Generalized repeated interaction model and transfer functions
aa r X i v : . [ m a t h . OA ] M a r Generalized Repeated Interaction Model andTransfer Functions
Santanu Dey and Kalpesh J. Haria
Abstract.
Using a scheme involving a lifting of a row contraction we in-troduce a toy model of repeated interactions between quantum systems.In this model there is an outgoing Cuntz scattering system involvingtwo wandering subspaces. We associate to this model an input/outputlinear system which leads to a transfer function. This transfer function isa multi-analytic operator, and we show that it is inner if we assume thatthe system is observable. Finally it is established that transfer functionscoincide with characteristic functions of associated liftings.
Mathematics Subject Classification (2010).
Primary 47A13; Secondary47A20, 46L53, 47A48, 47A40, 81R15.
Keywords. repeated interaction, quantum system, multivariate operatortheory, row contraction, contractive lifting, outgoing Cuntz scatteringsystem, transfer function, multi-analytic operator, input-output formal-ism, linear system, observability, scattering theory, characteristic func-tion.
1. Introduction
In page 287 of the article [9] the author has commented the following whilecomparing [9] with [4, 5]: In [4] a row contraction A on a Hilbert space H with a one-dimensional eigenspace is considered and the theory of minimalisometric dilations is used. The characteristic function introduced in [5] is amulti-analytic operator associated to a lifting and the ergodic case is studiedin detail in [4]. In [9] minimality is not considered but one starts with aninteraction U (which is a unitary operator) in a scheme similar to [4] andand obtains a multi-analytic operator which represents the transfer functionof an input-output system associated with the interaction. It is expected thatthe scheme developed [9] is more directly applicable to physical models. In thesetting of [5] the assumption of a one-dimensional eigenspace is dropped andthe theory is much more general in another direction. A further integrationof these schemes in the future may help to remove unnecessarily restrictive Santanu Dey and Kalpesh J. Hariaassumptions of the toy model considered in [9] and lead to the study of otherand of more realistic models.This paper achieves some of these objectives. In the model of repeatedinteractions between quantum systems, also called a noncommutative Markovchain, studied in [9] (cf. [8]) for given three Hilbert spaces H , K and P withunit vectors Ω H , Ω K and Ω P an interaction is defined to be a unitary operator U : H ⊗ K → H ⊗ P such that U (Ω H ⊗ Ω K ) = Ω H ⊗ Ω P . (1.1)Define K ∞ := N ∞ i =1 K and P ∞ := N ∞ i =1 P as infinite tensor products ofHilbert spaces with distinguished unit vectors. We denote m -th copy of K in K ∞ by K m and set K [ m,n ] := K m ⊗ · · · ⊗ K n for m ≤ n . Similar notations arealso used with respect to P . The repeated interaction is defined as U ( n ) := U n . . . U : H ⊗ K ∞ → H ⊗ P [1 ,n ] ⊗ K [ n +1 , ∞ ) where U i ’s are copies of U on the factors H ⊗ K i of the infinite tensor prod-ucts and U i ’s leaves other factors fixed. Equation (1.1) tells us that the tensorproduct of the vacuum vectors Ω H , Ω K (along with Ω P ) represents a stateof the coupled system which is not affected by the interaction U. This entiresetting represents interactions of an atom with light beams or fields. In par-ticular Ω H in [9] is thought of as the vacuum state of an atom, and Ω K andΩ P as a state indicating the absence of photons.In the generalized repeated interaction model that we introduce in thisarticle we use a pair of unitaries to encode the interactions instead of oneunitary as follows:Let ˜ H be a (closed) subspace of H , and U : H ⊗ K → H ⊗ P and ˜ U : ˜ H ⊗ K → ˜ H ⊗ P be two unitaries such that U (˜ h ⊗ Ω K ) = ˜ U (˜ h ⊗ Ω K ) for all ˜ h ∈ ˜ H . (1.2)We fix { ǫ , . . . , ǫ d } to be an orthonormal basis of P . The equation (1.2) is theanalog of the equation (1.1) for our model and thus our model can be usedfor the setting where a quantum system interacts with a stream of copies ofanother quantum system in such a way that there is no backaction (so weget a Markovian type of dynamics) and such that there is a certain kind ofsubprocess. In the model of [9] the vacuum state Ω H of an atom plays animportant role. For a model describing interaction of a quantum system witha stream of copies of another quantum sytem we need that the computationsdo not involve any fixed unit vector Ω H and we are able to achieve this inour model by using a pair of unitaries. Instead of Ω H we now have a kind ofsubprocess, described by ˜ U , which can be treated on the same level as thefull process, described by U. The main condition imposed on the unitary U : H ⊗ K → H ⊗ P inorder to get a generalized interaction model is that U ( ˜ H ⊗ Ω K ) ⊂ ˜ H ⊗ P (cf.Proposition 3.1 of [10] for an interesting consequence of this assumption). Wecan then define ˜ U restricted to ˜ H⊗ Ω K as U restricted to ˜ H⊗ Ω K , and assumethat H ⊗ P is big enough to allow a unitary extension ˜ U : ˜ H ⊗ K → ˜ H ⊗ P . eneralized Repeated Interaction Model and Transfer Functions 3The focus of the study done here, as also in [9], is to bring out that certainmulti-analytic operators of the multivariate operator theory are associated tononcommutative Markov chains and related models, and that these operatorscan be exploited as powerful tools. These operators occur as central objectsin various context such as in the systems theory related works (cf. [3]) andnoncommutative multivariable operator theory related works (cf. [14], [15]).A tuple T = ( T , . . . , T d ) of operators T i ’s on a common Hilbert space L is called a row contraction if P di =1 T i T ∗ i ≤ I. In particular if P di =1 T i T ∗ i = I ,then the tuple T = ( T , . . . , T d ) is called coisometric . We introduce the nota-tion ˜Λ for the free semigroup with generators 1 , . . . , d . Suppose T , . . . , T d ∈B ( L ) for a Hilbert space L . If α ∈ ˜Λ is the word α . . . α n with length | α | = n ,where each α j ∈ { , . . . , d } , then T α denote T α . . . T α n . For the empty word ∅ we define |∅| = 0 and T ∅ = I .The unitary U : H ⊗ K → H ⊗ P from our model can be decomposed as U ( h ⊗ Ω K ) = d X j =1 E ∗ j h ⊗ ǫ j for h ∈ H , (1.3)where E j ’s are some operators in B ( H ) , for j = 1 , . . . , d. Likewise there existsome operators C j ’s in B ( ˜ H ) such that˜ U (˜ h ⊗ Ω K ) = d X j =1 C ∗ j ˜ h ⊗ ǫ j for ˜ h ∈ ˜ H . (1.4)Observe that P dj =1 E j E ∗ j = I and P dj =1 C j C ∗ j = I, i.e., E and C are coiso-metric tuples. By equation (1.2) E ∗ j ˜ h = C ∗ j ˜ h for all ˜ h ∈ ˜ H , j = 1 , . . . , d. We recall from [5] that such tuple E = ( E , . . . , E d ) is called a lifting of C = ( C , . . . , C d ) . From a physicist perspective our model is a Markovian approximationof the repeated interaction between a quantum system and a stream of copiesof another quantum system in such a way that there is no backaction. Thechange of an observable X ∈ B ( H ) until time n, compressed to H , is writtenas Z n ( X ) := P H U ( n ) ∗ ( X ⊗ I ) U ( n ) | H . From equation (1.3) it follows that Z n ( X ) = Z n ( X ) where Z ( X ) = P di =1 E i XE ∗ i : B ( H ) → B ( H ) and Z is called the transition operator of thenoncommutative Markov chain.In section 2 we develop our generalized repeated interaction model andobtain a coisometric operator which intertwines between the minimal isomet-ric dilations of E and C, and which will be crucial for the further investigationin this article. Using this an outgoing Cuntz scattering system in the senseof [3] is constructed for our model in section 3. Popescu introduced the min-imal isometric dilation in [13] and the characteristic function in [14] of a row Santanu Dey and Kalpesh J. Hariacontraction, and systematically developed an extensive theory of row con-tractions (cf. [16], [17]). We use some of the concepts from Popescu’s theoryin this work.For the outgoing Cuntz scattering system in section 4 we give a ˜Λ-linearsystem with an input-output formalism. A multi-analytic operator appearshere as the transfer function and in the next section we show that this trans-fer function can be derived from the intertwining coisometry of section 2.In the scattering interpretation of the transfer function this now mediatesbetween two processes. This together with a nice product formula obtainedin Proposition 2.1 tells us that this identification of transfer function is areminiscent of the scattering operator construction using wave operators inLax-Phillips scattering theory [12], equation (1.5) (cf. [18]), with one of theprocesses moving forward combined with the other moving backward. In [20]and [7] there are other approaches to transfer functions. Several works ontransfer functions and on quantum systems using linear system theory canbe found in recent theoretical physics and control theory surveys.In section 5 we investigate in regard to our model what the notion ofobservability implies for the scattering theory and the theory of liftings. Sometechniques used here are similar to those of scattering theory of noncommu-tative Markov chains introduced in [11]. Characteristic functions for liftings,introduced in [5], are multi-analytic operators which classify certain class ofliftings. Our model generalizes the setting of [9], and a comparison is done insection 6 between the transfer function of our model and the characteristicfunction for the associated lifting using the series expansion of the transferfunction obtained in section 4. As a consequence mathematically generalizedinteraction models get firmly linked into the theory of functional models.
2. A Generalised Repeated Interaction Model
We begin with three Hilbert spaces H , K and P with unit vectors Ω K ∈ K andΩ P ∈ P , and unitaries U and ˜ U as in equation (1.2). In K ∞ = N ∞ i =1 K and P ∞ = N ∞ i =1 P define Ω K∞ := N ∞ i =1 Ω K and Ω P∞ := N ∞ i =1 Ω P respectively. Wedenote m -th copy of Ω K in Ω K∞ by Ω K m and in terms of this we introduce thenotation Ω K [ m,n ] := Ω K m ⊗ · · · ⊗ Ω K n . Identify K [ m,n ] with Ω K [1 ,m − ⊗ K [ m,n ] ⊗ Ω K [ n +1 , ∞ ) , H with H ⊗ Ω K∞ as a subspace of H ⊗ K ∞ and ˜ H with ˜ H ⊗ Ω K∞ as asubspace of ˜ H ⊗ K ∞ . Similar notations with respect to P are also used. Forsimplicity we assume that d is finite but all the results here can be derivedalso for d = ∞ .Associate a row contraction E to the unitary U as in equation (1.3) anddefine isometries b V Ej ( h ⊗ η ) := U ∗ ( h ⊗ ǫ j ) ⊗ η for j = 1 , . . . , d, on the elementary tensors h ⊗ η ∈ H ⊗ K ∞ and extend it linearly to obtain b V Ej ∈ B ( H ⊗ K ∞ ) for j = 1 , . . . , d . We recall that a lifting T = ( T , . . . , T d )of any row contraction S = ( S , . . . , S d ) is called its isometric dilation ifeneralized Repeated Interaction Model and Transfer Functions 5 T i ’s are isometries with orthogonal ranges. It can be easily verified that b V E = ( b V E , . . . , b V Ed ) on the space H ⊗ K ∞ is an isometric dilation of E =( E , . . . , E d ) . If h ∈ H and k ∈ K , then there exist h i ∈ H for i = 1 , . . . , d such that U ∗ ( P di =1 h i ⊗ ǫ i ) = h ⊗ k because U is a unitary. This implies d X i =1 b V Ei ( h i ⊗ Ω K∞ ) = h ⊗ k ⊗ Ω K [2 , ∞ ) . In addition if k ∈ K , then d X i =1 b V Ei ( h i ⊗ k ⊗ Ω K [2 , ∞ ) ) = U ∗ ( d X i =1 h i ⊗ ǫ i ) ⊗ k ⊗ Ω K [3 , ∞ ) = h ⊗ k ⊗ k ⊗ Ω K [3 , ∞ ) . By induction we conclude that
H ⊗ K ∞ = span { b V Eα ( h ⊗ Ω K∞ ) : h ∈ H , α ∈ ˜Λ } , i.e., b V E is the minimal isometric dilation of E. Note that the minimal iso-metric dilation is unique up to unitary equivalence (cf. [13]).Similarly, associate a row contraction C to the unitary ˜ U as in equation(1.4) and define isometries b V Cj (˜ h ⊗ η ) := ˜ U ∗ (˜ h ⊗ ǫ j ) ⊗ η for j = 1 , . . . , d (2.1)on the elementary tensors ˜ h ⊗ η ∈ ˜ H ⊗ K ∞ and extend it linearly to obtain b V Cj ∈ B ( ˜ H ⊗ K ∞ ) for j = 1 , . . . , d. The tuple b V C = ( b V C , . . . , b V Cd ) on thespace ˜ H ⊗ K ∞ is the minimal isometric dilation of C = ( C , . . . , C d ). Recallthat U m : H ⊗ K ∞ → H ⊗ K [1 ,m − ⊗ P m ⊗ K [ m +1 , ∞ ) is nothing but the operator which acts as U on H⊗K m and fixes other factorsof the infinite tensor products. Similarly, we define ˜ U m using ˜ U .
Proposition 2.1.
Let P n := P ˜ H ⊗ I P [1 ,n ] ⊗ I K [ n +1 , ∞ ) ∈ B ( H ⊗ P [1 ,n ] ⊗ K [ n +1 , ∞ ) ) for n ∈ N . Then sot − lim n →∞ ˜ U ∗ . . . ˜ U ∗ n P n U n . . . U exists and this limit defines a coisometry c W : H ⊗ K ∞ → ˜ H ⊗ K ∞ . Its adjoint c W ∗ : ˜ H ⊗ K ∞ → H ⊗ K ∞ is given by c W ∗ = sot − lim n →∞ U ∗ . . . U ∗ n ˜ U n . . . ˜ U . Here sot stands for the strong operator topology.Proof.
At first we construct the adjoint c W ∗ . For that consider the densesubset S m ≥ ˜ H⊗K [1 ,m ] of ˜ H⊗K ∞ and let an arbitrary simple tensor elementof this dense subset be ˜ h ⊗ k ⊗ . . . ⊗ k ℓ ⊗ Ω K [ ℓ +1 , ∞ ) for some ℓ ∈ N , ˜ h ∈ ˜ H and k i ∈ K i . Set a p = U ∗ . . . U ∗ p ˜ U p . . . ˜ U (˜ h ⊗ k ⊗ . . . ⊗ k ℓ ⊗ Ω K [ ℓ +1 , ∞ ) ) for p ∈ N . Santanu Dey and Kalpesh J. HariaSince U (˜ h ⊗ Ω K ) = ˜ U (˜ h ⊗ Ω K ) for all ˜ h ∈ ˜ H , we have a ℓ = a ℓ + n for all n ∈ N .Therefore we deduce thatlim n →∞ U ∗ . . . U ∗ n ˜ U n . . . ˜ U (˜ h ⊗ k ⊗ . . . ⊗ k ℓ ⊗ Ω K [ ℓ +1 , ∞ ) )exists. Because U and ˜ U are unitaries, we obtain an isometric extension c W ∗ tothe whole of ˜ H⊗K ∞ . Thus its adjoint is a coisometry c W : H⊗K ∞ → ˜ H⊗K ∞ .Now we will derive the limit form for c W as claimed in the statement ofthe proposition. If h ⊗ η ∈ H ⊗ K [1 ,k ] , ˜ h ⊗ ˜ η ∈ ˜ H ⊗ K [1 ,n ] and k ≤ n, then h c W ( h ⊗ η ) , ˜ h ⊗ ˜ η i = h h ⊗ ˜ η, c W ∗ (˜ h ⊗ ˜ η ) i = h h ⊗ η, U ∗ . . . U ∗ n ˜ U n . . . ˜ U (˜ h ⊗ ˜ η ) i = h ˜ U ∗ . . . ˜ U ∗ n P n U n . . . U ( h ⊗ η ) , ˜ h ⊗ ˜ η i . Consequently c W = sot − lim n →∞ ˜ U ∗ . . . ˜ U ∗ n P n U n . . . U on a dense subset andtherefore it can be extended to the whole of H ⊗ K ∞ . (cid:3) Observe that c W ∗ (˜ h ⊗ Ω K∞ ) = ˜ h ⊗ Ω K∞ for all ˜ h ∈ ˜ H . (2.2)Next we show that this coisometry c W intertwines between b V Ej and b V Cj forall j = 1 , . . . , d . For j = 1 , . . . , d , define S j : H ⊗ K ∞ → H ⊗ P ⊗ K [2 , ∞ ) ,h ⊗ η h ⊗ ǫ j ⊗ η. The following are immediate:(1) S ∗ j ( h ⊗ p ⊗ η ) = h ǫ j , p i ( h ⊗ η ) for ( h ⊗ p ⊗ η ) ∈ H ⊗ P ⊗ K [2 , ∞ ) .(2) b V Ej ( h ⊗ η ) = U ∗ S j ( h ⊗ η ) for h ⊗ η ∈ H ⊗ K ∞ .(3) b V Cj (˜ h ⊗ η ) = ˜ U ∗ S j (˜ h ⊗ η ) for ˜ h ⊗ η ∈ ˜ H ⊗ K ∞ . Proposition 2.2. If c W is as in Proposition 2.1, then c W b V Ej = b V Cj c W , b V Ej c W ∗ = c W ∗ b V Cj for all j = 1 , . . . , d. Proof. If h ∈ H , η ∈ K ∞ , ˜ h ∈ ˜ H and k i ∈ K i , then by the three observationsthat were noted preceding this proposition we obtain for j = 1 , . . . , d h c W b V Ej ( h ⊗ η ) , ˜ h ⊗ k ⊗ . . . ⊗ k ℓ ⊗ Ω K [ ℓ +1 , ∞ ) i = h U ∗ ( h ⊗ ǫ j ) ⊗ η, U ∗ . . . U ∗ ℓ ˜ U ℓ . . . ˜ U (˜ h ⊗ k ⊗ . . . ⊗ k ℓ ⊗ Ω K [ ℓ +1 , ∞ ) ) i . eneralized Repeated Interaction Model and Transfer Functions 7Substituting ˜ U (˜ h ⊗ k ) = P i ˜ h ( i ) ⊗ k ( i )1 where ˜ h ( i ) ∈ ˜ H and k ( i )1 ∈ K weobtain h c W b V Ej ( h ⊗ η ) , ˜ h ⊗ k ⊗ . . . ⊗ k ℓ ⊗ Ω K [ ℓ +1 , ∞ ) i = h h ⊗ ǫ j ⊗ η, U ∗ . . . U ∗ ℓ ˜ U ℓ . . . ˜ U ( X i (˜ h ( i ) ⊗ k ( i )1 ) ⊗ k ⊗ . . . ⊗ k ℓ ⊗ Ω K [ ℓ +1 , ∞ ) ) i = X i h ǫ j , k ( i )1 i h h ⊗ η, c W ∗ (˜ h ( i ) ⊗ k ⊗ . . . ⊗ k ℓ ⊗ Ω K [ ℓ +1 , ∞ ) ) i = h c W ( h ⊗ η ) , S ∗ j ˜ U (˜ h ⊗ k ⊗ . . . ⊗ k ℓ ⊗ Ω K [ ℓ +1 , ∞ ) ) i = h ˜ U ∗ S j c W (( h ⊗ η ) , ˜ h ⊗ k ⊗ . . . ⊗ k ℓ ⊗ Ω K [ ℓ +1 , ∞ ) i = h b V Cj c W ( h ⊗ η ) , ˜ h ⊗ k ⊗ . . . ⊗ k ℓ ⊗ Ω K [ ℓ +1 , ∞ ) i . Hence c W b V Ej = b V Cj c W for all j = 1 , . . . , d. To obtain the other equation ofthe proposition we again use the last two of the three observations as follows:For j = 1 , . . . , d c W ∗ b V Cj (˜ h ⊗ k ⊗ . . . ⊗ k ℓ ⊗ Ω K [ ℓ +1 , ∞ ) )= c W ∗ ˜ U ∗ (˜ h ⊗ ǫ j ⊗ k ⊗ . . . ⊗ k ℓ ⊗ Ω K [ ℓ +2 , ∞ ) )= U ∗ U ∗ . . . U ∗ ℓ +1 ˜ U ℓ +1 . . . ˜ U ˜ U ˜ U ∗ (˜ h ⊗ ǫ j ⊗ k ⊗ . . . ⊗ k ℓ ⊗ Ω K [ ℓ +2 , ∞ ) )= U ∗ U ∗ . . . U ∗ ℓ +1 ˜ U ℓ +1 . . . ˜ U S j (˜ h ⊗ k ⊗ . . . ⊗ k ℓ ⊗ Ω K [ ℓ +1 , ∞ ) )= U ∗ S j U ∗ . . . U ∗ ℓ ˜ U ℓ . . . ˜ U (˜ h ⊗ k ⊗ . . . ⊗ k ℓ ⊗ Ω K [ ℓ +1 , ∞ ) )= b V Ej c W ∗ (˜ h ⊗ k ⊗ . . . ⊗ k ℓ ⊗ Ω K [ ℓ +1 , ∞ ) ) (cid:3) Further define(
H ⊗ K ∞ ) ◦ := ( H ⊗ K ∞ ) ⊖ ( ˜ H ⊗ Ω K∞ ) , ( ˜ H ⊗ K ∞ ) ◦ := ( ˜ H ⊗ K ∞ ) ⊖ ( ˜ H ⊗ Ω K∞ ) and H ◦ := H ⊖ ˜ H . (2.3)Let P ki =1 ξ i ⊗ η i ∈ ( H ⊗ K ∞ ) ◦ and ˜ h ∈ ˜ H . Then for j = 1 , . . . , d h b V Ej ( X i ξ i ⊗ η i ) , ˜ h ⊗ Ω K∞ i = h X i U ∗ ( ξ i ⊗ ǫ j ) ⊗ η i , ˜ h ⊗ Ω K∞ i = h X i ξ i ⊗ ǫ j ⊗ η i , ˜ U (˜ h ⊗ Ω K ) ⊗ Ω K [2 , ∞ ) i = 0because ˜ U maps into ˜ H ⊗ P and P ki =1 ξ i ⊗ η i ⊥ ˜ H ⊗ Ω K . Therefore b V Ej ( H ⊗K ∞ ) ◦ ⊂ ( H ⊗ K ∞ ) ◦ for j = 1 , . . . , d. Similarly b V Cj ( ˜ H ⊗ K ∞ ) ◦ ⊂ ( ˜ H ⊗ K ∞ ) ◦ for j = 1 , . . . , d. Set V Ej := b V Ej | ( H⊗K ∞ ) ◦ and V Cj := b V Cj | ( ˜ H⊗K ∞ ) ◦ for j =1 , . . . , d. If we define W ∗ := c W ∗ | ( ˜ H⊗K ∞ ) ◦ , Santanu Dey and Kalpesh J. Hariathen by equation (2.2) it follows that W ∗ ∈ B (( ˜ H ⊗ K ∞ ) ◦ , ( H ⊗ K ∞ ) ◦ ). Theoperator W ∗ is an isometry because it is a restriction of an isometry and W ,the adjoint of W ∗ , is the restriction of c W to ( H⊗K ∞ ) ◦ , i.e., W = c W | ( H⊗K ∞ ) ◦ . Remark . It follows that
W V Ej = V Cj W for j = 1 , . . . , d .
3. Outgoing Cuntz Scattering Systems
In this section we aim to construct an outgoing Cuntz scattering system (cf.[3]) for our model. This will assist us in the next section to work with aninput-output formalism and to associate a transfer function to the model.Following are some notions from the multivariable operator theory:
Definition 3.1.
Suppose T = ( T , . . . , T d ) is a row contraction where T i ∈B ( L ) . (1) If T i ’s are isometries with orthogonal ranges, then the tuple T = ( T , . . . ,T d ) is called a row isometry .(2) If span j =1 ,...,d T j L = L and T = ( T , . . . , T d ) is a row isometry, then T is called a row unitary .(3) If there exist a subspace E of L such that L = L α ∈ ˜Λ T α E and T =( T , . . . , T d ) is a row isometry, then T is called a row shift and E iscalled a wandering subspace of L w.r.t. T .
Definition 3.2.
A collection ( L , V = ( V , . . . , V d ) , G + ∗ , G ) is called an outgoingCuntz scattering system (cf. [3]), if V is a row isometry on the Hilbert space L , and G + ∗ and G are subspaces of L such that(1) for E ∗ := L ⊖ span j =1 ,...,d V j L , the tuple V |G + ∗ is a row shift where G + ∗ = L α ∈ ˜Λ V α E ∗ .(2) there exist E := G ⊖ span j =1 ,...,d V j G with G = L α ∈ ˜Λ V α E , i.e., V | G is arow shift.In the above definition the part (1) is the Wold decomposition (cf. [13])of the row isometry V and therefore G + ∗ can be derived from V .
But G + ∗ is in-cluded in the data because it helps in describing the scattering phenomenon.We continue using the notations from the previous section. b V Ej ’s are isome-tries with orthogonal ranges and because ( ǫ j ) dj =1 is an orthonormal basis of P , we have span j =1 ,...,d b V Ej ( H ⊗ K ∞ ) = H ⊗ K ∞ . Thus b V E is a row unitary on H ⊗ K ∞ . Now using the fact that V Ej = b V Ej | ( H⊗K ∞ ) ◦ we infer that V Ej ’s are isometries with orthogonal ranges. There-fore V E is a row isometry on ( H ⊗ K ∞ ) ◦ .eneralized Repeated Interaction Model and Transfer Functions 9 Proposition 3.3. If Y := ˜ H ⊗ (Ω K ) ⊥ ⊗ Ω K [2 , ∞ ) ⊂ ˜ H ⊗ K ∞ , then W ∗ Y ⊥ span j =1 ,...,d V Ej ( H ⊗ K ∞ ) ◦ . Proof.
By Proposition 2.1 it is easy to see that W ∗ Y = U ∗ ˜ U Y ⊂ H ⊗ K ⊗ Ω K [2 , ∞ ) . (3.1)Let ˜ h i ∈ ˜ H and k i ⊥ Ω K for i = 1 , . . . , n, i.e., P i ˜ h i ⊗ k i ⊗ Ω K [2 , ∞ ) ∈ Y . For P k h k ⊗ η k ∈ ( H ⊗ K ∞ ) ◦ with h k ∈ H and η k ∈ K ∞ h W ∗ ( P i ˜ h i ⊗ k i ⊗ Ω K [2 , ∞ ) ) , V Ej ( P k h k ⊗ η k ) i = h U ∗ ˜ U ( P i ˜ h i ⊗ k i ) ⊗ Ω K [2 , ∞ ) , P k U ∗ ( h k ⊗ ǫ j ) ⊗ η k i = h ˜ U ( P i ˜ h i ⊗ k i ) ⊗ Ω K [2 , ∞ ) , P k h k ⊗ ǫ j ⊗ η k i = 0 . The last equality holds because P k h k ⊗ η k ⊥ ˜ H ⊗ Ω K∞ . Thus W ∗ Y ⊥ span j =1 ,...,d V Ej ( H ⊗ K ∞ ) ◦ . (cid:3) The following Proposition gives an explicit description of the Wold de-composition of V E : Proposition 3.4. If Y is defined as in the previous proposition, then W ∗ Y is awandering subspace of V E , i.e., V Eα ( W ∗ Y ) ⊥ V Eβ ( W ∗ Y ) whenever α, β ∈ ˜Λ , α = β , and W ∗ Y = ( H ⊗ K ∞ ) ◦ ⊖ span j =1 ,...,d V Ej ( H ⊗ K ∞ ) ◦ . Proof.
By Proposition 3.3 it is immediate that V Eα ( W ∗ Y ) ⊥ V Eβ ( W ∗ Y ) when-ever α, β ∈ ˜Λ, α = β and W ∗ Y ⊂ ( H ⊗ K ∞ ) ◦ ⊖ span j =1 ,...,d V Ej ( H ⊗ K ∞ ) ◦ .The only thing that remains to be shown is that( H ⊗ K ∞ ) ◦ ⊖ span j =1 ,...,d V Ej ( H ⊗ K ∞ ) ◦ ⊂ W ∗ Y . Let x ∈ ( H⊗K ∞ ) ◦ ⊖ span j =1 ,...,d V Ej ( H⊗K ∞ ) ◦ . Write down the decom-position of x as x ⊕ x w.r.t. W ∗ Y ⊕ ( W ∗ Y ) ⊥ . So x − x = x is orthogonal toboth span j =1 ,...,d V Ej ( H ⊗ K ∞ ) ◦ and W ∗ Y . Now we show that if any elementin ( H ⊗ K ∞ ) ◦ is orthogonal to span j =1 ,...,d V Ej ( H ⊗ K ∞ ) ◦ and W ∗ Y , then itis the zero vector. Let x be such an element. Because x ∈ ( H ⊗ K ∞ ) ◦ and x ⊥ W ∗ Y , x ⊥ U ∗ ( ˜ H ⊗ ǫ j ) ⊗ Ω K [2 , ∞ ) for j = 1 , . . . , d . This implies x ⊥ span j =1 ,...,d b V Ej ( ˜ H ⊗ Ω K∞ ) . We also knowthat x ⊥ span j =1 ,...,d V Ej ( H ⊗ K ∞ ) ◦ (= span j =1 ,...,d b V Ej ( H ⊗ K ∞ ) ◦ ) . Therefore x ⊥ span j =1 ,...,d b V Ej ( H ⊗ K ∞ ) . Since b V E is a row unitrary, x ⊥ H ⊗ K ∞ . So x = 0 and hence x = x ∈ W ∗ Y . We conclude that (
H ⊗ K ∞ ) ◦ ⊖ span j =1 ,...,d V Ej ( H ⊗ K ∞ ) ◦ ⊂ W ∗ Y . (cid:3) Proposition 3.5. If E := H ⊗ (Ω K ) ⊥ ⊗ Ω K [2 , ∞ ) ⊂ ( H ⊗ K ∞ ) ◦ , then V Eα E ⊥ V Eβ E whenever α, β ∈ ˜Λ , α = β and ( H ⊗ K ∞ ) ◦ = H ◦ ⊕ L α ∈ ˜Λ V Eα E .Proof. If | α | = | β | and α = β , then it is easy to see that V Eα E ⊥ V Eβ E because ranges of V Ej ’s are mutually orthogonal. If | α | 6 = | β | (without loss ofgenerality we can assume that | α | > | β | ), then by taking the inner productat the tensor factor K | α | +1 we obtain V Eα E ⊥ V Eβ E .To prove the second part of the proposition, observe that for n ∈ N , H ⊗ K [1 ,n ] ⊗ Ω K [ n +1 , ∞ ] = ( H ⊗ Ω K∞ ) ⊕ ( H ⊗ (Ω K ) ⊥ ⊗ Ω K [2 , ∞ ) ) ⊕ ( H ⊗ K ⊗ (Ω K ) ⊥ ⊗ Ω K [3 , ∞ ) ) ⊕ · · · ⊕ ( H ⊗ K [1 ,n − ⊗ (Ω K n ) ⊥ ⊗ Ω K [ n +1 , ∞ ) )= ( ˜ H ⊗ Ω K∞ ) ⊕ ( H ◦ ⊗ Ω K∞ ) ⊕ E ⊕ d M j =1 V Ej E ⊕ · · · ⊕ d M | α | = n − V Eα E . Taking n → ∞ we have the following: H ⊗ K ∞ = ( ˜ H ⊗ Ω K∞ ) ⊕ ( H ◦ ⊗ Ω K∞ ) ⊕ M α ∈ ˜Λ V Eα E . Since (
H ⊗ K ∞ ) ◦ = ( H ⊗ K ∞ ) ⊖ ( ˜ H ⊗ Ω K∞ ), it follows that( H ⊗ K ∞ ) ◦ = H ◦ ⊕ M α ∈ ˜Λ V Eα E . (cid:3) We sum up Propositions 3.3, 3.4 and 3.5 in the following theorem:
Theorem 3.6.
For a generalized repeated interaction model involving unitaries U and ˜ U as before set Y := ˜ H⊗ (Ω K ) ⊥ ⊗ Ω K [2 , ∞ ) and E := H⊗ (Ω K ) ⊥ ⊗ Ω K [2 , ∞ ) . If E ∗ := W ∗ Y , G + ∗ := L α ∈ ˜Λ V Eα E ∗ and G := L α ∈ ˜Λ V Eα E , then the collection (( H ⊗ K ∞ ) ◦ , V E = ( V E , . . . , V Ed ) , G + ∗ , G ) is an outgoing Cuntz scattering system such that ( H ⊗ K ∞ ) ◦ = H ◦ ⊕ G .Remark . Applying arguments similar to those used for proving the secondpart of the Proposition 3.5 one can prove the following:( ˜
H ⊗ K ∞ ) ◦ = M α ∈ ˜Λ V Cα Y . We refer the reader to Proposition 3.1 of [10] for a result in a similardirection.eneralized Repeated Interaction Model and Transfer Functions 11 ˜Λ -Linear Systems and Transfer Functions We would demonstrate that the outgoing Cuntz scattering system ((
H ⊗K ∞ ) ◦ , V E = ( V E , . . . , V Ed ) , G + ∗ , G ) from Theorem 3.6 has interesting relationswith a generalization of the linear systems theory that is associated to ourinteraction model. For a given model involving unitaries U and ˜ U as before,let us define the input space as U := E = H ⊗ (Ω K ) ⊥ ⊗ Ω K [2 , ∞ ) ⊂ ( H ⊗ K ∞ ) ◦ and the output space as Y = ˜ H ⊗ (Ω K ) ⊥ ⊗ Ω K [2 , ∞ ) ⊂ ( ˜ H ⊗ K ∞ ) ◦ . Here we assume that a quantum system A interacts with a stream ofcopies of another quantum system B and we assume H is the (quantummechanical) Hilbert space of A . Let K i be the Hilbert space of a part ofa stream of copies of B at time i immediately before the interaction with A . Let the Hilbert space P i be that the part of a stream of copies of B attime i immediately after the interaction with A . Ω K and Ω P denote statesindicating that no copy of quantum system B is present and so no interactionis taking place at time i. Then η ∈ U = H ⊗ (Ω K ) ⊥ ⊗ Ω K [2 , ∞ ) ⊂ H ⊗ K ∞ represents a vector state with copies of quantum system B arriving at time1 and stimulating an interaction between the stream of copies of A and B , but no further copy of B arriving at later times. But some activity is inducedwhich goes on for a longer period.Note that H ⊗ K = H ⊕ U and ˜
H ⊗ K = ˜
H ⊕ Y . So U maps H ⊕ U onto
H ⊗ P and ˜ U maps ˜ H ⊕ Y onto ˜
H ⊗ P . Using unitaries U and ˜ U we define F j : H → U and D j : ˜ H → Y for j = 1 , . . . , d by d X j =1 F ∗ j η ⊗ ǫ j := U (0 ⊕ η ) , d X j =1 D ∗ j y ⊗ ǫ j := ˜ U (0 ⊕ y ) for η ∈ U and y ∈ Y . (4.1)Combining equation (4.1) with equations (1.3) and (1.4) we have for h ∈H , η ∈ U , ˜ h ∈ ˜ H and y ∈ Y U ( h ⊕ η ) = d X j =1 ( E ∗ j h + F ∗ j η ) ⊗ ǫ j , (4.2)˜ U (˜ h ⊕ y ) = d X j =1 ( C ∗ j ˜ h + D ∗ j y ) ⊗ ǫ j (4.3)respectively. Using equation (4.3) it can be checked that˜ U ∗ (˜ h ⊗ ǫ j ) = C j ˜ h ⊕ D j ˜ h for ˜ h ∈ ˜ H ; j = 1 , . . . , d. (4.4)Let us define˜ C := d X j =1 D j P ˜ H E ∗ j : H → Y , ˜ D := d X j =1 D j P ˜ H F ∗ j : U → Y P ˜ H is the orthogonal projection onto ˜ H . It follows that P Y ˜ U ∗ P U ( h ⊕ η ) = ˜ Ch + ˜ Dη (4.5)where h ∈ H , η ∈ U , P is as in Proposition 2.1 and P Y is the orthogonalprojection onto Y .Define a colligation of operators (cf. [3]) using the operators E ∗ j ’s, F ∗ j ’s,˜ C and ˜ D by C U, ˜ U := E ∗ F ∗ ... ... E ∗ d F ∗ d ˜ C ˜ D : H ⊕ U → d M j =1 H ⊕ Y . From the colligation C U, ˜ U we get the following ˜Λ- linear system P U, ˜ U : x ( jα ) = E ∗ j x ( α ) + F ∗ j u ( α ) , (4.6) y ( α ) = ˜ Cx ( α ) + ˜ Du ( α ) (4.7)where j = 1 , . . . , d and α, jα are words in ˜Λ, and x : ˜Λ → H , u : ˜Λ → U , y : ˜Λ → Y . If x ( ∅ ) and u are known, then using P U, ˜ U we can compute x and y recursively.Such a ˜Λ-linear system is also called a noncommutative Fornasini-Marchesinisystem in [1] in reference to [6].Let z = ( z , . . . , z d ) be a d -tuple of formal noncommuting indetermi-nates. Define the Fourier transforms of x, u and y asˆ x ( z ) = X α ∈ ˜Λ x ( α ) z α , ˆ u ( z ) = X α ∈ ˜Λ u ( α ) z α , ˆ y ( z ) = X α ∈ ˜Λ y ( α ) z α respectively where z α = z α n . . . z α for α = α n . . . α ∈ ˜Λ. Assuming that z -variables commute with the coefficients the input-output relationˆ y ( z ) = Θ U, ˜ U ( z )ˆ u ( z )can be obtained on setting x ( ∅ ) := 0 whereΘ U, ˜ U ( z ) := X α ∈ ˜Λ Θ ( α ) U, ˜ U z α := ˜ D + ˜ C X β ∈ ˜Λ ,j =1 ,...,d ( E ¯ β ) ∗ F ∗ j z βj . (4.8)Here ¯ β = β . . . β n is the reverse of β = β n . . . β ∈ ˜Λ and Θ ( α ) U, ˜ U maps U to Y .The formal noncommutative power series Θ U, ˜ U is called the transfer function associated to the unitaries U and ˜ U . The transfer function is a mathematicaltool for encoding the evolution of a ˜Λ-linear system. For y ( α ) ∈ Y with P α ∈ ˜Λ k y ( α ) k < ∞ , any series P α ∈ ˜Λ y ( α ) z α stands for a series convergingto an element of ℓ (˜Λ , Y ).eneralized Repeated Interaction Model and Transfer Functions 13 Theorem 4.1.
The map M Θ U, ˜ U : ℓ (˜Λ , U ) → ℓ (˜Λ , Y ) defined by M Θ U, ˜ U ˆ u ( z ) := Θ U, ˜ U ( z )ˆ u ( z ) is a contraction.Proof. Observe that P Y ˜ U ∗ P U (˜ h ⊗ Ω K∞ ) = 0 for all ˜ h ∈ ˜ H . Consider anothercolligation which is defined as follows: C ◦ U, ˜ U := E ∗◦ F ∗◦ ... ... E ∗◦ d F ∗◦ d ˜ C ◦ ˜ D : H ◦ ⊕ U → d M j =1 H ◦ ⊕ Y where E ∗◦ j := P H ◦ E ∗ j | H ◦ : H ◦ → H ◦ , F ∗◦ j := P H ◦ F ∗ j : U → H ◦ and ˜ C ◦ :=˜ C | H ◦ : H ◦ → Y for j = 1 , . . . , d . Recall that H ◦ and ( H ⊗ K ∞ ) ◦ were definedin equation array (2.3). Consider the outgoing Cuntz scattering system (( H⊗K ∞ ) ◦ , V E = ( V E , . . . , V Ed ) , G + ∗ , G ), with ( H⊗K ∞ ) ◦ = H ◦ ⊕G , constructed byus in Theorem 3.6. In Chapter 5.2 of [3] it is shown that there is an associatedunitary colligation ˆ E ˆ F ... ...ˆ E d ˆ F d ˆ M ˆ N : H ◦ ⊕ E → d M j =1 H ◦ ⊕ E ∗ (4.9)such that ( ˆ E j , ˆ F j ) = P H ◦ ( V Ej ) ∗ | H ◦ ⊕E and ( ˆ M , ˆ N ) = P E ∗ | H ◦ ⊕E . Recall that E and E ∗ were introduced in Proposition 3.5 and Theorem 3.6 respectively.From equations (4.2) and (4.5) we observe that ( E ∗◦ j , F ∗◦ j ) = P H ◦ ⊗ ǫ j U | H ◦ ⊕E (identifying H ◦ with H ◦ ⊗ ǫ j ) and ( ˜ C ◦ , ˜ D ) = P Y ˜ U ∗ P U | H ◦ ⊕E . Using theseobservations we obtain the following relations: U ∗ ( E ∗◦ j , F ∗◦ j ) = U ∗ P H ◦ ⊗ ǫ j U | H ◦ ⊕E = P U ∗ ( H ◦ ⊗ ǫ j ) | H ◦ ⊕E = P V Ej H ◦ | H ◦ ⊕E = V Ej P H ◦ ( V Ej ) ∗ | H ◦ ⊕E = V Ej ( ˆ E j , ˆ F j ) (4.10)for j = 1 , . . . , d and U ∗ ˜ U ( ˜ C ◦ , ˜ D ) = U ∗ ˜ U P Y ˜ U ∗ P U | H ◦ ⊕E = U ∗ P ˜ U Y P U | H ◦ ⊕E = U ∗ P ˜ U Y U | H ◦ ⊕E = P U ∗ ˜ U Y | H ◦ ⊕E = P W ∗ Y | H ◦ ⊕E (by equation (3.1))= P E ∗ | H ◦ ⊕E = ( ˆ M , ˆ N ) . (4.11)Let ˆ u ( z ) = P α ∈ ˜Λ u ( α ) z α ∈ ℓ (˜Λ , U ) with u ( α ) ∈ U such that P α ∈ ˜Λ k u ( α ) k < ∞ . We would prove that k M Θ U, ˜ U ˆ u ( z ) k ≤ k ˆ u ( z ) k . Define x : ˜Λ → H by equation (4.6) such that x ( ∅ ) = 0 . Further, define x ◦ ( α ) := P H ◦ x ( α ) for all α ∈ ˜Λ . Now applying the projection P H ◦ to relation4 Santanu Dey and Kalpesh J. Haria(4.6) on both sides and using the fact ˜ H is invariant under E ∗ j for j = 1 , . . . , d we obtain the following relation: x ◦ ( jα ) = E ∗◦ j x ◦ ( α ) + F ∗◦ j u ( α ) for all α ∈ ˜Λ , j = 1 , . . . , d. (4.12)Because P Y ˜ U ∗ P U (˜ h ⊗ Ω K∞ ) = 0 for all ˜ h ∈ ˜ H we conclude by equation (4.5)that ˜ C ˜ h = 0 for ˜ h ∈ ˜ H . (4.13)This implies ˜ Cx ( α ) = ˜ C ◦ x ◦ ( α ) for all α ∈ ˜Λ . (4.14)Define y : ˜Λ → Y by y ( α ) := ˜ Cx ( α ) + ˜ Du ( α ) (4.15)for all α ∈ ˜Λ. Recall that the input-output relation stated just before thetheorem is ˆ y ( z ) = X α ∈ ˜Λ y ( α ) z α = Θ U, ˜ U ( z )ˆ u ( z )(= M Θ U, ˜ U ˆ u ( z )) . Using the unitary colligation given in equation (4.9) we have k x ◦ ( α ) k + k u ( α ) k = d X j =1 k ˆ E j x ◦ ( α ) + ˆ F j u ( α ) k + k ˆ M x ◦ ( α ) + ˆ N u ( α ) k = d X j =1 k E ∗◦ j x ◦ ( α ) + F ∗◦ j u ( α ) k + k ˜ C ◦ x ◦ ( α ) + ˜ Du ( α ) k = d X j =1 k x ◦ ( jα ) k + k ˜ Cx ( α ) + ˜ Du ( α ) k = d X j =1 k x ◦ ( jα ) k + k y ( α ) k for all α ∈ ˜Λ. In the above calculation equations (4.10), (4.11), (4.12), (4.14)and (4.15) respectively have been used. This gives us k u ( α ) k − k y ( α ) k = d X j =1 k x ◦ ( jα ) k − k x ◦ ( α ) k for all α ∈ ˜Λ. Summing over all α ∈ ˜Λ with | α | ≤ n and using the fact that x ◦ ( ∅ ) = 0 we obtain X | α |≤ n k u ( α ) k − X | α |≤ n k y ( α ) k = X | α | = n +1 k x ◦ ( α ) k ≥ n ∈ N . Therefore X | α |≤ n k y ( α ) k ≤ X | α |≤ n k u ( α ) k for all n ∈ N . Finally taking limit n → ∞ both the sides we get that M Θ U, ˜ U is a contraction. (cid:3) eneralized Repeated Interaction Model and Transfer Functions 15 M Θ U, ˜ U is a multi-analytic operator ([15]) (also called analytic intertwin-ing operator in [3]) because M Θ U, ˜ U ( X α ∈ ˜Λ u ( α ) z α z j ) = M Θ U, ˜ U ( X α ∈ ˜Λ u ( α ) z α ) z j for j = 1 , . . . , d, i.e., M Θ U, ˜ U intertwines with right translation. The noncommutative powerseries Θ U, ˜ U is called the symbol of M Θ U, ˜ U .
5. Transfer Functions, Observability and Scattering
We would now establish that the transfer function can be derived from thecoisometry W of section 2. In the last section d -tuple z = ( z , . . . , z d ) offormal noncommuting indeterminates were employed. Treat ( z α ) α ∈ ˜Λ as anorthonormal basis of ℓ (˜Λ , C ) . Assume Y and U to be the spaces associatedwith our model with unitaries U and ˜ U as in the last section. It follows fromRemark 3.7 that there exist a unitary operator ˜Γ : ( ˜ H ⊗ K ∞ ) ◦ → ℓ (˜Λ , Y )defined by ˜Γ( V Cα y ) := yz ¯ α for all α ∈ ˜Λ , y ∈ Y . We observe the following intertwining relation:˜Γ( V Cα y ) = (˜Γ y ) z ¯ α . (5.1)Similarly, using Theorem 3.6, we can define a unitary operator Γ : ( H ⊗K ∞ ) ◦ (= ( H ◦ ⊕ G )) → H ◦ ⊕ ℓ (˜Λ , U ) byΓ(˚ h ⊕ V Eα η ) := ˚ h ⊕ ηz ¯ α for all α ∈ ˜Λwhere ˚ h ∈ H ◦ , η ∈ U . In this case the intertwining relation isΓ( V Eα η ) = (Γ η ) z ¯ α . (5.2)Using the coisometric operator W , which appears in Remark 2.3, we defineΓ W by the following commutative diagram:( H ⊗ K ∞ ) ◦ W / / Γ (cid:15) (cid:15) ( ˜ H ⊗ K ∞ ) ◦ ˜Γ (cid:15) (cid:15) H ◦ ⊕ ℓ (˜Λ , U ) Γ W / / ℓ (˜Λ , Y ) , (5.3)i.e., Γ W = ˜Γ W Γ − . Theorem 5.1. Γ W defined by the above commutative diagram satisfies Γ W | ℓ (˜Λ , U ) = M Θ U, ˜ U . Proof.
Using the intertwining relation V Cj W = W V Ej from Remark 2.3, andequations (5.1) and (5.2) we obtainΓ W ( ηz β z j ) = ˜Γ W Γ − ( ηz β z j ) = ˜Γ W V Ej V E ¯ β η = ˜Γ V Cj V C ¯ β W η = (˜Γ
W η ) z β z j = Γ W ( ηz β ) z j η ∈ U , β ∈ ˜Λ , j = 1 , . . . , d . Hence, Γ W | ℓ (˜Λ , U ) is a multi-analytic operator.For computing its symbol we determine Γ W η for η ∈ U , where η is identifiedwith ηz φ ∈ ℓ (˜Λ , U ) . For α = α n − . . . α ∈ ˜Λ let P α be the orthogonalprojection onto˜Γ − { f ∈ ℓ (˜Λ , Y ) : f = yz α for some y ∈ Y} = V C ¯ α Y = ˜ U ∗ . . . ˜ U ∗ n − ( ˜ H ⊗ ǫ α ⊗ · · · ⊗ ǫ α n − ⊗ (Ω K n ) ⊥ ⊗ Ω K [ n +1 , ∞ ) )with ˜ U i ’s as in Proposition 2.1.Recall that the tuple E associated with the unitary U is a lifting of thetuple C (associated with the unitary ˜ U ) and so E can be written as a blockmatrix in terms of C as follows: E j = (cid:18) C j B j A j (cid:19) for j = 1 , . . . , d w.r.t. tothe decomposition H = ˜ H ⊕ H ◦ where B and A are some row contractions.Because E is a coisometric lifting of C we have d X j =1 C j C ∗ j = I and d X j =1 C j B ∗ j = 0(cf. [5]) . Now using these relations and equations (4.2), (4.3) and (4.4) it canbe easily verified that P α ˜ U ∗ . . . ˜ U ∗ n P n U n . . . U η = P α ˜ U ∗ . . . ˜ U ∗ m P m U m . . . U η for all m ≥ n, η ∈ U . Using the formula of W from Proposition 2.1 we obtain P α W η = P α ˜ U ∗ . . . ˜ U ∗ n P n U n . . . U η for η ∈ U . Finally for η ∈ U P α ˜ U ∗ . . . ˜ U ∗ n P n U n . . . U η = (cid:26) ˜ Dη if n = 1 , α = ∅ ,V C ¯ α ( ˜ CE ∗ α n − . . . E ∗ α F ∗ α η ) if n = | α | + 1 ≥ . This implies for η ∈ U ˜Γ W Γ − η = ˜Γ W η = ˜ Dη ⊕ X | α |≥ ( ˜ CE ∗ α n − . . . E ∗ α F ∗ α η ) z α . Comparing this with equation (4.8) we conclude that Γ W | ℓ (˜Λ , U ) = M Θ U, ˜ U . (cid:3) Note that the Theorem 4.1 and its proof concern the transfer functionof the ˜Λ-linear system and has nothing to do with the scattering theory.Theorem 5.1, on the other hand, is the scattering theory part in the senseof Lax-Phillips [12]. The same function M Θ U, ˜ U relates the outgoing Fourierrepresentation for a vector in the ambient scattering Hilbert space to theincoming Fourier representation for the same vector. This makes M Θ U, ˜ U thescattering function for the outgoing Cuntz scattering system. We introducea notion from the linear systems theory for our model: Definition 5.2.
The observability operator W : H ◦ → ℓ (˜Λ , Y ) is defined asthe restriction of the operator Γ W to H ◦ , i.e., W = Γ W | H ◦ .eneralized Repeated Interaction Model and Transfer Functions 17It follows that W ˚ h = ( ˜ C ( E ¯ α ) ∗ ˚ h ) α ∈ ˜Λ . Popescu has studied the similar typesof operators called Poisson kernels in [16]. Definition 5.3.
If there exist k, K > h ∈ H ◦ k k ˚ h k ≤ X α ∈ ˜Λ k ˜ C ( E ¯ α ) ∗ ˚ h k = k W ˚ h k ≤ K k ˚ h k , then the ˜Λ − linear system is called (uniformly) observable.We illustrate below that the notion of observability is closely related to thescattering theory notions of noncommutative Markov chains. Observabilityof a system for dim H < ∞ is interpreted as the property of the system thatin the absence of U -inputs we can determine the original state h ∈ H ◦ of thesystem from all Y -outputs at all times. Uniform observability is an analog ofthis for dim H = ∞ . We extend W to c W : ( ˜ H ⊕ H ◦ )(= H ) −→ ˜ H ⊕ ℓ (˜Λ , Y )by defining c W ˜ h := ˜ h for all ˜ h ∈ ˜ H . If W is uniformly observable, then usingˆ k = k and ˆ K = max { , K } the above inequalities can be extended to c W on H as ˆ k k h k ≤ k c W h k ≤ ˆ K k h k for all h ∈ H .Before stating the main theorem of this section regarding observabilitywe recall from [5] the following: Let C be a row contraction on a Hilbert space H C . The lifting E of C is called subisometric [5] if the minimal isometricdilations b V E and b V C of E and C respectively are unitarily equivalent andthe corresponding unitary, which intertwines between b V Ei and b V Ci for all i = 1 , , . . . , d, acts as identity on H C . Some of the techniques used here arefrom the scattering theory of noncommutative Markov chains (cf. [11], [8]).
Theorem 5.4.
For any ˜Λ -linear system associated to a generalized repeatedinteraction model with unitaries U, ˜ U the following statements are equivalent: (a) The system is (uniformly) observable. (b)
The observability operator W is isometric. (c) The tuple E associated with the unitary U is a subisometric lifting ofthe tuple C (associated with the unitary ˜ U ). (d) W : ( H ⊗ K ∞ ) ◦ → ( ˜ H ⊗ K ∞ ) ◦ is unitary.If one of the above holds, then (e) The transfer function Θ U, ˜ U is inner, i.e., M Θ U, ˜ U : ℓ (˜Λ , U ) → ℓ (˜Λ , Y ) is isometric.If we have additional assumptions, viz. dim H < ∞ and dim P ≥ , then theconverse holds, i.e., ( e ) implies all of ( a ) , ( b ) , ( c ) and ( d ) . Proof.
Clearly ( d ) ⇒ ( b ) ⇒ ( a ). We now prove ( a ) ⇒ ( d ). Because the systemis (uniformly) observable there exist k > h ∈ H ◦ k k ˚ h k ≤ k W ˚ h k . Since S m ≥ H⊗K [1 ,m ] is a dense subspace of H⊗K ∞ , for any 0 = η ∈ H⊗K ∞ there exist n ∈ N and η ′ ∈ H ⊗ K [1 ,n ] such that k η − η ′ k < √ k √ k + 1 k η k . Let η ∈ H ⊗ K [1 ,n ] . Suppose U n . . . U η = h ⊗ p ⊗ Ω K [ n +1 , ∞ ) , where h ∈ H , p ∈ P [1 ,n ] . Then clearlylim N →∞ k ˜ U ∗ . . . ˜ U ∗ n ˜ U ∗ n +1 . . . ˜ U ∗ N P N U N . . . U n +1 U n . . . U η k = k c W h kk p k and thus by Proposition 2.1 it is equal to k c W η k . Because the system is(uniformly) observable, k c W h kk p k ≥ √ k k h kk p k . Therefore k c W η k ≥ k k η k . However, in general U n . . . U η = P j h ( j )0 ⊗ p ( j )0 ⊗ Ω K [ n +1 , ∞ ) with h ( j )0 ∈ H and some mutually orthogonal vectors p ( j )0 ∈P [1 ,n ] . By using the above inequality for each term of the summation andthen adding them we find that in general for all η ∈ H ⊗ K [1 ,n ] k c W η k ≥ k k η k . In particular, for η ′ ∈ H ⊗ K [1 ,n ] we have the above inequality. Therefore k c W η k ≥ k c W η ′ k − k c W ( η ′ − η ) k≥ √ k k η ′ k − k η − η ′ k≥ √ k k η k − ( √ k + 1) k η − η ′ k > . This implies c W η = 0 for all 0 = η ∈ H ⊗ K ∞ and hence c W is injective. Re-call that c W is a coisometry and an injective coisometry is unitary. Further,because c W (˜ h ⊗ Ω K∞ ) = ˜ h ⊗ Ω K∞ for all ˜ h ∈ ˜ H it follows that W is unitary.This establishes ( a ) ⇒ ( d ) and we have proved ( a ) ⇔ ( b ) ⇔ ( d ).Next we prove ( d ) ⇔ ( c ). Assume that ( d ) holds. Since W is unitary,clearly c W is unitary. We know that c W intertwines between the minimalisometric dilations b V E and b V C of E and C respectively. Hence E is a subi-sometric lifting of C .Conversely, if we assume ( c ), then by the definition of subisometric lift-ing there exist a unitary operator c W : H ⊗ K ∞ −→ ˜ H ⊗ K ∞ which intertwines between b V E and b V C , and c W acts as an identity on ˜ H⊗ Ω K∞ . To prove W is unitary it is enough to prove c W is unitary. We show that c W = c W . By the definition of the minimal isometric dilation we know thateneralized Repeated Interaction Model and Transfer Functions 19˜ H ⊗ K ∞ = span { b V Cα (˜ h ⊗ Ω K∞ ) : ˜ h ∈ ˜ H , α ∈ ˜Λ } . For j = 1 , . . . , d and ˜ h ∈ ˜ H , by equation (2.2) and Proposition 2.2, c W ∗ b V Cj (˜ h ⊗ Ω K∞ ) = b V Ej c W ∗ (˜ h ⊗ Ω K∞ ) = b V Ej (˜ h ⊗ Ω K∞ )= c W ∗ b V Cj c W (˜ h ⊗ Ω K∞ ) = c W ∗ b V Cj (˜ h ⊗ Ω K∞ ) . Thus c W ∗ = c W ∗ and hence c W = c W .To prove ( d ) ⇒ ( e ) we at first note that since W is unitary, Γ W is alsounitary. By Theorem 4.2, we have M Θ U, ˜ U = Γ W | ℓ (˜Λ , U ) . Since a restriction ofa unitary operator is an isometry, M Θ U, ˜ U is isometric.Finally with the additional assumptions dim H < ∞ and dim P ≥
2, weshow ( e ) ⇒ ( b ). Define H scat := H ∩ c W ∗ ( ˜ H ⊗ K ∞ ) = ˜ H ⊕ { ˚ h ∈ H ◦ : k W ˚ h k = k ˚ h k} . Since k c W h k = lim n →∞ k ˜ U . . . ˜ U n ˜ P n U n . . . U h k by Proposition 2.1, the follow-ing can be easily verified: U ( H scat ⊗ Ω K ) ⊂ H scat ⊗ P . (5.4)Because M Θ U, ˜ U = Γ W | ℓ (˜Λ , U ) is isometric by (e), it can be checked that U ( H ⊗ (Ω K ) ⊥ ) ⊂ H scat ⊗ P . (5.5)Combining equations (5.4) and (5.5) we have U ∗ (( H ⊖ H scat ) ⊗ P ) ⊂ ( H ⊖ H scat ) ⊗ Ω K . Since dim H < ∞ and dim P ≥
2, we obtain
H⊖H scat = { } , i.e., H = H scat .This implies W is isometric and hence ( e ) ⇒ ( b ). (cid:3)
6. Transfer Functions and Characteristic Functions of Liftings
Continuing with the study of our generalized repeated interaction model,from equations (2.1) and (4.4) we obtain b V Cj (˜ h ⊗ Ω K∞ ) = ( C j ˜ h ⊕ D j ˜ h ) ⊗ Ω K [2 , ∞ ) for ˜ h ∈ ˜ H and j = 1 , . . . , d. (6.1)Let D C := ( I − C ∗ C ) : L di =1 ˜ H → L di =1 ˜ H denote the defect operator and D C := Range D C . The full Fock space over C d ( d ≥
2) denoted by F is F = C ⊕ C d ⊕ ( C d ) ⊗ ⊕ · · · ⊕ ( C d ) ⊗ m ⊕ · · · . The vector e ∅ := 1 ⊕ ⊕ · · · is called the vacuum vector. Let { e , . . . , e d } bethe standard orthonormal basis of C d . For α ∈ ˜Λ and | α | = n, e α denote thevector e α ⊗ e α ⊗ · · · ⊗ e α n in the full Fock space F . We recall that Popescu’sconstruction [13] of the minimal isometric dilation ˜ V C = ( ˜ V C , . . . , ˜ V Cd ) on˜ H ⊕ ( F ⊗ D C ) of the tuple C is˜ V Cj (˜ h ⊕ X α ∈ ˜Λ e α ⊗ d α ) = C j ˜ h ⊕ [ e ∅ ⊗ ( D C ) j ˜ h + e j ⊗ X α ∈ ˜Λ e α ⊗ d α ]0 Santanu Dey and Kalpesh J. Hariafor ˜ h ∈ ˜ H and d α ∈ D C where ( D C ) j ˜ h = D C (0 , . . . , ˜ h, . . . ,
0) (˜ h is embeddedat the j th component). So˜ V Cj ˜ h = C j ˜ h ⊕ ( e ∅ ⊗ ( D C ) j ˜ h ) for ˜ h ∈ ˜ H and j = 1 , . . . , d. (6.2)From equations (6.1) and (6.2) it follows that k d X j =1 D j ˜ h j k = k d X j =1 ( D C ) j ˜ h j k (6.3)where ˜ h j ∈ ˜ H for j = 1 , . . . , d . Let Φ C : span { D j ˜ h : ˜ h ∈ ˜ H , j = 1 , . . . , d } →D C be the unitary given byΦ C ( d X j =1 D j ˜ h j ) = d X j =1 ( D C ) j ˜ h j for ˜ h j ∈ ˜ H and j = 1 , . . . , d. Similarly for E i ’s and F i ’s obtained from interaction U in equation (4.2) weset D E := ( I − E ∗ E ) : L di =1 H → L di =1 H and D E := Range D E , anddefine another unitary operator Φ E : span { F j h : h ∈ H , j = 1 , . . . , d } → D E by Φ E ( d X j =1 F j h j ) = d X j =1 ( D E ) j h j for h j ∈ H and j = 1 , . . . , d. The second equation of (4.1) yields d X j =1 D j D ∗ j y = y for y ∈ Y . This implies span { D j ˜ h : ˜ h ∈ ˜ H , j = 1 , . . . , d } = Y . Similarly, we can show that span { F j h : h ∈ H , j = 1 , . . . , d } = U . Thus Φ C is a unitary from Y onto D C and Φ E is a unitary from U onto D E . As aconsequence we have for i, j = 1 , . . . , dD ∗ j D i = ( D C ) ∗ j ( D C ) i = δ ij I − C ∗ j C i , (6.4) F ∗ j F i = ( D E ) ∗ j ( D E ) i = δ ij I − E ∗ j E i . (6.5)Define unitaries ˜ M Φ C : ℓ (˜Λ , Y ) → F ⊗ D C and ˜Φ E : U z ∅ → e ∅ ⊗ D E by˜ M Φ C (cid:0) X α ∈ ˜Λ y α z α (cid:1) := X α ∈ ˜Λ e ¯ α ⊗ Φ C ( y α ) , ˜Φ E ( uz ∅ ) := e ∅ ⊗ Φ E u which would be useful in comparing transfer functions with characteristicfunctions.Define D ∗ ,A := ( I − AA ∗ ) : H ◦ → H ◦ and D ∗ ,A := Range D ∗ ,A . Because E is a coisometric lifting of C, using Theorem 2.1 of [5] we concludeeneralized Repeated Interaction Model and Transfer Functions 21that there exist an isometry γ : D ∗ ,A → D C with γD ∗ ,A h = B ∗ h for all h ∈ H ◦ . Further, for h ∈ H ◦ Φ C ˜ Ch = Φ C d X j =1 D j P ˜ H E ∗ j h = Φ C d X j =1 D j P ˜ H ( B ∗ j h ⊕ A ∗ j h )= Φ C d X j =1 D j B ∗ j h = d X j =1 ( D C ) j B ∗ j h = D C B ∗ h = B ∗ h. The last equality holds because for the coisometric tuple C the operator D C is the projection onto D C and Range B ∗ ⊂ D C . This impliesΦ C ˜ Ch = γD ∗ ,A h. (6.6)The characteristic function M C,E : F ⊗ D E → F ⊗ D C of lifting E of C, which was introduced in [5], and its symbol Θ C,E has the following expansion:For i = 1 , . . . , d and h ∈ ˜ H Θ C,E ( D E ) i h = e ∅ ⊗ [( D C ) i h − γD ∗ ,A B i h ] − X | α |≥ e α ⊗ γD ∗ ,A ( A α ) ∗ B i h, (6.7)and for h ∈ H ◦ Θ C,E ( D E ) i h = − e ∅ ⊗ γD ∗ ,A A i h + d X j =1 e j ⊗ X α e α ⊗ γD ∗ ,A ( A α ) ∗ ( δ ji I − A ∗ j A i ) h. (6.8) Theorem 6.1.
Let U and ˜ U be unitaries associated with a generalized repeatedinteraction model, and the lifting E of C be the corresponding lifting. Thenthe characteristic function M C,E coincides with the transfer function Θ U, ˜ U , i.e., ˜ M Φ C Θ U, ˜ U ( z ) = Θ C,E ˜Φ E . Proof. If h ∈ H and i = 1 , . . . , d, then by equation (4.8)˜ M Φ C Θ U, ˜ U ( z )( F i hz ∅ )= ˜ M Φ C [ ˜ D z ∅ + X β ∈ ˜Λ ,j =1 ,...,d ˜ C ( E ¯ β ) ∗ F ∗ j z βj ]( F i hz ∅ )= ˜ M Φ C [ ˜ DF i h z ∅ + X β ∈ ˜Λ ,j =1 ,...,d ˜ C ( E ¯ β ) ∗ F ∗ j F i h z βj ] . (6.9)2 Santanu Dey and Kalpesh J. HariaCase 1. h ∈ ˜ H :˜ DF i h = d X j =1 D j P ˜ H F ∗ j F i h = d X j =1 D j P ˜ H ( δ ij I − E ∗ j E i ) h = D i h − (cid:0) d X j =1 D j P ˜ H E ∗ j (cid:1) E i h = D i h − ˜ CE i h = D i h − ˜ C ( C i h ⊕ B i h ) = D i h − ˜ CB i h. Second and last equalities follows from equations (6.5) and (4.13) respectively.By equation (6.5) again we obtain X β ∈ ˜Λ ,j =1 ,...,d ˜ C ( E ¯ β ) ∗ F ∗ j F i h z βj = X β ∈ ˜Λ ,j =1 ,...,d ˜ C ( E ¯ β ) ∗ ( δ ij I − E ∗ j E i ) h z βj = X β ∈ ˜Λ ˜ C ( E ¯ β ) ∗ h z βi − X β ∈ ˜Λ ,j =1 ,...,d ˜ C ( E ¯ β ) ∗ E ∗ j E i h z βj = − X β ∈ ˜Λ ,j =1 ,...,d ˜ C ( E ¯ β ) ∗ E ∗ j E i h z βj (because ˜ C ( E ¯ β ) ∗ h = ˜ C ( C ¯ β ) ∗ h = 0 by equation (4.13))= − X β ∈ ˜Λ ,j =1 ,...,d ˜ C ( E ¯ β ) ∗ (cid:0) ( C ∗ j C i + B ∗ j B i ) h ⊕ A ∗ j B i h (cid:1) z βj = − X β ∈ ˜Λ ,j =1 ,...,d ˜ C ( A ¯ β ) ∗ A ∗ j B i h z βj (by equation (4.13))= − X | α |≥ ˜ C ( A ¯ α ) ∗ B i h z α . So by equation (6.9) we have for all i = 1 , . . . , d and h ∈ ˜ H ˜ M Φ C Θ U, ˜ U ( z )( F i hz ∅ )= ˜ M Φ C [( D i h − ˜ CB i h ) z ∅ − X | α |≥ ˜ C ( A ¯ α ) ∗ B i h z α ]= e ∅ ⊗ Φ C ( D i h − ˜ CB i h ) − X | α |≥ e ¯ α ⊗ Φ C ( ˜ C ( A ¯ α ) ∗ B i h )= e ∅ ⊗ [( D C ) i h − γD ∗ ,A B i h ] − X | α |≥ e ¯ α ⊗ γD ∗ ,A ( A ¯ α ) ∗ B i h. By equation (6.7) it follows that˜ M Φ C Θ U, ˜ U ( z )( F i hz ∅ ) = Θ C,E ( e ∅ ⊗ ( D E ) i h )= Θ C,E ˜Φ E ( F i hz ∅ ) . eneralized Repeated Interaction Model and Transfer Functions 23Case 2. h ∈ H ◦ :˜ DF i h = d X j =1 D j P ˜ H F ∗ j F i h = d X j =1 D j P ˜ H ( δ ij I − E ∗ j E i ) h = D i P ˜ H h − (cid:0) d X j =1 D j P ˜ H E ∗ j (cid:1) E i h = − ˜ CA i h Second equality follows from equation (6.5). By equations (6.5) and (4.13)again we obtain X β ∈ ˜Λ ,j =1 ,...,d ˜ C ( E ¯ β ) ∗ F ∗ j F i h z βj = X β ∈ ˜Λ ,j =1 ,...,d ˜ C ( E ¯ β ) ∗ ( δ ij I − E ∗ j E i ) h z βj = X β ∈ ˜Λ ,j =1 ,...,d ˜ C ( A ¯ β ) ∗ ( δ ij I − A ∗ j A i ) h z βj . So by equation (6.9) we have for all i = 1 , . . . , d and h ∈ H ◦ ˜ M Φ C Θ U, ˜ U ( z )( F i hz ∅ )= ˜ M Φ C [ − ˜ CA i h z ∅ + X β ∈ ˜Λ ,j =1 ,...,d ˜ C ( A ¯ β ) ∗ ( δ ij I − A ∗ j A i ) h z βj ]= − e ∅ ⊗ Φ C ( ˜ CA i h ) + X β ∈ ˜Λ ,j =1 ,...,d e j ⊗ e ¯ β ⊗ Φ C ( ˜ C ( A ¯ β ) ∗ ( δ ij I − A ∗ j A i ) h )= − e ∅ ⊗ γD ∗ ,A A i h + X β ∈ ˜Λ ,j =1 ,...,d e j ⊗ e ¯ β ⊗ γD ∗ ,A ( A ¯ β ) ∗ ( δ ij I − A ∗ j A i ) h. By equation (6.8) it follows that˜ M Φ C Θ U, ˜ U ( z )( F i hz ∅ ) = Θ C,E ( e ∅ ⊗ ( D E ) i h )= Θ C,E ˜Φ E ( F i hz ∅ ) . Hence we conclude that˜ M Φ C Θ U, ˜ U ( z ) = Θ C,E ˜Φ E . (cid:3) The transfer function is a notion affiliated with the input/state/outputlinear system, while the scattering function is a notion affiliated with thescattering theory in the sense of Lax-Phillips. For our repeated interactionmodel Theorem 6.1 elucidates that the transfer function is identifiable withthe characteristic function of the associated lifting. This establishes a strongconnection between a model for quantum systems and the multivariate op-erator theory. Connections between them were also endorsed in other workslike [2], [8], [4] and [10], and this indicates that such approaches to quantumsystems using multi-analytic operators are promising.4 Santanu Dey and Kalpesh J. Haria
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The first author received a support from UKIERI to visit Aberystwyth Uni-versity, UK in July 2011 which was helpful for this project.
Santanu DeyDepartment of Mathematics,Indian Institute of Technology Bombay,Powai, Mumbai- 400076, Indiae-mail: [email protected]