CONSERVATIVE DILATIONS OF DISSIPATIVE N-D SYSTEMS: THE COMMUTATIVE AND NON-COMMUTATIVE SETTINGS. Dmitry S. Kalyuzhnyĭ-Verbovetzkiĭ

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1 CONSERVATIVE DILATIONS OF DISSIPATIVE N-D SYSTEMS: THE COMMUTATIVE AND NON-COMMUTATIVE SETTINGS Dmitry S. Kalyuzhnyĭ-Verbovetzkiĭ Department of Mathematics Ben-Gurion University of the Negev P.O. Box Beer-Sheva, Israel ABSTRACT We establish the existence of conservative dilations for various types of dissipative non-commutative N-D systems. As a corollary, the criteria for existence of conservative dilations of corresponding dissipative commutative N-D systems are obtained. We show that such a criterion is fulfilled for all dissipative commutative Givone Roesser systems, however there are also types of commutative N-D systems for which this criterion is fulfilled not for all dissipative systems. New results announced in this short paper appear without proofs, which we intend to present elsewhere. 1. PRELIMINARIES We consider N-dimensional linear systems of the form [ [ x(z) x(z) Σ : = G y(z) z, (1.1) u(z) where x(z), u(z), y(z) are formal power series (FPSs) in N indeterminates (z 1,..., z N ), which commute (in this case we write Σ = Σ c ) or not commute (in this case we write Σ = Σ nc ), with coefficients in separable Hilbert spaces X (the state space), U (the input space), Y (the output space), and G G 0 + N [ Az B G k z k = z k=1 C z D z (1.2) is a linear polynomial with coefficients in the Banach space L(X U, X Y) of bounded linear operators from X U to X Y. We use the convention that the product of two FPSs (in particular, one or both of The author is supported by the Center for Advanced Studies in Mathematics, Ben-Gurion University of the Negev. them may be polynomial) is well defined, i.e., indeterminates formally commute with coefficients. In the commutative case, x(z) = x t z t X[[z 1,..., z N, t Z + N where Z + N := {t ZN : t k 0, k = 1,..., N} and z t := N k=1 zt k k for t = (t 1,..., t N ) Z + N. Similarly for u(z) U[[z 1,..., z N, y(z) Y[[z 1,..., z N. In the non-commutative case, x(z) = x w z w X z 1,..., z N, w F N where F N is the free semigroup with N generators g 1,..., g N and neutral element ; z w := z i1 z im for w = g i1 g im, and z = 1. Similarly for u(z) U z 1,..., z N, y(z) Y z 1,..., z N. In this paper we concentrate on the following three types of N-D systems. KV systems. These are systems (1.1) for which [ G G KV za zb zc zd, (1.3) where za = N k=1 z ka k for A = (A 1,..., A N ) L(X) N (:= L(X, X) N ), and similarly for zb, zc, zd. The corresponding N-D time-domain equations are obtained as equations for the coefficients of FPSs in (1.1) with G G KV Z given in (1.3). In the commutative case, they were considered first in [1 (see also [2, 3, 4). Fornasini Marchesini (FM) systems. These are systems (1.1) for which [ G G FM za zb, (1.4)

2 where C and D are constant. The corresponding N- D time-domain equations have appeared first in the commutative case in [5, and in the non-commutative case in [6. Givone Roesser (GR) systems. These are a special case of FM systems where G G GR [ zp 0 0 I Y [ A B, (1.5) with a constant operator-block matrix [ A B G = L(X U, X Y), (1.6) and zp := N k=1 z kp k (here P = (P 1,..., P N ) L(X) N is an N-tuple of the orthogonal projections P k onto the subspaces X k X such that X = N k=1 X k). The corresponding N-D time-domain equations have appeared first in the commutative case in [7, 8, and in the non-commutative case in [6. If A z in (1.2) is linear homogeneous, i.e., A za (which is the case for the three types of systems above), then it follows from system equations (1.1) that where y(z) = T Σ (z)u(z), T Σ (z) = D z + C z A j zb z j=0 is a FPS (commutative or non-commutative, depending on the setting) with coefficients in L(U, Y), which is called the transfer function of the system Σ. We will use a notation Σ = (N; G z ; X, U, Y). 2. DISSIPATIVE AND CONSERVATIVE N-D SYSTEMS A commutative system Σ c = (N; G z ; X, U, Y) will be said to be dissipative (resp., conservative) if for every ζ T N := {z C N : z k = 1, k = 1,..., N} (the N-dimensional unit torus) the operator G ζ is contractive (resp., unitary). It is clear that a commutative GR system Σ GR,c is dissipative (resp., conservative) if and only if the operator G in (1.6) is contractive (resp., unitary). The class B N (U, Y) consists of commutative FPSs which become holomorphic functions on the open unit polydisk D N := {z C N : z k < 1, k = 1,..., N} and take therein contractive values from L(U, Y). Its subclass B 0 N (U, Y) consists of FPSs f(z) B N(U, Y) satisfying f 0 = 0. Proposition 2.1 The transfer function T Σ c(z) of a dissipative commutative system Σ c = (N; G z ; X, U, Y) belongs to the class B N (U, Y) for the cases Σ c = Σ FM,c and Σ c = Σ GR,c, and to the class B 0 N (U, Y) for the case Σ c = Σ KV,c. For the case Σ c = Σ KV,c Proposition 2.1 has been proved in [1, Theorem 2.3. Recall [9 that the Schur Agler class SA N (U, Y) consists of FPSs f(z) L(U, Y)[[z 1,..., z N which become holomorphic L(U, Y)-valued functions on D N satisfying the property that for an arbitrary N-tuple T = (T 1,..., T N ) of commuting strict contractions on a common separable Hilbert space H (i.e., T k < 1, k = 1,..., N) the series f(t) = f t T t t Z N + converges in the operator norm topology to a contractive operator f(t) L(U H, Y H). Its subclass SA 0 N(U, Y) consists of FPSs f(z) SA N (U, Y) satisfying f 0 = 0. It is clear that SA N (U, Y) B N (U, Y) and SA 0 N(U, Y) B 0 N (U, Y). It is known due to [10, 11 that for the cases N = 1, 2 one has actually equality in these inclusions, however [12 implies that for N > 2 these inclusions are proper. Let us note that the Schur Agler class was extended to a wider class of domains in [13, 14. Proposition 2.2 The class of transfer functions of conservative commutative systems Σ c with the input space U and the output space Y for the cases Σ c = Σ FM,c and Σ c = Σ GR,c coincides with SA N (U, Y), and for the case Σ c = Σ KV,c with SA 0 N(U, Y). For the case of GR systems Proposition 2.2 has been proved in [9 (in its present form it can be found in [15). For the case of FM systems it follows then from the relationship between commutative conservative FM and GR systems established in [4. For the case of KV systems it has been proved in [1, Theorem 3.2. It has been shown in [3 that the class of transfer functions of dissipative commutative KV systems Σ nc with the input space U and the output space Y is wider than SA 0 N(U, Y), however the question on whether this class coincides with B 0 N (U, Y) or is a proper subclass of the latter is still open. It can be easily shown that the situation for the case of dissipative commutative FM systems is analogous, with the classes SA 0 N(U, Y) and B 0 N (U, Y) replaced by SA N(U, Y) and B N (U, Y), respectively. In this paper we will show however that

3 the class of transfer functions of dissipative commutative GR systems with the input space U and the output space Y coincides with SA N (U, Y). Let us define some classes of N-tuples of (generally speaking, non-commuting) operators. The class U N (resp., D N ) consists of N-tuples of unitary (resp., strictly contractive) operators on a common separable Hilbert space. The subclass U N matr U N (resp., D N matr D N ) consists of N-tuples of unitary (resp., strictly contractive) n n matrices, n = 1, 2,.... The classes D N and D N matr are non-commutative analogues of D N, the classes U N and U N matr are non-commutative analogues of T N. We shall call a non-commutative system Σ nc = (N; G z ; X, U, Y) dissipative (resp., conservative) if for every U = (U 1,..., U N ) U N ( L(H) N ), G U := G 0 I H + N G k U k k=1 L((X U) H, (X Y) H) is a contractive (resp., unitary) operator. Let us remark that we can consider the same collection of data Σ = (N; G z ; X, U, Y) both for the commutative system Σ c and for the non-commutative system Σ nc. Then [1, Proposition 2.4 and [16, Proposition 2.1 imply that Σ nc is conservative if and only if Σ c is conservative. It is obvious that if Σ nc is dissipative then so is Σ c. The opposite is true for N = 1, 2 and also in the case of GR systems. It follows from the main result of [17 that for N > 2 in the case of KV and FM systems the dissipativity of Σ c, in general, doesn t imply the dissipativity of Σ nc. Recall that the non-commutative Schur Agler class SA nc N (U, Y) has been defined in [18 as the class of FPSs f(z) L(U, Y) z 1,..., z N which satisfy the condition that for an arbitrary T = (T 1,..., T N ) D N ( L(H) N ) the series f(t) = w F N f w T w converges in the operator norm topology to a contractive operator f(t) L(U H, Y H). It has been shown in [19 that in order that f(z) SA nc N (U, Y) it suffices to verify the convergence and the property f(t) 1 only for all T D N matr. The subclass (U, Y) consists of FPSs f(z) SAnc N (U, Y) satisfying f = 0. Let us note that the non-commutative Schur Agler class has been defined in [18 for more general non-commutative domains than D N. SA nc,0 N Proposition 2.3 The transfer function T Σ nc(z) of an arbitrary dissipative non-commutative system Σ nc = (N; G z ; X, U, Y) belongs to the class SA nc N (U, Y) for the cases Σ nc = Σ FM,nc and Σ nc = Σ GR,nc, and to the class SA nc,0 N (U, Y) for the case Σnc = Σ KV,nc. For the case of non-commutative GR systems Proposition 2.3 has been proved in [18, Theorem 4.2. Proposition 2.4 The class of transfer functions of conservative non-commutative systems Σ nc with the input space U and the output space Y for the cases Σ nc = Σ FM,nc and Σ nc = Σ GR,nc coincides with SA nc N (U, Y), and for the case Σ nc = Σ KV,nc with SA nc,0 (U, Y). N For the case of non-commutative GR systems Proposition 2.4 has been proved in [18, Theorem 5.3. The two last propositions above imply that the classes of transfer functions for dissipative non-commutative KV, FM or GR systems are the same as for conservative ones. 3. CONSERVATIVE DILATIONS OF DISSIPATIVE N-D SYSTEMS We will say that the non-commutative system Σ nc = (N; G z ; X, U, Y) is a dilation of the non-commutative system Σ nc = (N; G z ; X, U, Y) if D D z and for every n Z + \ {0} and Z (C n n ) N there exist subspaces D Z and D,Z in X C n such that X C n = D Z (X C n ) D,Z, (3.1) Ã Z D Z D Z, CZ D Z = {0}, Ã ZD,Z D,Z, B Z D,Z = {0}, (3.2) A Z = (P X I n )ÃZ X C n, C Z = C Z X C n, B Z = (P X I n ) B Z. (3.3) For dilations of non-commutative GR systems we require additionally that D Z = N k=1 D Z,k, D,Z = N k=1 D,Z,k, and (3.1) is replaced by a stronger condition X k C n = D Z,k (X k C n ) D,Z,k, k = 1,..., N. (3.4) The dilation Σ nc of a system Σ nc is called uniform if the subspaces D Z and D,Z are independent of Z and have the form D Z = D C n, D,Z = D C n (in the case of GR systems, D Z,k = D k C n, D,Z,k = D,k C n, k = 1,..., N).

4 Proposition 3.1 A system Σ nc is a dilation of a system Σ nc if and only if Σ nc is a uniform dilation of Σ nc. Proposition 3.2 The transfer functions T Σ nc(z) and T Σnc(z) of a system Σ nc and of its dilation Σ nc, respectively, coincide. The main result of the present paper is the following. Theorem 3.3 Every dissipative non-commutative KV, FM or GR system has a (uniform) conservative dilation. We will say that the commutative system Σ c = (N; G z ; X, U, Y) is a dilation of the commutative system Σ c = (N; G z ; X, U, Y) if D D z, and all other properties of dilations from the case of non-commutative systems are fulfilled for n = 1 and not necessarily fulfilled for n > 1. It is clear that if Σ nc is a dilation of Σ nc then, correspondingly, Σ c is a dilation of Σ c, however the opposite is, in general, not true. The dilation Σ c of a commutative system Σ c is called uniform if the subspaces D Z and D,Z are independent of Z C N and have the form D Z = D, D,Z = D (in the case of GR systems, D Z,k = D k, D,Z,k = D,k, k = 1,..., N). Thus, we can say that if Σ nc is a (uniform) dilation of Σ nc then, correspondingly, Σ c is a uniform dilation of Σ c. Proposition 3.4 The transfer functions T Σ c(z) and T Σc(z) of a system Σ c and of its dilation Σ c, respectively, coincide. From Theorem 3.3 we deduce the following criterion for existence of a uniform conservative dilation of a dissipative commutative system. Theorem 3.5 A dissipative commutative system Σ c = (N; G z ; X, U, Y) has a uniform conservative dilation if and only if for the cases Σ c = Σ FM,c and Σ c = Σ GR,c one has G z SA N (X U, X Y), and for the case Σ c = Σ KV,c one has G z SA 0 N(X U, X Y). In the case of commutative KV systems, Theorem 3.5 improves the result of [2, Theorem 4.2, a criterion for existence of (not necessarily uniform) conservative dilations. More precisely, the criterion in Theorem 3.5 is formulated in the same way as one in [2, Theorem 4.2, however additionally in this case the existence of a uniform conservative dilation is established. For the case of commutative GR systems the criterion from Theorem 3.5 is always fulfilled, i.e., every dissipative commutative GR system has a uniform conservative dilation. In particular, this implies the following result. Theorem 3.6 The class of transfer functions of dissipative commutative GR systems with the input space U and the output space Y coincides with SA N (U, Y). 4. OPEN QUESTIONS 1. It is of interest to know whether the main results of this paper, Theorems 3.3 and 3.5, are true for general dissipative N-D systems of the form (1.1), with our definitions of dissipative and conservative N-D systems. 2. The latter notions are related with the polydisk D N in the commutative setting, and with the noncommutative polydisk D N in the non-commutative setting, or respectively with their essential boundaries T N and U N. However, there exist other possibilities to define dissipative and conservative N-D systems. For instance, structured non-commutative systems from [6 also can be written in the form (1.1), however the conditions of dissipativity or conservativity for them which appear in [18 are related to other non-commutative domains. (Only in the case of GR systems, such a structured dissipative or conservative non-commutative system is related to D N.) It would be useful to develop a theory of conservative dilations for these cases, too. 5. REFERENCES [1 D. S. Kalyuzhniy, Multiparametric dissipative linear stationary dynamical scattering systems: discrete case, J. Operator Theory, vol. 43, no. 2, pp , [2 D. S. Kalyuzhniy, Multiparametric dissipative linear stationary dynamical scattering systems: discrete case. II. Existence of conservative dilations, Integral Equations Operator Theory, vol. 36, no. 1, pp , [3 D. S. Kalyuzhniy-Verbovetzky, Does any analytic contractive operator function on the polydisk have a dissipative scattering nd realization?, in Unsolved problems in mathematical systems and control theory, Vincent D. Blondel and Alexandre Megretski, Eds., pp Princeton University Press, Princeton, NJ, [4 J. A. Ball, C. Sadosky, and V. Vinnikov, Conservative input-state-output systems with evolution on a multidimensional integer lattice, To appear in Multidimens. Syst. Signal Process. [5 E. Fornasini and G. Marchesini, Doublyindexed dynamical systems: state-space models

5 and structural properties, Math. Systems Theory, vol. 12, no. 1, pp , 1978/79. [6 J. A. Ball, G. Groenewald, and T. Malakorn, Structured noncommutative multidimensional linear systems, Preprint. [7 D. D. Givone and R. P. Roesser, Multidimensional linear iterative circuits general properties, IEEE Trans. Computers, vol. C-21, pp , [8 D. D. Givone and R. P. Roesser, Minimization of multidimensional linear iterative circuits, IEEE Trans. Computers, vol. C-22, pp , [17 D. S. Kalyuzhnyĭ, The von Neumann inequality for linear matrix functions of several variables, Mat. Zametki, vol. 64, no. 2, pp , 1998, (Russian); translation in Math. Notes 64 (1998), no. 1-2, (1999). [18 J. A. Ball, G. Groenewald, and T. Malakorn, Conservative structured noncommutative multidimensional linear systems, To appear in Oper. Theory Adv. Appl. [19 D. Alpay and D. S. Kalyuzhnyĭ-Verbovetzkiĭ, Matrix-J-unitary non-commutative rational formal power series, To appear in Oper. Theory Adv. Appl. [9 J. Agler, On the representation of certain holomorphic functions defined on a polydisc, in Topics in operator theory: Ernst D. Hellinger memorial volume, vol. 48 of Oper. Theory Adv. Appl., pp Birkhäuser, Basel, [10 J. von Neumann, Eine Spektraltheorie für allgemeine Operatoren eines unitären Raumes, Math. Nachr., vol. 4, pp , [11 T. Andô, On a pair of commutative contractions, Acta Sci. Math. (Szeged), vol. 24, pp , [12 N. Th. Varopoulos, On an inequality of von Neumann and an application of the metric theory of tensor products to operators theory, J. Functional Analysis, vol. 16, pp , [13 C.-G. Ambrozie and D. Timotin, A von Neumann type inequality for certain domains in C n, Proc. Amer. Math. Soc., vol. 131, no. 3, pp (electronic), [14 J. A. Ball and V. Bolotnikov, Realization and interpolation for Schur-Agler-class functions on domains with matrix polynomial defining function in C n, J. Funct. Anal., vol. 213, no. 1, pp , [15 J. A. Ball and T. T. Trent, Unitary colligations, reproducing kernel Hilbert spaces, and Nevanlinna-Pick interpolation in several variables, J. Funct. Anal., vol. 157, no. 1, pp. 1 61, [16 D. S. Kalyuzhnyĭ-Verbovetzkiĭ, Carathéodory interpolation on the non-commutative polydisk, To appear in J. Funct. Anal.