projection onto z n ⊗ H for n ≥ 0, where z n stands for the monomial {z → z n } in this context. In particular, 1 ⊗ H denotes constant functions in be the Cartesian product Hilbert space with ·, · K T = ·, · H + ·, · H 2 (T)⊗D T and the respective norms. By the Nagy result the minimal isometric dilation of T may by defined as an operator on K T given by the matrix where E : there exists a bounded operator C S ∈ B(K, K ) satisfying the conditions: By (ii) and (iii) the operator C S is a lifting of S.
Parrott noticed in [11] that the lifting theorem is equivalent to the existence of a unitary dilation for a pair of contractions. Indeed, Theorem 2.1 for H = H and T = T ∈ B(H) provides a lifting of any operator in the commutant of T . By conditions (i) -(iv) of Theorem 2.1 the pair (V T , C S ) is a dilation of (T, S). It is

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Zbigniew Burdak and Wiesław Grygierzec not yet an isometric dilation, but V T is an isometry. However, if we lift V T with respect to the isometric dilation of C S , since a lifting of an isometry is an isometry, we get a pair of isometries (C V T , V C S ), so an isometric dilation of (T, S).
A similar construction could be used for an n-tuple of commuting contractions provided the respective contractions can be lifted to commuting operators. Indeed, for a given n-tuple of commuting contractions T = (T 1 , . . . , T n ) let k 0 ≤ n + 1 be the maximal integer, such that T i is an isometry for i < k 0 (k 0 = 1 if T 1 is not an isometry). Let C T = (C T1 , . . . , C Tn ) consist of an isometric dilation of T k0 and corresponding commuting liftings of the remaining contractions. Since a lifting of an isometry is an isometry, C Ti are isometries for i < k 0 + 1, so at least C T1 is an isometry. If k 1 is the maximal integer such that C Ti are isometries for i < k 1 , then k 0 + 1 ≤ k 1 . If we repeat the construction with C T we get C C T which is a lifting of C T , so also a lifting of T and C C T i are isometries for i < k 1 + 1, so at least C C T 1 , C C T 2 are isometries. Repeating the construction at most n-times we get an n-tuple of commuting isometries which is an isometric dilation of T. However, the lifting theorem does not extend to n-tuples of operators. For S, R ∈ B(H) commuting with T there are C S , C R by Theorem 2.1 but not necessarily commuting, which is required in the construction of an isometric dilation described above. Another idea is to construct a dilation of a pair and add a lifting of the third operator afterwards. However, if we replace T , T by n-tuples T, T the operator C S may not exist even if T, T admit regular dilations -see [10] for details.
Summing up, the problem of the existence of a dilation (isometric, unitary) reduces to the problem of the existence of commuting liftings. More precisely, we focus on conditions under which the lifting theorem holds for an n-tuple of contractions to be understood as the existence of commuting liftings of n − 1 contractions with respect to the dilation of the remaining one. A triple is representative for n-tuples.

A lifting of a contraction
The lifting theorem does not hold for an arbitrary n-tuple. However, for an n-tuple for which commuting liftings exist and the liftings may be lifted to commuting contractions and so on, there is an isometric dilation of the n-tuple as it was described in the last but one paragraph of the previous section. Lifting C S may be constructed of the same norm as S by Theorem 2.1. However, for the purposes of the construction of an isometric dilation it is enough if C S is a contraction. Hence, if needed, we may use a wider class of liftings than constructed in the proof of Theorem 2.1. Let us define: Let T, S ∈ B(H) be two commuting contractions and V T ∈ B(K) be the minimal isometric dilation of T . The operator C S satisfying conditions
is called a contractive lifting of S with respect to V T or simply a contractive lifting of S if the operator V T is clear.
The starting point for a construction of C S is a characterization of 2 by 2 block matrices in [8,Chap. 4]. Since H ⊂ K, we may assume that K = H ⊕ H 1 and Indeed, the matrix of C S is lower triangular by S * = C * S | H . Since C S is a contraction, by [8, Theorem 3.1, Chap. 4] and C S is lower triangular we get where  (2) commutes with V T , but if such Γ exists, then it is unique. Hence any contractive lifting C S is determined by the contraction Y ∈ B(D S , H 1 ) and may be denoted by C S,Y . In other words, there is a one-to-one correspondence between contractive liftings of S commuting with V T and some subfamily of contractions in B(D S , H 1 ). (1). Any lifting of S commuting with V T is of the form

Lemma 3.2 Let S, T ∈ B(H) be a pair of commuting contractions and let V T be an isometric dilation of T as in
To be clear, the lemma considers an existing lifting and investigates its form. Hence (4) describes the existing contraction S Y on a dense subset of its domain. By the continuity, S Y is determined by (4) on the whole domain.
Proof. By (2) only the formula for S Y needs an explanation. By the commutativity [126]

Zbigniew Burdak and Wiesław Grygierzec
By the equality of (2, 1) entries the operator S Y is defined on 1 ⊗ D T by However, by equality of (2, 2) entries S Y commutes with M z . Hence We have already emphasized that Lemma 3.2 does not show the existence of a contractive lifting, but only investigates its form. Indeed, if we take an arbitrary Y and try to use (4) as a definition of S Y two problems appear. The first one is that S Y acts on H 2 (T) ⊗ D T , so among others on vectors of the form z n ⊗ D T h for h ∈ H, while the right hand side of (4) acts directly on h ∈ H. Hence (4) properly defines a linear operator if the right hand side does not depend on the choice of h ∈ H representing a certain k ∈ D T H. In other words, the right hand side of (4) has to vanish on ker D T . Then S Y is a properly defined operator on z n ⊗ D T H and as the right hand side is a bounded, so a continuous operator, the definition extends to z n ⊗ D T . However, this does not imply that S Y defined by (4) is a bounded operator, while it should be a contraction. Indeed, the norm of a value of the right hand side operator in (4) Summing up, Lemma 3.2 does not provide a proof independent on Theorem 2.1 that there is Y properly defining a contractive lifting. However, since by Theorem 2.1 a lifting of S exists, by Lemma 3.2 it has to be of the form (3) and so there is at least one operator Y properly defining S Y and so C S,Y .

Remark 3.3 A contraction Y properly defines by (4) a linear operator S Y if and only if
Then the lifting C S,Y is a contraction if and only if S Y is a contraction.
In the following example Y = 0 does not define a lifting. In Remark 3.7 we describe all contractions Y defining a lifting for this example. It is inspired by Parrotts example [11]. The example is the case when Note that ker Hence and by (5) for Y = 0 we get so S 0 is not a linear operator.
From (2)  Proof. By Lemma 3.2 the extension is given by where, by (4), S 0 (z n ⊗ D T h) = z n ⊗ D T Sh = (I ⊗S)(z n ⊗ D T h). Hence S 0 is a well defined contraction if and only ifS is a well defined contraction, which in turn is equivalent to D T Sh ≤ D T h for any h ∈ H. This fact and Remark 3.3 finish the proof.
In particular, the only lifting of an isometry is an extension.

Corollary 3.6
An isometry S commuting with a contraction T admits a unique lifting with respect to V T and the lifting is an isometry. The lifting is an extension of the form Proof. Since S is an isometry, D S = 0 and by (2) the only lifting is an extension.
Recall that D T h 2 = h 2 − T h 2 . Hence D T h = D T Sh and soS is a well defined isometry. Since S andS are isometries, C S is an isometry. [128]

Zbigniew Burdak and Wiesław Grygierzec
Since not all contractions Y create a commuting lifting, it is natural to ask which of them do. Let us investigate Y more thoroughly. Since Y maps D S to where the convergence is in the strong operator topology. Subsequently (4) may be reformulated as Let us describe liftings of S in Example 3.4. Hence, by (7), we get for an arbitrary contraction Y generating a lifting. On the other hand, S Y (z n ⊗ D T (h, 0)) = 0 and D S T (h, 0) = (0, V h), which, by (7), implies for i ≥ 1 and any h ∈ H. Hence we get

On dilation and commuting liftings of n-tuples [129]
Summing up, In other words, an arbitrary contraction Y defining a lifting of S is determined by its restriction Y | D B ⊕ker V * . However, an arbitrary contractionỸ : which may not be a contraction. More precisely, if B < 1, thenỸ of sufficiently small norm (any contraction multiplied by a sufficiently small constant) generates by (9) a contraction Y . On the other hand, since 1 H⊕H and the estimate may not be improved if B = 1 as ranV * = H. Consequently, Hence, if B = 1, then Y may not be a contraction even if Ỹ is very small. By a direct calculation one may check that any operator of the form (9) satisfies Hence, since ker D T = H ⊕ {0} and by (8) we get by Remark 3.3 a one-to-one correspondence between liftings of S and contractions in B(D B ⊕ ker V * , H 2 (T) ⊗ D T ) such that (9) defines a contraction. Let us give a few hints how to find the proper contractionsỸ . The case B < 1 was explained. For an arbitrary contraction B,Ỹ = 0 generates Y = 0, which in turn generates the lifting C S,Y which is not an extension. The other way is to get orthogonality of summands in (9), which may be obtained using decomposition (6) ofỸ . For example, letỸ n = 0 for odd n or simplyỸ n = 0 for all n except one n 0 = 0.

Triples
In Section 2 there was described a construction of an isometric dilation of an n-tuple, which in the case of the triple R, S, T ∈ B(H) starts with the dilation of T to the isometry V T and liftings of R, S to contractions C R , C S commuting with V T . The problem that appeared was that C R , C S may not commute with each other and if so, the further construction fails. We investigate the condition under which C R , C S commute.
be the liftings of R, S and an isometric dilation of T , respectively. Operators Proof. The commutativity between C R,X and C S,Y leads to the equality The equality of entries (2, 1) yields the condition (10) to be necessary. To show it is a sufficient condition we need to show that if it holds, then R X S Y = S Y R X . However, since R X , S Y commute with M z it is enough to show commutativity on 1 ⊗ D T . Note that 1 ⊗ D T = ran(ED T ). On the other hand, from (5) we get and, similarly, Let us now describe precisely a dilation of the triple (R, S, T ) via liftings. We start with the dilation to a pair of contractions and an isometry (C R , C S , V T ). Next denote by V C S the isometric dilation of C S . Since V T is an isometry, by Corollary 3.6 it admits the extension C V T with respect to V C S . Assume that C R may be lifted to a contraction C C R with respect to V C S commuting with C V T . Then we get the commuting triple (C C R , V C S , C V T ). Since V C S , C V T are isometries, they admit by Corollary 3.6 extensions with respect to V C C R -the isometric dilation On dilation and commuting liftings of n-tuples (R, S, T ), where the operators commute by Proposition 4.1. Indeed, extensions are defined by X = 0, Y = 0, so they satisfy the condition (10). In other words, the dilation is constructed in three steps where the third one is always possible.
The second step of the construction makes the case when one contraction is an isometry interesting. The remaining part follows by the decomposition of Y .
As mentioned before, the last step of the described construction of an isometric dilation of a triple, which under the assumptions of Corollary 4.2 is the second step, is always possible. This follows from Corollary 3.6 and the fact that extensions always commute. Indeed, in this case we get an isometric dilation of a contraction and extensions of two remaining isometries.
The interesting case is when the triple admits a dilation to an isometry and liftings to contractions by some X, Y , where X = 0 or Y = 0 and it is not possible to get an isometry and two extensions. Instead of a single example of such a case we examine a class of triples based on the Parrott's idea. The class is defined in Example 4.3 below, where the Parrott's example is obtained by taking V = I and assuming B to be an isometry (denoted by V in the Parrott's work) not commuting with A. We are not going to give an equivalent condition for a triple in the considered class to admit a dilation to an isometry and two contractions. Our aim is rather to show some constructive approach to the dilation problem following from the method described in Section 3. We give a necessary condition which covers the Parrott's result. Some sufficient condition is also presented.
With the notation of Proposition 4.1 let the liftings of R and S be determined by contractions Zbigniew Burdak and Wiesław Grygierzec respectively. As noted in Remark 3.7, D T = {0} ⊕ H and so (8) describes S Y on a (linearly) dense subset of its domain. In particular, since ran( ·). Taking advantage of Remark 3.7 we get the left hand side of the condition (10) as Similarly, the right hand side has the form Hence X, Y generate commuting liftings if and only if they satisfy and for i ≥ 1. In particular, (12) for i = 1 and h = 0 yields On dilation and commuting liftings of n-tuples

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It turns out that (11) and (13) are not only necessary but also sufficient conditions for the existence of commuting liftings of R, S with respect to the dilation of T . More precisely, if X and Y are contractions generating liftings of R and S, respectively, and satisfying (11) and (13), then the liftings do not necessarily commute. However, the other two contractions X, Y defined as Y 0 vanish on the first coordinate) generate liftings of R and S, respectively, which do commute. Indeed, by Remark 3.7, X , Y satisfy (9) for the respectiveX , Y and are contractions. Then one can check that also X, Y satisfy (9) with X = 1 ⊗X 0 andỸ = 1 ⊗Ỹ 0 , whereX 0 (h, k) =X 0 (0, k),Ỹ 0 (h, k) =Ỹ 0 (0, k) and X ≤ X , Y ≤ Y , so X, Y are contractions defining liftings of R, S. Since X, Y satisfy also (12) (note that X i = 0, Y i = 0 for i ≥ 1), the liftings commute. Summing up, without lost of generality we may assume that X = 1 ⊗ X 0 and and both operators vanish on the first coordinate. By virtue of (9) our aim is to constructX 0 ,Ỹ 0 such that and satisfy (11) and (13). However, note that (11) is precisely Hence we look for a commuting pair X 0 , Y 0 of the form above. In particular, since A, B commute with V , the condition (13) for k = V 2 h where h ∈ H is arbitrary yields commutativity of A and B. Conclusion: A necessary condition for the existence of commuting liftings of R, S with respect to the dilation of T is commutativity of A, B.
Let us finish the remark by giving also a sufficient condition. Obviously we assume that A, B commute. If AV * commutes with BV * , then takingX 0 = Y 0 = 0 we get commuting liftings. Let us generalize this condition. Note that since H = V * H, the commutativity AV * BV * − BV * AV * = 0 is equivalent to and similarly Y 0 (h, k) = (0, V * Bk).
Such X 0 and Y 0 are clearly contractions and they commute if V * A commutes with V * B. The latter is equivalent to V * (AV * B − BV * A) = 0 which is more general than previous commutativity of AV * and BV * (check B = V ).
In particular, if V is a unitary operator, commutativity of A and B is an equivalent condition for the existence of commuting liftings of R and S with respect to the dilation of T .
are the dilation of T 1 and commuting extensions of T 2 , . . . , T n ; (ii) operators T i satisfy Moreover, if the conditions above are satisfied and C T = (C T2 , . . . , C Tn ), then for Proof. Note that operators C Ti considered in (i) are contractions. Hence we require for i = 2, . . . , n. If (i) holds, thenT i are contractions. Hence and by (15) we get (ii). Conversely, condition (ii), by (15), implies D T1 T i h ≤ D T1 h and so ker D T1 is invariant under T i . HenceT i is a well defined contraction which may be extended to D T1 . We get Proof. Since extensions are obtained by zero operators which clearly satisfy the condition (10), each n-tuple T k in (i) is a commuting n-tuple. Hence indeed, T n is an isometric dilation. Moreover, if T k is defined, then T i,k are isometries for i ≤ k. In particular, the last but one tuple T n−1 consists of isometries T i,n−1 (for i ≤ n − 1) and a contraction T n,n−1 . Hence, if the tuple T n−1 is obtained, each T i,n−1 may be extended with respect to the isometric dilation of T n,n−1 by Corollary 3.6. In other words, the condition (i) holds if and only if the sequence T k may be constructed up to the term n − 1. The last term may always be obtained. We use induction on n. The base step n = 2 follows by Lemma 5.1. Indeed, by the first paragraph of the proof, condition (i) is equivalent to the existence of T 1 , which in turn is the condition (i) of Lemma 5.1. On the other hand, since T 1 , T 2 are contractions, the pair (T 1 , T 2 ) is positive definite if and only if v⊂{1,2} (−1) |α(v)| T α(v) h 2 ≥ 0, which is condition (ii) of Lemma 5.1. The inductive step. Assume the conditions are equivalent for n − 1 tuples and consider an n-tuple T. If T 1 exists, then T satisfies conditions of Lemma 5.1, where C T = (T 2,1 , . . . , T n,1 ) is an (n−1)-subtuple of T 1 . Moreover, if T satisfies (i) then C T satisfies (i) with (C T ) k = (T 2,k+1 , . . . , T n,k+1 ). Then, by the assumption of the inductive step, C T is positive definite. Hence, by Corollary 5.2, T is positive definite.
We can conclude from Theorem 5.3 that a positive definite n-tuple admits an isometric, so also a unitary dilation. As we mentioned in Introduction such a result is known ([16, Theorem 9.1, Chap.1]) in a more precise version where an n-tuple is showed to admit a regular dilation if and only if it is positive definite. It is not our aim to provide a new proof of [16, Theorem 9.1, Chap.1], so we do not check whether the constructed dilation is regular or minimal. Our aim was to show the construction of the dilation and to emphasise that using extensions instead of liftings (which is a convenient choice by virtue of Proposition 4.1) has limited usability.