Vol. 28 (1989) REPORTS ON ,MATHE.MATlCdL PHYSICS No. 1 UNBOUNDED OPERATORS SATISFYING RELATIONS NON-LIE COMMUTATION V. L. OSTROVSKII and Yu. S. SAMOILENKO Institute of Mathematics, Ukrainian Academy of Sciences, Kiev-l, Repin Str. 3, USSR (Received March 2, 1989) We study two unbounded operators A = A* and B on a Hilbert space H satisfying non-Lie relations AB = BF(A) (F(.): R’ + R 1 is a fixed function). We construct for A, B commutative models using a technique of decomposition with respect to the generalized eigenvectors. For the operators A and B, under the assumption that B is self-adjoint, we formulate a structure theorem that describes the pair of operators as integrals of irreducible ones. The possibility to prove the structure theorem under the assumption that B is normal depends on the form of F(.). Different examples and generalizations for the families of unbounded operators A = (A,),,x and B = (BJpeY satisfying the corresponding relations are given. 0. Introduction In recent times, due to its applications in mathematical physics, there has been interest in the theory of, generally speaking, unbounded operators which satisfy some non-Lie commutation relations (see, for example, Cl], [2], and others). In this article we study unbounded operators A and B defined on a separable complex Hilbert space H and satisfying the relation AB = M(A) (1) (here A is self-adjoint, F(.) is a fixed measurable function on R’). We also consider a family of commuting self-adjoint operators A = (A,),,x which satisfy, together with the operators of the family B = (By),,EY,the relations 4B, = B,Kq,(A) (2) (here RX3 A( .) -+ F,., (A (.))g RX is a fixed mapping measurable with respect to the cylindrical o-algebra C,(RX), and F,,,,(A) = J F,,,(i (.)) dE,(;I. ( .)), where E, (.) is a joint resolution of the identity for the fagily A, X and Y gre arbitrary sets). If the operators are unbounded, there is’ an ambiguity in the definition of the relations of the type (1) or (2) ( even if F(i) = 3,). In Sec. 1, we shall give and study such a definition of relations (l), (2) that follows from the definition of a represen- c9t1
Transcript
Vol. 28 (1989) REPORTS ON ,MATHE.MATlCdL PHYSICS No. 1
UNBOUNDED OPERATORS SATISFYING RELATIONS
NON-LIE COMMUTATION
V. L. OSTROVSKII and Yu. S. SAMOILENKO
Institute of Mathematics, Ukrainian Academy of Sciences, Kiev-l, Repin Str. 3, USSR
(Received March 2, 1989)
We study two unbounded operators A = A* and B on a Hilbert space H satisfying non-Lie relations AB = BF(A) (F(.): R’ + R 1 is a fixed function). We construct for A,
B commutative models using a technique of decomposition with respect to the generalized eigenvectors. For the operators A and B, under the assumption that B is self-adjoint, we formulate a structure theorem that describes the pair of operators as integrals of irreducible ones. The possibility to prove the structure theorem under the assumption that B is normal depends on the form of F(.). Different examples and generalizations for the families of unbounded operators A = (A,),,x and B = (BJpeY satisfying the corresponding relations are given.
0. Introduction
In recent times, due to its applications in mathematical physics, there has been interest in the theory of, generally speaking, unbounded operators which satisfy some non-Lie commutation relations (see, for example, Cl], [2], and others).
In this article we study unbounded operators A and B defined on a separable complex Hilbert space H and satisfying the relation
AB = M(A) (1)
(here A is self-adjoint, F(.) is a fixed measurable function on R’). We also consider a family of commuting self-adjoint operators A = (A,),,x which satisfy, together with the operators of the family B = (By),,EY, the relations
4B, = B,Kq,(A) (2)
(here RX 3 A( .) -+ F,., (A (.))g RX is a fixed mapping measurable with respect to the cylindrical o-algebra C,(RX), and F,,,,(A) = J F,,,(i (.)) dE,(;I. ( .)), where E, (.) is
a joint resolution of the identity for the fagily A, X and Y gre arbitrary sets). If the operators are unbounded, there is’ an ambiguity in the definition of the
relations of the type (1) or (2) ( even if F(i) = 3,). In Sec. 1, we shall give and study such a definition of relations (l), (2) that follows from the definition of a represen-
c9t1
92 V. L. OSTROVSKII AND Yu. S. SAMOILENKO
tation of a Lie algebra that can be extended to a representation of the corresponding Lie group [3, 41.
Relations (1) (or (2)) give a necessary and sufficient condition for reducing the study of the operators A, B (the families A, B) to the study of quasi-invariant measures and cocycles of the dynamic system (o(A), F) (o(A), (Fx,y)xEX,,ysY), where o(A) (c(A)) is the spectrum of the operator A (the joint spectrum of the family A). In what follows, according to [S], the operators A, B (families A, B) with the dynamic
system (fl (A), F) ((0 (A), (FX.JXtX,YEY)) will b e called a commutative model for the operators A, B (A, B). In some works [6, 7, 8, 93 and others, commutative models were constructed for operators and families of operators which satisfy some specific relations. In Sec. 2, for the unbounded operators satisfying (l), (2), following [lo], [l 11, [12], we construct commutative models using a technique of decomposition of a famiiy of commuting operators with respect to the joint generalized eigenvectors [13, 141.
In Sec. 3, we study the operators A and B under the assumption that B is self-adjoint. For these operators, we formulate a structure theorem that describes the operators in the form of integrals of elementary ones (see [15], [16]). If F(i) = %, this is a classic spectral theorem for commuting self-adjoint operators A and B (family of commuting self-adjoint operators A and B), if F(A) = -,I, it is a structure theorem for anticommuting unbounded self-adjoint operators [17], and if, for example, F(i) = 1 --A, it becomes a structure theorem for self-adjoint operators A and B satisfying the relation AB+B,4 = B.
In Sec. 4, it is shown that the possibility to prove the structure theorem under the assumption that B is unitary depends on the form of F (.). For F(A) = OA (OERI) considered in Example 2 in connection with *-representations of the algebras &?‘, with quadratic relations [a], we give a structure theorem in Sec. 4. For the function F(A) = (A-c(*)~ that appears in dealing with *-representations of algebras with relations of 4th order, we give a reasoning in Example 3 indicating that the proof of such a theorem is a “wild” problem. The representation problem for relation (1) with a normal operator B is “tame” or “wild” if and only if it is so for the case when the operator B is unitary.
1. Exact meaning of relations (1) and (2)
To make sense out of relation (1) if A and B are unbounded, it is natural to require that the operators A, F(A), B be defined on a dense linear subset Cp c H invariant under these operators. But if we require, as in the case of a Lie algebra, that its representation could be integrated to get a representation of the corresponding group, it must be demanded that @ c H” (A, B) (@ should consist of vectors analytic for the operators A and B satisfy (1) if
ABu = BF(A)u
UNBOUNDED OPERATORS 93
for all UE @, where (a) @ is invariant with respect to A, F(A), B, B*; (b) @ is a base for B, B*; (c) @ c H” (A, F(A)).
THEOREM 1. Let conditions (a), (b), (c) hold for @ c H. Then the ,followiny is equivalent:
Due to uniqueness of Fourier transform for complex measures we can deduce that
(E,(d)Bu, u) = (E,(F-’ (A))u, B* v)
for all U, VE @, A E 23 (R’). Since Q, is a base for B*, we have
E,(F-‘(d))=D(@, E,(A)Bu = BE,(F-‘(A))u.
The implications (2) * (1) and (2)0(3) follow by considering the spectral decomposition for the operators A and f(A). W
Relations (1) could be also formulated in terms of bounded operators without using @. To do this, consider the polar decomposition of the operator B = UC and the projection into the initial space of the isometry U, P = sing C.
94 V. L. OSTROVSKlI AND Yu. S. SAMOILENKO
THEOREM 2. For the operators A and B, the relations (1) are equivalent to the relations
Since the operator C2 = B* B is essentially self-adjoint on @, it commutes with A in the sense of resolution of the identity. Consider the polar decomposition of the
operator B = UC, where C = ,:IB*B, U is an isometry, and introduce P = sing C = U* U, a projection into the initial space of the isometry U. There, the operators P and C commute with A. We have
Since the operator C is invertible on the image of
E,(d) UP = UE, (F- ’ (A)) P.
Conversely, from (3) we have VUE @
E,(d) UCu = UCE, (F- ’ (A)) u
or
the projection $3 (P),
for all UE @.
E,(d)Bu = BE,(F-‘(d))u
n
Remark 1. If the operator B is bounded and the image and preimage of
F: R’ + R’ of any compact set is compact, then there always exists a nuclear space @ c H (a rigging) that satisfies conditions (a), (b) and (c). We give a method of constructing such a space @.
For the operator A, choose a bounded vector e, EH and set
A, = EA(C-m, ml)A, F,,,(A) = EAK-m, ml)F(A),
@‘;’ = LS{S, . ..S.ellSi is any of the operators
Q1 = i, @I”’ n=1
B, B*, A,, F,(A) (m E N)},
(here LS is the linear span of the sets).
UNBOUNDED OPERATORS 95
Endow the set Q1 with the nuclear topology of the inductive limit of the finite-dimensional spaces (9 ‘;) It can be checked directly that @r is invariant with .
respect to B, B* and consists of the vectors bounded for the operators A and F(A). But then for any UE@,,, there exists such rnE N that Au = A,u, F(A)u = F,(A)u, i.e., CD, is invariant with respect to A, F(A). If Q1 is not dense in H, choose a vector e2 E Q?:, bounded for the operators A, F(A), and construct Q2, and so on. We can
?j take @, to be @ = u @)m with the nuclear topology of the inductive limit of the
m= 1
nuclear spaces Qm, mEN.
Remark 2. The given method to construct a nuclear rigging works also for an unbounded normal operator B or, more generally, for an operator B satisfying the relation
E,(A)UU = uE,(G-~(A))u (AE’~~(P)),
where G: R’ -+ R’ is a one-to-one measurable mapping which together with its inverse maps compact sets into compact sets.
Remark 3. One can define the topology on Qi to be the topology of the projective limit of Hilbert spaces [l 1 J.
Remark 4. One could consider families of operators A = (A&% 1 and B = (Bj)s 1 such that (A,J~=, are commuting self-adjoint operators, and for all UE @,
A, Bj u = Bj F,,j (A) u,
where F, = (Fk,j)s 1: R” 4 R” is a mapping measurable with respect to the cylindrical a-algebra. It was assumed there that @ is invariant with respect to the above mentioned operators and it consists of the vectors analytic for the operators
A,, Fk,j(A) (k,j = 1, 2, .**). For such operator families, Theorem 1 holds together with Remarks 1,2, 3 if we
replace EA( .) with the joint resolution of the identity EA( .) of the commutative family A.
2. Commutative models
In this section, we construct a commutative model for the operators A and B which satisfy relation (1) and find their realization in the space of Fourier images of the operator A (the family of operators A). The method of constructing commutative models for different families of operators goes back to the works L. Girding, A. Wightman [7, 83, I. M. Gel’fand, N. Ya. Vilenkin [9], and others. The term “commutative model” was introduced in [S], where a commutative model was constructed for representations of a current group SL(2, R)X.
96 V. L. OSTROVSKII AND Yu. S. SAMOILENKO
In this paper, following [l&12], the commutative models are constructed using decomposition of families of commutating self-adjoint operators with respect to generalized eigenvectors.
First recall (see [13], [14]) the construction of the Fourier transform related to a family of commuting self-adjoint operators A = (AJXEX. Let H _ 3 H 3 H, be a Hilbert rigging of the space H with the imbedding 0: H, G H being quasi-nuclear. Let e(d) = Tr Of E,(d) 0 be a spectral measure, P(i): H, + H generalized projec- tions (do’ E,(I) 0 = P@)&(I)). For e (.)-almost all i, I+ P(i)I: H --* H (here I: H+H+, I+: H _ --f H are the isometries constructed using the rigging) is a Hilbert-Schmidt operator on H such that the Hilbert-Schmidt norm lI+P(1)]] = 1. Let $,(A), y = 1, . . . . N, < co, be a collection of the normed eigenvectors corresponding to the non-zero egienvalues vy(3L) of the operator I’P(i)I. To every vector UEH, corresponds its Fourier transform
u-(n) = (W.), 62 (A), . . .)E 1, (NJ,
u”? (4 = (v, w- 1’2 (4 p (4 ‘ti, (4).
The space H is isomorphic to the direct integral
H = 0 f UN&W). RX
Also Parseval’s equality
holds. In the space of Fourier images, the operator A, has the form
(A,u)” (A) = &C(A).
The following theorem was announced in [lo].
THEOREM 3. Suppose there exists a nuclear rigging Qi’ 3 H 3 @ standardly related to the operators B, B*, A, F(A). Let the operators A, B satisfy relations (1). Then, for UE @, in the space of Fourier images of the operator A,
VW” (A) = b (2) xAo (2) (d@‘se(:::“)))1’20(F-l(~)),
(4)
(B* u)” (A) = b* (F(n)) XF- ltdo) @) I).
Here, A, = {A E R'I P(1) B # 0}, the measure x,,(,I)dg(F-’ (2)) is absolutely con- tinuous with respect to dg (A), b(a) is a weakly measurable operator-valued function
b(i): 12 W,-q,,) -+ 12WJ-
UNBOUNDED OPERATORS 97
If B considered as an operator Q, -+ @ is invertible, then NF-ICI, = N, for &.)-almost all ;1. If the operator B is bounded, then representation (4) holds for all u E H, and the operator-valuedfunction B (. ) is essentially bounded. [f B is unitary, then the values of b(A) are unitary operators for Q( .)-almost all I..
Proof: Here we give a proof in the case when the operator B: @ --+ @ is invertible (in this case, Q (A,) = 1) (see [I 11, [12]).
First of all note that the measure @(F-l (-)) is equivalent to the measure Q (.). Indeed, since the operator B: @ + @ is invertible, it follows that E, (F-l (A)) u = B-l E,(d) B and @(F-‘(d)) = 0 if and only if ,o(d) = 0.
Furthermore, for any U, v 6 @, A E 23(R1), we have
for Q( .)-almost all IERI. Next we will need the chain of densely imbedded Hilbert spaces
H__~>H_~H~H+r>H++~@,whereH__=H’+.,H,=H’+andtheimbed- ding 0,: H + 4 H is quasi-nuclear, the imbeddings 0,: H + + 4 H, ? 0: @ G H + + are continuous, and the operator B acts from H, + into H, continuously. We can always construct such a chain using the fact that @ is nuclear.
Each of the spaces H + , H, + is quasi-nuclearly imbedded in H. This allows to construct the Fourier transform for every space. Let Q+ (A) = Tr 0: E,(d) 0, ,’ Q+ + (A) = Tr 0: 0: E,(d) 0, 0, be equivalent spectral measures, d0: E,(I)O, = P+(i)dQ+(i), d0: 0:E,(1)0,02 =P++(E.)~Q++(I). There P++(I) =(de+(A)/
/de+ + (A)) 0: P, (A) 0, for Q ( .)-almost all A. For a vector u E @, let 6, (. ) denote its image under the Fourier transform corresponding to the rigging H _ 1 H I H + and 6, + (4) corresponding to the rigging H_ _ 2 H 2 H, +.
We find out how the operator B acts in the space of Fourier and g(o)-almost all /ZER’,
images. For all u E @
(Bu)Y+,j(i) = (V+ +,j (J+))-“‘(Bu, p++(n)I++ ti++,j(J*)) z (V ++,j(l))-1’2(u> B*P++(‘)‘G++,j(‘b))
c (V + +,j(A))-1'2 de+ (4 de+ (I:-‘(4) de++ (4 de+ (4
(~3 O:p+ (F-‘(A))B*‘++ ti++,j(l))
98
where
V. L. OSTROVSKII /jND Yu. S. SAMOILENKO
X(‘Y$+,k(F-l(A))~ B*l++ $++,j(l*))ff+.
Thus, setting p(A) = (pjk (A)): 1, (NF- I(iJ) --+ 1, (NJ, we get
For Q, (.)-almost all IER’, introduce an operator c(A): 1, (N,) -+ I, (N,) such that c(A) u”, + (A) = u”, (A). Setting b(A) = c(A) b (A), we get
(Bu); (/I) = b(l) (“p+&@)J)‘i’p, p-1 (A)).
Similarly, we can get the formula for B *. The operator B(L) is an isomorphism from 1, (N,-lt,,) onto &(NJ for Q( .)-almost all i so IV,- I(A) = NVn. The remaining statements can be easily derived using the proved representation (4).
For an arbitrary B see [l 11. n
Remark 5. The given constructions can be generalized to the case of a family of commuting self-adjoint operators A = (AJxex which together with the operator B satisfy relations (2) [10-121. There one uses the theory of decomposition of a family of commuting self-adjoint operators with respect to the joint eigenvectors developed in [13], [14].
Remark 6. Constructing a commutative model, the space H, used for the Fourier transform can be chosen independently of B (B is used to choose H, + , which plays an auxiliary role). Due to this fact, we can construct a commutative model not only for a single operator B, but for all the operators in the family B = (By),,, which are related to A = (AJxex by means of (2).
Remark 7. Theorem 3 holds for the normal operator A (family A). The relation must have the form (3).
3. Structure theorem (B is self-adjoint)
To describe up to unitary equivalence all the solutions A, B of equation (1) with a fixed function F ( a) or to describe up to unitary equivalence all irreducible solutions (such that the von Neumann algebra generated by A, B, B* is L(H)) of (1) could be in a general setting a very difficult (“wild”) problem. For example, it contains a problem to describe up to unitary equivalence all non-self-adjoint operators. However, if
UNBOUNDED OPERATORS 99
A = A*, B = B*, then it is possible to give a description of all the solutions of (1) by means of an integral of irreducible ones (prove a structure theorem). If F(A) = A, then this is the classical spectral theorem for a pair of commuting self-adjoint operators, if F(A) = - 2, it becomes a structure theorem form anticommutating operators [17].
The definition of relations (1) given in Sec. 1 can be restated for a self-adjoint B in terms of bounded functions of self-adjoint operators A and B.
THEOREM 4. For a self-adjoint B, relations (1) are wquivalent to any of the following relations:
(a) E,(A)B, = B,E,(F-‘(A)), Vd~‘lj(R’), b’l2 0, where B, = s bdE,(B) =
3(cp(+Y(--)); (d) ABM = BF (A) u, Vu E @, where @ is a domain dense in H, invariant with respect
to the operators A, F(A), B consisting of the vectors analytic for A, F(A), B.
Proof: One can use functional calculus similarly to the proof of Theorem 1 to show that (a)o (b) o(c). To prove (a)-(d) we use the following lemmas:
LEMMA 1. F: a(A) + o(A) is one-to-one. LEMMA 2. The self-adjoint (unbounded) operators A, F(A) and B2 commute. The set
@ = u E,(d,)E,(,,(d2)Es2(d3)H (A,, A,, A, are compact subsets of R’) has AlrAz.A3
the properties (a), (b), (c) of Definition 1. H
Now we state a structure theorem (see [16]). Its proof essentially follows the proof of an analogous theorem for bounded A, B (see [15]).
THEOREM 5. Let A, B be self-adjoint operators on H satisfying (1). Then the decomposition H = H, @ H, @ (C2 @ H,) and the following orthogonal resolutions of the indentity are uniquely defined: (1) E,( .) defined on R’ with the values in the projections onto the subspaces of H,; (2) E, (*;) dejked on M, = {(A, b)ER21F(i) = /1, b # 0}, with the values in the projections onto the subspaces of
H,; (3) E2C.7.) df d e me on M, = {(I, b)ER2 1 F (F (,I)) = A, F (I.) > IV, b > 0}, with the values in the projections onto the subspaces of H, such that
A = JAdE&)+ Jj.dE,(k b)+ j[(: F;~&dE2(i, b),
R’ Ml
B = bdE,(1, b)+ s
100 V. L. OSTROVSKII AND Yu. S. SAMOILENkO
Remark 8. The given construction can be also carried out for a family of commuting self-adjoint operators A = (A,&% 1 which satisfy (2) with the operator B. In this case, E, ( .) should be replaced by the joint resolution of the identity E, ( .) on R” of the commuting family A and F (.) = (F&Z= 1: R” -+ R”.
Remark 9. One can also get a structure theorem for ,families of self-adjoint operators A = (A,Jr= 1, B = (Bj)lz 1, where commution relations with each of the other operators (Bj)J=, and (A,&L1 satisfy (2). To get the structure theorems, it is essential that the family B be finite and the operators Bj be self-adjoint. A structure theorem for representations of the C*-algebra of local observables of a spin system would follow from the structure theorem for countable collections A, B = (B,)j”,, .
But this algebra is not of type I.
4. Structure theorem (B is normal)
Consider a self-adjoint, generally speaking, unbounded operator A and a unitary operator U satisfying
AU = UF(A). (5)
Throughout this section we assume that F ( .) is continuous and one-to-one on the spectrum o(A) ((5) implies that F: o(A) -+ a(A)). Whether we can get a structure theorem now depends on the mapping F: R’ -+ R’, more exactly, on the dynamic system (o(A), F(a)). First consider some examples.
EXAMPLE 1. If U = U*, then the structure of the dynamic system is simple:
o(A) c c1 u Go, where or = {JlF(i) = A}, c‘2 = (11 F(F(1)) = i, F(A) # A>. The ir- reducible unitarily non-equivalent representations of (5) could be of two types: (1) one-dimensional: H = C’, A = I (AEcT~), U = ) 1, or (2) two-dimensional: H = C2,
A= ’ ’ , (ho2, F(A) > il), U = ;.; . L 1 0 F(i) i I
An arbitrary representation can be represented in the form of an integral of irreducible ones (Theorem 5).
EXAMPLE 2. Consider an algebra BP, (a # 0, t_ 1) generated by two self-adjoint generators X = X*, Y = Y*, and the quadratic relation
f[X, Y] = a(X2+ Y2).
Introduce the non-self-adjoint generators B = X+iY, B* = X-iY. The last equality is equivalent to
l-a B* B = pBB*.
l+a
UNROUNDED OPERATORS 101
From this it follows that Ker B = Ker B* is a subspace invariant with respect to B, B*. So we can assume that the operators B and B* are non-degenerate and in the polar decomposition of the operator B = U 1B1, the operator U is unitary. So we consider the relation
AU = BUA (6)
where A = B* B is self-adjoint and U is unitary.
PROPOSITION 1. There do not exist a bounded positive self-adjoint operator A and a unitary operator U such that AU = UUA (OER~\(O, l}).
Proof: If 3, E r~ (A) as U* AU = OA, Ok d E CJ (A) (k E Z). But this is impossible since A is bounded. W
As in Sec. 1, when considering relation (6), we assume that V’d E% (R’),
E, (A) U = UE,(F- ’ (A)) (h ere F(I”) = 02). The condition A > 0 and the invariance of o(A) relative to F(e) implies that 8 > 0 and a(A) c [0, 00). To be specific, let 0 > 1. The irreducible unitary non-equivalent representations of relation
(6) are : (1) one-dimensional - H = Cl, A = 0, U = eip (pi [0, 27c)), (2) infinite-dimensional - H = 1, (Z), Ae, = Ok 1, ek (I E [ 1, O)), Ue, = ek+ 1. The structure of an arbitrary representation of (6) is given by the following
proposition.
PROPOSITION 2. There is one-to-one correspondence between the operators A, U on H satisfying (6) and the orthogonal decompositions H = H, 0 H, = H, 0 (1, (Z) 0 ~4’~) with the resolutions of the identity
(a) E, (.) dejined on [0, 27~) with the values in the projections onto subspaces ofH,; (b) E, (0) defined on [l, 0) with the values in the projections onto subspaces of .X,
such that
A=
0 2n .I.()
0 dE, (4, ‘YO @I,_. . .
1 ’ 0 .
. .
EXAMPLE 3 (a d 0). The study of *-representations of an algebra with two self-adjoint generators satisfykg the polynomial relation
XZ+YZ+f[x, Y] = P X2+Y+X, Y] >
102 V. L. OSTROVSKll AND Yu. S. SAMOILCNKC>
leads, as in Example 2, to the relation
AU = CTP(A), (7)
where A is positive, II unitary and P( .) is a polynomial of the second order. Using a real linear change of coordinates one can reduce (7) to the form
/qLi = C!(,L-21)’ (MR’). (8)
(a) For N < ‘-’ 114, (8) does not have representations. (b) n = - l/4. The spectrum of the operator 0 (‘4) c [l/4, xl). The mapping
P(j+) = (A+ l/4)2 is one-to-one on [l/4, a,). Denote PG(i) = Pf. p(i). ._)>
(here 1, > X, is fixed). For l/4 < cc < 0, similarly to Proposition _. 3 any representation of relation (8) can
be represented as an integral of i?reducible representations. n
The formulation of a structure theorem for a certain class of (bijective and non-bijective) mappings F: R l + R’ is given in [1X]. However, there are many F( -) for which the description of all unitarily non-equivalent irreducible representations of (2) is a very difficult (“wild”) problem.
UNBOUNDED OPERATORS 103
EXAMPLE 3 (a = a*). Consider relation (8) for a = cx* (2” = 1, 4, . . is a certain number such that the mapping P(L) = (A --a*)“: [0, X,] -j [0, X,] has cycles of all the periods equal to 2k, k = 1, 2, . . . , and does not have cycles of other periods). The spectrum of the operator a(A) c [0, m). Then
(a) The mapping P( .) is bijective on [X,, co), and so H’, = E, ((X1, m)) H is a subspace invariant with respect to A and U. The operators A and U restricted to H’, have a very simple structure, they are “glued” from irreducible ones: unbounded Ae, = P@(A) ek and the unitary shift operator Ue, = ek+ 1 are both defined on 1, (Z)
(AE(&, WQ), kEZ). (b) The structure of representations of (8) with bounded operators on H 0 H’, is
more complicated. The mapping P( .): [0, X,] -+ [O, X,] is not bijective, however,
P(.): K + K, where K = {P@(a*), n = 0, 1, . . .> is homeomorphic to Cantor set, is one-to-one (see, for example [19]). The dynamic system (K, P( . )) has a unique ergodic invariant probability measure pO( .) [20]. Following [21], we can use the measure cl,, (.) to construct factor representations of type II,. This shows that the problem of describing an infinite-dimensional representation of (8) with bounded operators is “wild” for a = M*.
Remark 10. The possibility of proving a structure theorem for a family of commuting self-adjoint operators A = (AJXEX and a unitary operator U satisfying (2) also depends on the structure of the dynamic system (R”, F(.) = (F,( .))XEX). n
Consider now, generally speaking, an unbounded operator A and a normal B satisfying (1). Since Ker B = Ker B* and (Ker B)l are the subspaces invariant with respect to A and B, we can suppose that B is non-degenerate. In the polar decomposition B = UC, the operator U is unitary, C is positive and it commute with U. Then it follows from Theorem 2 that AU = UF(A), AC = CA. Since the commuting self-adjoint operators A and C, together with the unitary operator U, satisfy relation (l), the description problem for the pairs A, B is “tame” or “wild” if and only if it is so for the pairs A, U. Here, the proof of the structure theorem for a pair A, B can be obtained from the structure theorem for A, U.
REFERENCES
[l] Sklyanin, E. K.: On some algebraic structures related to Yang--Baxter’s equation (in Russian), Funkt. And. i Pribzhen., 16 (1982), 27-34.
[2] Vershik, A. M.: Algebras with quadratic relations, in: Spectral Throrey of operators and Infinite- Dimensional Analysis, Institute of Mathematics, Akademy of Sciences of Ukr. SSR, Kiev, 1984, 32-56.
[S] Nelson, E.: Analytic vectors, Ann. Math., 70 (1959), 272--615. [4] Flato, M., Simon, J., Snellman, H., Sternheimer, D.: Simple facts about analytic vectors and
integrability, Ann. Sci. de I’Ecole Norm. Sup., 4 serie, 5 (1972) .423434. [S] Vershik, A. M.. Gel’fand, I. M., Graev, M. I.: Commutative model for representations of the current
group SL(2, R)X related to a unipotent subgroups (in Russian), Funkt. Anal. i Prilozhen., 17 (1983), 70-72.
104 V. L. OSTROVSKII AlriD Yu. S. SAMOILENKO
[6] Mackey, G.: Imprimitivity for representations of locally compact groups, Proc. Nat. Acud. Sci. USA, 3s (1949), 537-545.
c71
PI II91
Cl01
1111
Cl21
Cl31
Cl41
Cl51
Cl61
Cl71
WI
Cl91
1201
WI
Garding, L., Wightman, A.: Representations of the anticommutation relations, Ibid., 40 (1954), 617-622. GHrding, L., Wightman, A.: Representations of the commutation relations, Ibid., 623-626. Gel’fand, I. M., Vilenkin, N. Ya.: Some Applications ojHarmonic Analysis. Rigged Hilbert Spaces (in Russian), Fizmatgiz, Moscov, 196 1. Berezanskii, Yu. M., Ostrovskii, V. L., Samoilenko, Yu. S.: Decomposition of a family of commuting operators with respect to the eigenfunctions and representations of commutation relations (in Russian), Ukruin. Mut. Zh., 40 (1988) 106 -109. Ostrovskii, V. L., Samoilenko, Yu. S.: Application of the spectral projection theorem to non-commutative families of operators (in Russian), Ibid., 469481. Berezanskii, Yu. M., Kondrat’ev, Yu. G.: Spectral Methods in Infinite-Dimensional Analysis (in Russian), Naukova Dumka, Kiev, 1988. Berezanskii, Yu. M.: Self-adjoint operators in spaces of functions of infinitely many variables, Truns. Much. Monographs, 63, AMS, Providence, 1986. Berezanskii, Yu. M.: The spectral projection theorem (in Russian), Uspekhi Mat. Nuuk, 39 (1984), 3-52. Samoilenko, Yu. S., Kharitonskii, A. M.: On a representation of relation AB = By(A) by bounded self-adjoint operators, Application of Functionui Analysis Methods in Mathematical Physics Problems, Institute of Mathematics of Sciences of Ukr. SSR, Kiev, 1987, 53361. Ostrovskii, V. L., Samoilenko, Yu. S.: Representations of *-algebras with two generators and polynomial relations, in: Diflerential Geometry, Lie Groups and Mechanics, X (in Russian), Zap. Nuuchn. Semin. LOMI, v. 173, Leningrad, Nauka, 1988. Samoilenko Yu. S., Spectral theory of collections of self-adjoint operators (in Russian), Naukova Dumka, Kiev, 1984. Samoilenko, Yu. S., Vaisleb, E. E.: On a representation of relations AU = UF(A) by unbounded self-adjoint and unitary operators, in: Boundary Problems for Differential Equations (in Russian), Institute of Mathematics of Academy of Sciences of Ukr. SSR, Kiev, 1988, 30-52. Sharkovskii, A. N., Maistrenko, Yu. L., Romanenko, E. Yu.: DSfferentiul Equalions and their
Applications (in Russian), Naukova Dumka, Kiev, 1986. Misiurewicz M.: Absolutely continuous measures for certain maps of an interval, Pub/. Math, Inst. HUUES Etud. Sci., 53 (1981), 17-51. Von Neumann, J.: Collected Works. III., On Rings of‘ Operators, Princeton University Press, Princeton, 1960.
