Isometric Expansions of Quantum Algebra of Linear Bounded Operators

Isometric Expansions of Quantum Algebra of Linear Bounded Operators

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Автор: Zolotarev, V.A.
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Опубліковано: Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України 2006
Назва видання:Журнал математической физики, анализа, геометрии
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Цитувати:Isometric Expansions of Quantum Algebra of Linear Bounded Operators / V.A. Zolotarev // Журнал математической физики, анализа, геометрии. — 2006. — Т. 2, № 2. — С. 207-224. — Бібліогр.: 11 назв. — англ.

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spelling irk-123456789-1065922016-10-01T03:02:21Z Isometric Expansions of Quantum Algebra of Linear Bounded Operators Zolotarev, V.A. 2006 Article Isometric Expansions of Quantum Algebra of Linear Bounded Operators / V.A. Zolotarev // Журнал математической физики, анализа, геометрии. — 2006. — Т. 2, № 2. — С. 207-224. — Бібліогр.: 11 назв. — англ. 1812-9471 http://dspace.nbuv.gov.ua/handle/123456789/106592 en Журнал математической физики, анализа, геометрии Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України
institution Digital Library of Periodicals of National Academy of Sciences of Ukraine
collection DSpace DC
language English
format Article
author Zolotarev, V.A.
spellingShingle Zolotarev, V.A.
Isometric Expansions of Quantum Algebra of Linear Bounded Operators
Журнал математической физики, анализа, геометрии
author_facet Zolotarev, V.A.
author_sort Zolotarev, V.A.
title Isometric Expansions of Quantum Algebra of Linear Bounded Operators
title_short Isometric Expansions of Quantum Algebra of Linear Bounded Operators
title_full Isometric Expansions of Quantum Algebra of Linear Bounded Operators
title_fullStr Isometric Expansions of Quantum Algebra of Linear Bounded Operators
title_full_unstemmed Isometric Expansions of Quantum Algebra of Linear Bounded Operators
title_sort isometric expansions of quantum algebra of linear bounded operators
publisher Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України
publishDate 2006
url http://dspace.nbuv.gov.ua/handle/123456789/106592
citation_txt Isometric Expansions of Quantum Algebra of Linear Bounded Operators / V.A. Zolotarev // Журнал математической физики, анализа, геометрии. — 2006. — Т. 2, № 2. — С. 207-224. — Бібліогр.: 11 назв. — англ.
series Журнал математической физики, анализа, геометрии
work_keys_str_mv AT zolotarevva isometricexpansionsofquantumalgebraoflinearboundedoperators
first_indexed 2025-07-07T18:44:34Z
last_indexed 2025-07-07T18:44:34Z
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fulltext Journal of Mathematical Physics, Analysis, Geometry 2006, vol. 2, No. 2, pp. 207�224 Isometric Expansions of Quantum Algebra of Linear Bounded Operators V.A. Zolotarev Department of Mechanics and Mathematics, V.N. Karazin Kharkov National University 4 Svobody Sq., Kharkov, 61077, Ukraine E-mail:Vladimir.A.Zolotarev@univer.kharkov.ua Received March 29, 2005 The isometric expansion � Vs; + Vs �2 s=1 for a quantum algebra of linear bounded nonunitary operators fT1; T2g, which is given by commutative re- lation T1T2 = qT2T1 (jqj = 1), is constructed. Basic properties of charac- teristic function S(z) corresponding to isometric expansion � Vs; + Vs �2 s=1 for given quantum algebra fT1; T2g are described. Key words: isometric expansion, quantum algebra, the Rie�el torus. Mathematics Subject Classi�cation 2000: 47A45. Spectral decompositions of the selfadjoint operators have its origin in quantum mechanics and are among the major achievements of the functional analysis. They play the key role in many �elds of mathematics and physics. It is common to consider functional models [3, 4] as an analogue of such spectral decompositions for non-selfadjoint and nonunitary operators. Construction of these models is based on the theory of isometric and unitary expansions (dilations) of the opera- tor semigroups. Given �eld of analysis is well investigated and has a number of nontrivial and substantial applications. Solution of the problem of construction of the spectral decomposition for the linear selfadjoint operator systems fAkg n 1 satisfying certain commutative relations (e.g. that of the Lie algebra), that also emerged in quantum mechanics, stimulated mutually enriching development of di�erent �elds of mathematics, [1]. For the commutative systems of linear non- selfadjoint operators fAkg n 1 , the construction of the functional models is based [5, 6] on the basic idea of M.S. Liv�sic [5] who showed that construction of the isometric (unitary) expansions of the semigroups for such operator systems is based on the study of the consistency conditions for the systems of di�erential equations. Given consistency conditions signify closedness of the di�erential forms c V.A. Zolotarev, 2006 V.A. Zolotarev and are written in terms of the external parameters of the expansion. The author [7] has successfully generalized this approach for the Lie algebras of the linear non-selfadjoint operators fAkg n 1 . This fact led to the problems of the harmonic analysis on the Lie groups and also to the noncommutative Lax�Fillips scattering scheme on groups. At the same time, the important connections with theory of functions on the Riemann surface have been established. The formation [8] of the similar constructions for the commutative systems of linear operators fTkg n 1 that are close to the unitary ones (e.g. contractions) allowed to �nd the constructive approach to the solution of the problem (which is more than 30 years old) of the construction of the unitary dilation and of the corresponding functional model for the commutative system of the contractive operatorsfTkg n 1 , [3]. The search for the sensible discrete noncommutative analogue of the Lie algebras for the operator systems fTkg n 1 that are close to the unitary ones has led the author to the quantum operator algebras [10, 11]. In this work, the simplest quantum algebra of the linear operators fT1; T2g, that represent the so-called Rie�el torus [9�11], the commutative relation for which is T1T2 = qT2T1, q 2 C , jqj = 1, is studied. Really, if q = eih, h 2 R, and T2 = exp fhA2g then the corresponding classical limit process in the relation T1T2 = qT2T1 when q ! 1, h! 0, leads us to the Lie algebra of the linear operators [A1; A2] = iA1 (T1 def = A1) which represents the Lie algebra of a�ne line transformations [7]. Thus, the quantum algebra fT1; T2g, T1T2 = qT2T1 is nothing else than the quanti�cation of the operator Lie algebra fA1; A2g, [A1; A2] = iA1 [11]. In this work following the constructions [8], the construction of the isometric expansion� Vs; + Vs �2 s=1 for the quantum algebra of linear bounded nonunitary operators fT1; T2g given by the relation T1T2 = qT2T1 is o�ered. In Part 3, the totality of the invariants of the given quantum operator algebra is presented. 1. The Preliminary Information I. An arbitrary linear bounded operator T acting in the Hilbert space H has the isometric (in the inde�nite, generally speaking, metric) expansion [3, 4, 8]. This fact means that there exist the Hilbert spaces E and ~E and such operators � : E ! H, : H ! ~E, K : E ! ~E that the expansion operator VT = � T � K � : H �E ! H � ~E (1) has properties V � T � I 0 0 ~� � VT = � I 0 0 � � ; VT � I 0 0 ��1 � V � T = � I 0 0 ~��1 � ; (2) 208 Journal of Mathematical Physics, Analysis, Geometry, 2006, vol. 2, No. 2 Isometric Expansions of Quantum Algebra of Linear Bounded Operators where � and ~� are the selfadjoint boundedly invertible operators in E and ~E respectively. The expansion VT (1) is built by the operator T in an ambiguous way. Consider one of the methods of the construction of such an expansion which we will need later on [4, 8]. Let D = T �T � I and ~D = TT � � I be the de�cient operators [3, 4, 8] corresponding to the operator T , and E = ~DH, ~E = DH be respective de�cient subspaces. De�ne operators: � = ~D : E ! H; = P ~E : H ! ~E is an orthoprojector in H on ~E; K = T �: E ! ~E; and, �nally, � = � ~D, ~� = �D, [4]. It is easy to see that the operator VT (1) constructed in such a manner will satisfy relations (2). II. Denote by hn 2 H, un 2 E, vn 2 ~E the vector-functions of discrete argument n 2 Z+ = fn 2 Z : n � 0g in the respective Hilbert spaces. Further, consider the equation system which is commonly [4] known as the open expansion system VT (1),� hn+1 = Thn +�un; h0 = h; vn = hn +Kun; n 2 Z+; VT � hn un � = � hn+1 vn � : (3) The following conservation law results from the �rst relation in (2): khnk 2 + h�un; uni = khn+1k 2 + h~�vn; vni : (4) Note that if un � 0 then hn is generated by the semigroup Tn = T n of the discrete argument n 2 Z+, i.e. hn = Tnh and vn = Tnh. Now consider the vector-functions ~hn 2 H, ~un 2 E, ~vn 2 ~E of the argument n 2 Z� = fn 2 Z : n < 0g and specify the dual open system (regarding (3)) that is generated by the operator V � T ,� ~hn�1 = T �~hn + �~vn; ~h�1 = ~h; ~un = ��~hn +K�~vn; n 2 Z�; V � T � ~hn ~vn � = � ~hn�1 ~un � : (5) Then, if ~vn � 0 then the vector-function ~hn�1 has the form of ~hn�1 = T � jnj ~h and is generated by the semigroup T � jnj = (T �)jnj where (�n) 2 Z+ and ~un�1 = ��T � jnj ~h. The conservation law for system (5) has the former type (4) if hn = ~h�n�1, vn = ~��1v�n�1, un = ��1~u�n�1 (n 2 Z+). Let un = znu0 (z 2 C , u0 2 E) and assume that hn and vn depends on n 2 Z+, similarly hn = znh0, vn = znv0. Then from equations (3) for the open system, we obtain that h0 = (zI � T1) �1�u0, v0 = S(z)u0, where S(z) = S�(z) = K + (zI � T )�1� (6) is the characteristic function of M.S. Liv�sic of the expansion VT (1) [4, 8], which is de�ned for all z outside the spectrum of operator T . Journal of Mathematical Physics, Analysis, Geometry, 2006, vol. 2, No. 2 209 V.A. Zolotarev Study of the isometric expansions VT (1) of the open systems (3), (5) plays the fundamental role in the construction of the unitary dilation U of the contraction T (kTk � 1) [3, 4], and also in the construction of the functional models U and T [3, 4]. Sensible generalization of the constructions mentioned above on the case of the commutative systems of linear nonunitary operators fTkg n 1 , [Tk; Ts] = 0, 1 � k, s � n, was proposed in work [8]. So, in [8] the commutative isometric expansion for the system of the commutative linear operators has been built. It turned out that the construction of the isometric expansion stated in [8] has natural generalization on the quantum algebra of linear operators (the Rie�el torus). 2. Isometric Expansions and Open Systems I. In the Hilbert space H, consider the system of the linear bounded operators fT1; T2g satisfying the relation T1T2 = qT2T1; q = eih; h 2 R: (7) In the same way as for the commutative operator systems q = 1 [8], de�ne the isometric expansion corresponding to the case of q 6= 1 that generalizes the concept of the expansion VT (1). De�nition 1. The totality of the mappings Vs = � Ts �Ns K � : H �E ! H � ~E; s = 1; 2; + Vs= � T �s � ~N� s �� K� � : H � ~E ! H �E; s = 1; 2; (8) is said to be the isometric expansion of the system of linear bounded operators fT1; T2g in H satisfying (7) if there are such operators �s, �s, Ns, � and ~�s, ~�s, ~Ns, ~� in the Hilbert spaces E and ~E respectively (�s, �s and ~�s, ~�s are selfadjoint (s = 1, 2)) that equalities 1) V � s � I 0 0 ~�s � Vs = � I 0 0 �s � ; s = 1; 2; 2) + V � s � I 0 0 �s � + Vs= � I 0 0 �s � ; s = 1; 2; 3) qT2�N1 � T1�N2 = q��; ~N1 T2 � q ~N2 T1 = q~� ; 4) q ~N2 �N1 � ~N1 �N2 = qK�� q~�K; 5) KNs = ~NsK; s = 1; 2; (9) 210 Journal of Mathematical Physics, Analysis, Geometry, 2006, vol. 2, No. 2 Isometric Expansions of Quantum Algebra of Linear Bounded Operators are true. First of all, show that there exists such an expansion Vs, + Vs (8) for every operator system fT1; T2g satisfying (7). Let Ds = T �s Ts � I and ~Ds = TsT � s � I be the defect operators (s = 1; 2). Consider the operators N1 = ~D1T � 2 ; N2 = q ~D2T � 1 ; ~N1 = qT �2D1; ~N2 = T �1D2; ~�1 = �D1; ~�2 = �D2; �1 = �T2 ~D1T � 2 ; �2 = �T1 ~D2T � 1 ; �1 = � ~D1; �2 = � ~D2; ~�1 = �T �2D1T2; ~�2 = �T �1D2T1; � = ~D1 � ~D2; ~� = D1 �D2; K = T �1 T � 2 : (10) Further, de�ne the Hilbert spaces E = span n ~DkH +N� sH : k; s = 1; 2 o ; ~E = span n DkH + ~NsH : k; s = 1; 2 o : (11) It is easy to see that the operators Vs = � Ts PENs P ~E K � ; + VS= � T �s P ~E ~N� s PE K� � ; s = 1; 2; (12) satisfy the conditions 1), 2) (9). Relations 3) (9) follow from the identities T2 ~D1T � 2 � T1 ~D2T � 1 = ~D1 � ~D2; T �2D1T2 � T �1D2T1 = D1 �D2; (13) that are easily veri�ed if one takes into account (7) and the fact that jqj = 1. Relations 4), 5) in (9) are veri�ed directly in view of equality (7). The inclusion KE � ~E is an obvious corollary of the relations T �1 T � 2 ~D1H = qT �2D1T � 1H � ~N1T � 1H � ~E; T �1 T � 2N � 1H = D1T � 1H + ~N2 ~D1H � E and of relations similar to them. O b s e r v a t i o n 1. The external parameters f�s; �s; Ns;�g in E andn ~�s; ~�s; ~Ns; ~� o in ~E of the expansion Vs, + Vs (8) are not independent. So from (10) and (13), it follows that �2 � �1 = �2 � �1 = �; ~�2 � ~�1 = ~�2 � ~�1 = ~�: From relation 1) (9) and from the presentation of Vs, + Vs (8), it follows that, in the case of invertibility of �s and ~�s, operators �s and ~�s have the form of Journal of Mathematical Physics, Analysis, Geometry, 2006, vol. 2, No. 2 211 V.A. Zolotarev �s = Ns� �1 s N� s and ~�s = ~Ns~� �1 s ~N� s , s = 1; 2, but, as can be seen from (10), the invertibility of the operators �s and ~�s, generally speaking, is not taking place. O b s e r v a t i o n 2. It is obvious that the following relation is true:� I 0 0 ~Ns � Vs = + V � s � I 0 0 Ns � ; s = 1; 2; (14) for the operators Vs and + Vs (8). II. De�ne the vector-functions of discrete argument hn 2 H, un 2 E, vn 2 ~E at the points of integer-valued grid n = (n1; n2) 2 Z 2 + = Z+ � Z+. Consider the di�erential linear operators @1 ø @2 that on the vector-function hn act in the following way: @1hn = h(n1+1;n2); @2hn = q�n1h(n1;n2+1); (15) besides it is obvious that q@1@2 = @2@1: (16) Further, de�ne the analogue with two variables of the open system (3)8< : @1hn = T1hn +�N1un; h(0;0) = h0; @2hn = T2hn +�N2un; n 2 Z2 +; vn = hn +Kun; Vs � hn un � = � @shn vn � ; s = 1; 2; (17) where f@1; @2g have the form of (15). Using relations (7) and (16), we obtain the following statement. Theorem 1. The equation system (17) is consistent if the vector-function un is the solution of the equation fqN2@1 �N1@2 + q�gun = 0: (18) The proof of the theorem easily follows from (7), (16) taking into account the �rst relation 3) in (9). Theorem 2. Suppose that un is the solution of equation (18) and the vector- functions hn and un are de�ned by relations (17), then vn satis�es the following equation n q ~N2@1 � ~N1@2 + q~� o vn = 0: (19) P r o o f. Really, from (17) and 3)�5) (9), it follows thatn ~N1@2 � q ~N2@1 o vn = � ~N1 T2 � q ~N2 T1 � h+ � ~N1 �N2 � q ~N2 �N1 � u 212 Journal of Mathematical Physics, Analysis, Geometry, 2006, vol. 2, No. 2 Isometric Expansions of Quantum Algebra of Linear Bounded Operators +K (N1@2 � qN2@1)u = q~� h+ q(~�K �K�)u+ qK�u = q~�v; this fact ends the proof. � Calculate the expression @2@1hn = T1T2hn + T1�N2un +�N1@2un = qT2T1hn + qT2�N1un � q��un +�N1@2u = q fT2T1hn + T2�N1un +�N2@1ung = q@1@2hn in view of relations (7), 3) (9) and equation (18), and thus k@2@1hnk = k@1@2hnk since jqj = 1. Using the isometric property 1) (9) of the expansions Vs (8), it is easy to see that k@2@1hnk 2 + h~�2@1vn; @1vni+ h~�1vn; vni = khnk 2 + h�2@1un; @1uni + h�1un; uni ; k@1@2hnk 2 + h~�1@2vn; @2vni+ h~�2vn; vni = khnk 2 + h�1@2un; @2uni + h�2un; uni : (20) Therefore similarly to (4), the following conservation laws are true for the open system (17): 1) k@shnk 2 + h~�svn; vni = khnk 2 + h�sun; uni ; s = 1; 2; 2) h(~�1 � ~�2) vn; vni+ h~�2@1vn; @1vni � h~�1@2vn; @2vni = h(�1 � �2) un; uni+ h�2@1un; @1uni � h�1@2un; @2uni : (21) Note that equality 2) (21) follows from (20) after the subtraction and will play the important role hereinafter. III. The equation system (17) corresponding to the operators Vs describes �the dynamics� of the outgoing waves that are de�ned on Z2 +. To study the dual situation, consider the vector-functions ~hn 2 H, ~un 2 E, ~vn 2 ~E at the points of the integer-valued grid n = (n1; n2) 2 Z2 � = Z� � Z� De�ne the di�erential linear operators ~@1 and ~@2 that act on the function ~hn in the following way: ~@1~hn = ~h(n1�1;n2); ~@2~hn = qn1~h(n1;n2�1); (22) besides, as is easy to see, q ~@1 ~@2 = ~@2 ~@1: (23) Similarly to (5), consider the analogue with two variables of the dual open system8< : ~@1~hn = T �1 ~hn + � ~N� 1 ~vn; h(�1;�1) = ~h�1; ~@2~hn = T �2 ~hn + � ~N� 2 ~vn; n 2 Z2 �; ~un = ��~hn +K�~vn; + Vs � ~hn ~vn � = � ~@s~hn ~un � ; s = 1; 2; (24) Journal of Mathematical Physics, Analysis, Geometry, 2006, vol. 2, No. 2 213 V.A. Zolotarev and the operators ~@1, ~@2 are de�ned by formulas (23). For the equation system (24), the statements similar to Theorems 1 and 2 are true. Theorem 3. The consistency of the equation system (24) for ~hn takes place if the function ~vn satis�es the equationn q ~N� 2 ~@1 � ~N� 1 ~@2 + ~�� o ~vn = 0: (25) Theorem 4. The vector-function ~un (24) is the solution of the di�erence equa- tion n qN� 2 ~@1 �N� 1 ~@2 + �� o ~un = 0 (26) if ~vn satis�es equation (25) and ~hn is the solution of system (24). As in the case of system (17), in this case ~@2 ~@1~hn = ~@1 ~@2hn . Taking into account the isometric property 2) (9), it is easy to show that the conservation laws 1) ~@s~hn 2 + h�s~un; ~uni = ~hn 2 + h~�s~vn; ~vni ; s = 1; 2; 2) h(�1 � �2) ~un; ~uni+ D �2 ~@1~un; ~@1~un E � D �1 ~@2~un; ~@2~un E = h(~�1 � ~�2) ~vn; ~vni+ D ~�2 ~@1~vn; ~@1~vn E � D ~�1 ~@2~vn; ~@2~vn E (27) take place. As it will be shown later, the relations of isometric nature 2) (28) and 2) (27) result in the nontrivial relations for the characteristic function S(z) of the operator T1. 3. The Basic Properties of the Characteristic Function I. Suppose the vector-functions hn, un, vn from the equations of the open system (17) have the form of hn = zn1hn2 , un = zn1un2 , vn = zn1un2 where n = (n1; n2) 2 Z2 + and z 2 C . Then from the �rst equation in (17) and the last one in (17) it follows that vn2 = S(z)un2 ; where S(z) = K + (zI � T1) �1 �N1 is the characteristic function (6) of the expansion V1 (8) of the operator T1. Theorem 5. Suppose that the operators N1 and ~N1 of the isometric expansion� Vs; + Vs �2 s=1 (8) are such that N�1 1 and ~N�1 1 exist and are bounded. Then the 214 Journal of Mathematical Physics, Analysis, Geometry, 2006, vol. 2, No. 2 Isometric Expansions of Quantum Algebra of Linear Bounded Operators characteristic function S(z) = K+ (z � T1) �1 �N1 of the expansion V1 satis�es the �intertwining� condition, ~N�1 1 � ~N2z + ~� � S(z) = S(zq)N�1 1 (N2z + �) : (28) P r o o f. First of all, note that from (7) it follows that (T1 � zqI) T2 = qT2 (T1 � zI) and so T2 (T1 � zI)�1 = q (T1 � zqI)�1 T2: (29) Rewrite the �rst equality in 3) (9) in the following way: qT2�N1 = (T1 � zqI) �N2 +q�(N2z + �), then T2 (T1 � zI)�1 �N1 = �N2 + q (T1 � zqI)�1�(N2z + �) (30) by virtue of (29). Similarly rewriting the second equality in 3) (9) in the form of ~N1 T2 = q ~N2 (T1 � zI) + q � ~� + ~N2z � , we obtain that ~N1 T2 (T1 � zI)�1 = q ~N2 + q � ~� + ~N2z � (T1 � zI)�1 : (31) Multiplying equality (30) by ~N1 from the left and subtracting from it equality (31) multiplied by �N1 from the right, we obtain the relation ~N2 �N1 � q ~N1 �N2 + q ~N1 (T1 � zqI)�1�(N2z + �) = q � ~N2z + ~� � (T1 � zI)�1�N1: Now using 4) (9) and the form of the characteristic function S(z), we can rewrite the last equality in the following way: ~�K �K� + ~N1(K � S(zq))N�1 1 (N2z + �) = � ~N2z + ~� � (K � S(z)): To complete the proof, one needs to take into account 5) (9). � Using similar considerations as applied to the dual equation system (24), we obtain the characteristic function + S (z) = K� + �� (zI � T �1 ) �1 � ~N� 1 that also will satisfy the intertwining relation, besides the last one will coincide with (22) up to conjunction. From the conservation laws 1) (21) and 1) (27) when s = 1, the following formulas result: S�(w)~�1S(z) � �1 1� z �w = N� 1� � ( �wI � T �1 ) �1 (zI � T1) �1�N1; Journal of Mathematical Physics, Analysis, Geometry, 2006, vol. 2, No. 2 215 V.A. Zolotarev + S� (w)�1 + S (z)� ~�1 1� z �w = ~N1 ( �wI � T1) �1 (zI � T �1 ) �1 � ~N� 1 : Therefore the kernel K(z; w) = 2 6664 S�(w)~�1S(z)� �1 1� z �w N� 1 + S (z)� + S ( �w) �w � z ~N1 S(z) � S( �w) �w � z + S� (w)�1 + S (z)� ~�1 1� z �w 3 7775 (32) is the positively de�ned kernel [4, 8] since for all the �nite sets fzk; fkg N 1 , N <1, where zk 2 C ðnd fk 2 E � ~E, NX k;s=1 hK (zk; zs) fk; fsi � 0 takes place, for K(z; w) = Y �(w)Y (z) besides Y (z) = h (zI � T1) �1 �N1; (zI � T �1 ) �1 � ~N� 1 i . II. Consider the following subspace in H: H1 = span n T (n)�g + T �(m) �f : f 2 ~E; g 2 E;n;m 2 Z 2 + o ; (33) where T (n) = T n1 1 T n2 2 , n = (n1; n2) 2 Z2 +. Theorem 6. Suppose that the operators N1 and ~N1 of the isometric expansion� Vs; + Vs �2 s=1 (8) are invertible. Then subspace (33) reduces both operators T1 and T2, besides the contractions of the operators T1 and T2 on the subspace H0 = H H1 are the unitary operators. P r o o f. First, prove that the subspace H1 reduces the operator T1. To prove that, it is enough to show that T1T �(m) � ~E � H1; T �1 T (n)�E � H1 for all n, m 2 Z2 +. Prove the �rst inclusion (the second one is proved similarly). From 3) (9) by virtue of invertibility of ~N1, it follows that T � 2 � = �qT �1 ~N� 2 ~N��1 1 + �q �~�� ~N��1 1 , therefore using (7) we obtain that T �(m) � ~E � span n T �n1 �g : n 2 Z+; g 2 ~E o : 216 Journal of Mathematical Physics, Analysis, Geometry, 2006, vol. 2, No. 2 Isometric Expansions of Quantum Algebra of Linear Bounded Operators And since T1T � 1 + ��1� � = I and T1 � + ��1K � ~N��1 1 = 0 (in virtue of 2) (9)) then it is easy to prove by induction that T1 (T � 1 ) n � ~E � H1 for all n 2 Z+. Thus, the space H1 reduces the operator T1 and since H0 � Ker (T �1 T1 � I) and H0 � Ker (T1T � 1 � I), for (I � T �1 T1)H = �~�1 H � H1. and (I � T1T � 1 )H = ��1� �H � H1. Then we obtain that the operator T1 induces the unitary operator on the subspace H0. To prove that the subspace H1 also reduces the operator T2 (in virtue of the similar considerations), it is su�cient to prove that T2T �n 1 � ~E � H1; T �2 T m 1 �E � H1 for all n, m 2 Z+. Prove, for example, the �rst inclusion. Let Ln = T2T �n 1 � ~E and let Ln = Ln1 � Ln0 where Lns = PsL n, besides Ps is the orthoprojector on Hs, s = 0; 1. From the reducibility of the operator T1 by the subspaces H1 and H0, it follows that T1L n 1 � H1, T1L n 0 � H0. And on the other hand, T1T2T �n 1 � ~E = qT2T1T �n 1 � ~E and so T2T1 � ~E � H1 when n = 0, since T1 � ~N� 1 + ��1K � = 0, consequently L0 0 = f0g, and using T1T � 1 + ��1� � = I when n � 1, we will have that T1L n � span � Ln�1 +H1 (by virtue of 3) (9)). Therefore P0T1L n 0 � Ln�10 and thus P0T n 1 L n 0 � L0 0 = f0g. Using now the unitarity of the contraction of the operator T1 on H0, we obtain that Ln0 = f0g. Thus Ln � H1 for all n 2 Z+. The unitarity of the contraction of operator T2 on H0 is proven in the same way as for T1. � O b s e r v a t i o n 3. From the proof of Theorem 6 it follows that in the case of the reversibility of the operators N1 and ~N1 the operator T1 generates the simple component (33) [4, 8], i.e. H1 = span n T n 1 �g + T �m1 �f : g 2 E; f 2 ~E;n;m 2 Z+ o : (34) The isometric expansion Vs, + Vs (8) is said to be the simple expansion if H1 = H, where H1 has the form of (33). Along with the operator system fT1; T2g in H satisfying relation (7), consider another system of linear bounded operators fT 01; T 0 2g in H 0 satisfying (7). The iso- metric expansion � Vs; + Vs �2 s=1 (8) of the system fT1; T2g is said to be the unitarily equivalent to the expansion � V 0 s ; + V 0 s �2 s=1 (8) of the system fT 01; T 0 2g if: 1) the external spaces E = E0 and ~E = ~E0 coincide and the respective opera- tors �s, �s, Ns, � and ~�s, ~�s, ~Ns, ~� acting in them coincide also; 2) there exists the unitary operator U from H into H 0 such that UTs = T 0sU (s = 1; 2); U� = �0; 0U = : Journal of Mathematical Physics, Analysis, Geometry, 2006, vol. 2, No. 2 217 V.A. Zolotarev It is obvious that the characteristic functions S(z) of the unitarily equivalent expansions coincide. The following theorem of unitary equivalency is true [4, 8]. Theorem 7. Suppose that simple (33) isometric expansion � Vs; + Vs �2 s=1 (8) of the operator system fT1; T2g that satis�es equality (7) and the simple isometric expansion V 0 s , + V 0 s (8) of the system fT 01; T 0 2g, for which also (7) takes place, are de�ned. And let the external spaces of these expansions E = E0, E = ~E0 coincide and the corresponding operators be equal, � = �0, Ns = N 0 s, �s = �0s, �s = � 0s and also ~� = ~�0, ~Ns = ~N 0 s, ~�s = ~�0s, ~�s = ~� 0s, s = 1; 2. Then if in some general region of holomorphy S(z) = S0(z) takes place and operators N1 and ~N1 are invertible then the commutative expansions Vs, + Vs and V 0 s , + V 0 s are unitarily equivalent. P r o o f. Denote formulas (29) and (30) in the following way: (zqI � T1) �1�(N2z + �) = �N2 + q (zqI � T1) �1 T2�N1; q � ~N2z + ~� � (zI � T1) �1 = q ~N2 + ~N1 T2 (zI � T1) �1: (35) Therefore after the necessary number of iterations from the right N�1 1 (N2z + �) and from the left (N� 2 �w + ��)N��1 1 of the block K 1;1 (z;w) of the kernel K(z; w) (32) and also after the appropriate substitutions z ! zq, we obtain that the expressions ��T �m2 ( �wI � T �1 ) �1 (zI � T1) �1 T n 2 �; 8n; m 2 Z+; for the extensions Vs, + Vs and V 0 s , + V 0 s (8) coincide since S(z) = S0(z). Use of the similar considerations for other blocks of the kernel K(z; w) (32) leads to the coincidence of expressions T n 2 ( �wI � T1) �1 (zI � T �1 ) �1 T �n2 �; Tm 2 ( �wI � T1) �1 (zI � T1) �1 T n 2 � for all n, m 2 Z+. De�ne the operator U from H into the space H 0 by the formulas UT (n)�g = T 0(n)�0g (g 2 E); UT �(m) �f = � T 0(m) �� 0�f (f 2 ~E); where T (n) = T n1T n2 2 for all n, m 2 Z2 +. The unitarity of the operator U , as well as the ful�llment of the unitary equivalence conditions, follows from the coincidence of the expressions listed above. � 218 Journal of Mathematical Physics, Analysis, Geometry, 2006, vol. 2, No. 2 Isometric Expansions of Quantum Algebra of Linear Bounded Operators III. The characteristic function S(z) of the isometric expansion � Vs; + Vs �2 s=1 (8) has not only the traditional hermitian nonnegativeness of the kernel K(z; w) (32) [4, 8], but also the additional properties that are inherited by the fact that T1 and T2 satisfy equality (7). Suppose that the operators N1 and ~N1 of the expansion � Vs; + Vs �2 s=1 (8) are invertible, then the following relations: 1) S(zq)N�1 1 (N2z + �) = ~N�1 1 � ~N2z + ~� � S(z); 2) K 1;1 (z;w) � (N� 2 �w + ��)N��1 1 K 1;1 (zq;wq) N�1 1 (N2z + �) = S�(w)~�2S(z)� �2; 3) K 2;2 (z;w) � � ~N2 �w�q + ~� � ~N�1 1 K 2;2 (zq;wq) ~N��1 1 � ~N� 2 zq + ~�� � = + S� (w)�2 + S (z)� ~�2; (36) where n K p;s (z;w) o are the corresponding blocks of the kernelK(z; w) (32), are taking place. While relation 1) (36) follows from Theorem 5, equalities 2) and 3) in (36) follow from the conservation laws 2) (21) and 2) (27) on condition that (18), (19) and (25), (26) are taking place. For example, prove that relation 2) (36) follows from 2) (21). Really, the fact that @2vn = q ~N�1 1 � ~N2z + ~� � vn; @2un = qN�1 1 (N2z + �) un follows from (18) and from (19) by virtue of the dependence of un and vn by the variable n1 chosen in Paragraph I of this section and also from the form of @1 (15). Therefore relation 2) (21) we can represent in the form S�(w) (~�1 � ~�2)S(z) + z �wS�(w)~�2S(z)� S�(w) � ~N� 2 �w + ~�� � ~N��1 1 ~�1 � ~N�1 1 � ~N2z + ~� � S(z) = �1 � �2 + z �w�2 � (N� 2 �w + ��)N��1 1 �1N �1 1 (N2z + �) : Now using the intertwining relation 1) (36), we obtain that (1� z �w) [S�(w)~�2S(z)� �2] = S�(w)~�1S(z)� �1 � � N� 2 �w + ~� � ~N��1 1 [S�(wq)~�1S(zq)� �1]N �1 1 (N2z + �) ; Journal of Mathematical Physics, Analysis, Geometry, 2006, vol. 2, No. 2 219 V.A. Zolotarev this fact proves 2) (36) by virtue of the form of the kernel K 1;1 (z;w) (32). O b s e r v a t i o n 4. Equating the coe�cients of equal powers in (33), we can obtain the additional relations between the external parameters of expansion (8), �s, Ns, �s, � and ~�s, ~Ns, ~�s, ~�, for example, K� � ~�2 � ~N� 2 ~N��1 1 ~�1 ~N �1 1 ~N2 � K = �2 �N� 2N ��1 1 �1N �1 1 N2; K � �2 �N2N �1 1 �1N ��1 1 N� 2 � K� = ~�2 � ~N2 ~N�1 1 ~�1 ~N ��1 1 ~N2: The description of the independent parameters of expansion (8) is the separate problem, and we will not touch it here. IV. Consider now the description of the class of functions S(z) that are the characteristic functions of the expansion � Vs; + Vs �2 s=1 (8) of the operator system T1, T2 satisfying (7). First of all, note several obvious properties of the expansion operators Vs and + Vs (8). It is easy to see that for all the vectors u from E, V1 � (zI � T1) �1�N1u u � = � z (zI � T1) �1�N1u S(z)u � ; V2 � (zI � T1) �1�N1u u � = � q (zqI � T1) �1 �N1N �1 1 (N2z +�) S(z)u � (37) take place by virtue of (30) and 3) (9). It is important that the isometric property of the expansion V1, 1) (9), results in the fact that the block K 1;1 (z;w) of the kernel K(z; w) (32) has the form of K 1;1 (z;w) = N� 1� � ( ~wI � T �1 ) �1 (zI � T1) �1�N1, and the isometric condition 1) (9) in terms of K 1;1 (z;w) exactly gives equality 2) (36) obtained above. Similarly, the operators + Vs (8) will act in the dual situation, + V1 � (zI � T �1 ) �1 � ~N� 1 v v � = " z (zI � T �1 ) �1 � ~N� 1 v + S (z)v # ; (38) + V2 � (zI � T �1 ) �1 � ~N� 1 v v � = 2 4 (zqI � T �1 ) �1 � ~N� 1 ~N��1 1 � ~N� 2 zq + ~�� � v + S (z)v 3 5 for all v 2 ~E by virtue of (31) and 3) (9). As before, the isometric property 2) (9) of the operator + V1 (38) results in the fact that the block K 2;2 (z;w) of the kernel K(z; w) (32) has the form of K 2;2 (z;w) = ~N1 ( �wI � T1) �1 (zI � T �1 ) �1 � ~N� 1 , and the isometric condition 2) (9) of the operator + V2 results in equality 3) (36) for the 220 Journal of Mathematical Physics, Analysis, Geometry, 2006, vol. 2, No. 2 Isometric Expansions of Quantum Algebra of Linear Bounded Operators block K 2;2 (z;w) . Finally, if we take into the consideration the connection between Vs and + Vs (14) in these terms, then we obtain that ~N1S(z) = + S� (z)N1 when s = 1, and we obtain the intertwining condition 1) (36) again when s = 2. Really, from (14) and from the form of the block K 2;1 (z;w) , we obtain that q ~N S(qz)� S( �w) �w � zq N�1 1 (N2z +�) +N2S(z) = � ~N2wq + ~� � S(z)� S(wq) wq � z + ~N1S( �w)N �1 1 N2: As a result of the elementary calculations, we obtain that ~N�1 1 � ~N2z + ~� � S(z) � S(zq)N�1 1 (N2z + �) = ~N�1 1 � ~N2wq + ~� � S(wq) �S( �w)N�1 1 (N2wq + �) ; so the expression ~N�1 1 � ~N2z + ~� � S(z)�S(zq)N�1 1 (N2z + �) is a constant, which obviously equals to zero if z !1 by virtue of 4), 5) (9). Thus, all relations (36) have the natural origin that results from the isometric property and from the consistency of the expansion Vs, + Vs (8). Theorem 8. Suppose that the operator-function S(z) mapping E into ~E is such that there exist such operators �s, �s, Ns, � in the Hilbert space E, and, correspondingly, the operators ~�s, ~�s, ~Ns, ~�, s = 1; 2, exist in the space ~E, besides the operators N1 and ~N1 are invertible, and, moreover, the following conditions are true: 1) the kernel K(z; w) (32) is positively de�ned, besides + S (z) = N��1 1 S�(z) ~N�; 2) for the function S(z) and for the kernel blocks K(z; w) (32), relations (36) are true; 3) the function S(z) and the kernel K(z; w) are analytical by z and by �w in the region D = fz 2 C : jzj � Rg for some R >> 1, besides S(1) 6= 0 and K(1;1) 6= 0. Then there exist the Hilbert space H and the system of linear bounded operators T1, T2 in H satisfying (7), such that for its isometric expansion � Vs; + Vs �2 s=1 (8), relations 1)�5) (9), with operators f�s; �s; Ns;�g and n ~�s; ~�s; ~Ns; ~� o de�ned above, are true, besides the characteristic function of the expansion V1 of the operator T1 coincides with S(z). P r o o f. On the Cartesian product D� (E � ~E), de�ne the vector-functions ezh, the carrier of which is amassed at the point z, and hT = (u; v) where u 2 E, Journal of Mathematical Physics, Analysis, Geometry, 2006, vol. 2, No. 2 221 V.A. Zolotarev v 2 ~E. On the set of formal linear combinations NX 1 ezkhk (N < 1), de�ne the nonnegative bilinear form with the use of the kernel K(z; w) (32), hezh; ewgiK = hK(z; w)h; giE� ~E : After the closing and factorization by the kernel of the given metric, we obtain the Hilbert space H. By HE and H ~E , de�ne the subspaces in H generated by the elements of the type ez(u; 0) T = ezu and ez(0; v) T = ezv correspondingly. First, de�ne the expansions V1 and V2 on HE �E in the following way: V1 � ezu u � = � zezu S(z)u � ; V2 � ezu u � = � ezqqN �1 1 (N2z + �) u S(z)u � ; (39) where u 2 E, and Vs (39), per se, have the form of (37). It is easy to see, that for Vs (37) relations 1) (9) will take place by virtue of the form of the block K 1;1 (z;w) of the kernel K(z; w) (32) and by virtue of equality 2) (36). Similarly, de�ne the expansions + Vs on H ~E � ~E, + V1 � ezv v � = " zezv + S (z)v # ; + V2 � ezv v � = 2 4 ezq ~N ��1 1 � ~N� 2 zq + ~�� � v + S (z)v 3 5 ; (40) where + S (z) = N��1 1 S�(�z) ~N� 1 and v 2 ~E; it is obvious that formulas (40) and (38) have the identical nature. For the operators + Vs (40), relations 2) (9) also take place, this fact easily follows from the structure of the block K 2;2 (z;w) of the kernel K(z; w) (32) and from relation 3) (36). It is easy to prove that from the intertwining relation 1) (36) it follows that�� I 0 0 ~Ns � Vs � ezu u � ; � ezv v �� K = �� I 0 0 Ns � � ezu u � ; + Vs � ewv v �� K : (41) This equality allows to continue Vs (39) (as well as + Vs (40)) onto the whole H�E (correspondingly onto the H � ~E) correctly. Really, from (41) it follows that� I 0 0 ~Ns � Vs ���� H�E = + V � s � I 0 0 Ns ����� H�E : It is easy to test that 1) and 2) (9) take place as a result of such a continuation for Vs (39) and for + Vs (40). It is easy to prove that the operators Ts, �, , K (by 222 Journal of Mathematical Physics, Analysis, Geometry, 2006, vol. 2, No. 2 Isometric Expansions of Quantum Algebra of Linear Bounded Operators virtue of (39)�(41)), that are the block elements of Vs (39) and of + Vs (40), have the form of T1ezu = zezu� e0u; T1ezv = 1 z � ezv � e0�1 + S (z)v � ; T �1 ezv = zezv � e0v; T �2 ezv = 1 z fezu� e0~�1S(z)ug ; T2ezu = ezqq � N�1 1 (N2z + �) � u� e0qN �1 1 N2u; T2ezq h ~N��1 1 � ~N� 2 zq + ~�� �i v = ezv � ��2 + S (z)v; T �2 ezv = ezq h ~N��1 1 � ~N� 2 zq + � �i v � e0 ~N��1 1 ~N� 2 v; T �2 ezq � qN�1 1 (N2z + �) � u = ezu� e0�2S(z)u; K = S(1); ezu = [S(z)� S(1)]u; ezv = 1 z ~N��1 1 � ~�1 �K�1 + S (z) � v; �v = e0 ~N ��1 1 v; �u = e0N �1 1 u; �ezv = [ + S (z)� + S (1)]v; ��ezu = 1 z N��1 1 f�1 �K�~�1S(z)g u; (42) besides e0f = s � lim z!1 zezf 2 H where fT = (u; v) 2 E � ~E. It is easy to test that this limit exists and belongs to the space H considering the analyticity of the kernel K(z; w) (32) in D�D. Finally, the trivial testing proves that relations 3)� 5) (9) are true, and the characteristic function of the expansion V1 of the operator T1 coincides with S(z) (for example, by virtue of (31)). � O b s e r v a t i o n 5. From Theorems 7 and 8, it follows that, in the case of the invertibility of the operators N1 and ~N1, the totalityn S(z);�s; �s;Ns; �; ~�s; ~�s; ~Ns; ~� o s=1;2 is the total set of the invariants of the quantum algebra of linear operators fT1; T2g de�ned by the commutation relation (7). Journal of Mathematical Physics, Analysis, Geometry, 2006, vol. 2, No. 2 223 V.A. Zolotarev References [1] Yu.S. Samoilenko, The Spectral Theory of the Sets of Selfadjoined Operators. Naukova dumka, Kiev, 1984. (Russian) [2] M.S. Liv�sic and A.A. Yantsevich, The Theory of Operator Colligations in the Hilbert Spaces. Publ. Kharkov Univ., Kharkov, 1971. 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Connes, Noncommutative Geometry. Acad. Press, New York�London, 1994. [11] A. Klimyk and K. Schmutgen, Quantum Groups and their Representations. Springer, Berlin, 1997. 224 Journal of Mathematical Physics, Analysis, Geometry, 2006, vol. 2, No. 2

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