About *-Representations of Polynomial Semilinear Relations
In the present paper we study *-representations of semilinear relations with polynomial characteristic functions. For any finite simple non-oriented graph Г we construct a polynomial characteristic function such that Г is its graph. Full description of graphs which satisfy polynomial (degree one and...
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irk-123456789-57082010-02-03T12:01:16Z About *-Representations of Polynomial Semilinear Relations Omel'chenko, P.V. In the present paper we study *-representations of semilinear relations with polynomial characteristic functions. For any finite simple non-oriented graph Г we construct a polynomial characteristic function such that Г is its graph. Full description of graphs which satisfy polynomial (degree one and two) semilinear relations is obtained. We introduce the G-orthoscalarity condition and prove that any semilinear relation with quadratic characteristic function and condition of G-orthoscalarity is *-tame. This class of relations contains, in particular, *-representations of Uq(so(3)). 2009 Article About *-representations of polynomial semilinear relations / P.V. Omel'chenko // Methods of Functional Analysis and Topology. — 2009. — Т. 15, № 2. — С. 168-176. — Бібліогр.: 14 назв. — англ. 1029-3531 http://dspace.nbuv.gov.ua/handle/123456789/5708 en Інститут математики НАН України |
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In the present paper we study *-representations of semilinear relations with polynomial characteristic functions. For any finite simple non-oriented graph Г we construct a polynomial characteristic function such that Г is its graph. Full description of graphs which satisfy polynomial (degree one and two) semilinear relations is obtained. We introduce the G-orthoscalarity condition and prove that any semilinear relation with quadratic characteristic function and condition of G-orthoscalarity is *-tame. This class of relations contains, in particular, *-representations of Uq(so(3)). |
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Omel'chenko, P.V. |
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Omel'chenko, P.V. About *-Representations of Polynomial Semilinear Relations |
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Omel'chenko, P.V. |
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Omel'chenko, P.V. |
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About *-Representations of Polynomial Semilinear Relations |
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About *-Representations of Polynomial Semilinear Relations |
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About *-Representations of Polynomial Semilinear Relations |
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About *-Representations of Polynomial Semilinear Relations |
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About *-Representations of Polynomial Semilinear Relations |
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about *-representations of polynomial semilinear relations |
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Інститут математики НАН України |
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About *-representations of polynomial semilinear relations / P.V. Omel'chenko // Methods of Functional Analysis and Topology. — 2009. — Т. 15, № 2. — С. 168-176. — Бібліогр.: 14 назв. — англ. |
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1836524279302193152 |
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Methods of Functional Analysis and Topology
Vol. 15 (2009), no. 2, pp. 168–176
ABOUT ∗-REPRESENTATIONS OF POLYNOMIAL SEMILINEAR
RELATIONS
P. V. OMEL’CHENKO
Dedicated to the memory of A. Ya. Povzner
Abstract. In the present paper we study ∗-representations of semilinear relations
with polynomial characteristic functions. For any finite simple non-oriented graph Γ
we construct a polynomial characteristic function such that Γ is its graph. Full de-
scription of graphs which satisfy polynomial (degree one and two) semilinear relations
is obtained. We introduce the G-orthoscalarity condition and prove that any semilin-
ear relation with quadratic characteristic function and condition of G-orthoscalarity is
∗-tame. This class of relations contains, in particular, ∗-representations of Uq(so(3)).
1. Introduction
Pairs of self-adjoint linear operators in a Hilbert space which satisfy semilinear rela-
tions arise in different problems of mathematics and physics (see, for example [3, 5]) and
were studied in [2, 10, 11, 12, 13, 14] and others.
Following [2, 10, 14] we will say that a pair of bounded selfadjoint operators (A, B) in
a Hilbert space H satisfies a semilinear relation if
(1.1)
m
∑
i=1
f i(A)B g i(A) = h(A),
where f i(t), g i(t), h(t) are polynomials on R for all i = 1, m. A pair of operators (A, B),
which satisfies (1.1) is called a representation of the semilinear relation.
With any semilinear relation the following two objects are associated:
• a characteristic function, which is a polynomial of two variables on R2 :
Pn(t, s) :=
m
∑
i=1
f i(t)g i(s) =
∑
0≤i+j≤n
aij t is j , ( t , s ) ∈ R 2, aij ∈ R.
In the present paper we assume that for the characteristic function, Pn(t, s) = 0
if and only if Pn(s, t) = 0;
• a simple non-oriented graph Γ, for which the set of vertices in R, and a vertex t ∈
R is connected with a vertex s ∈ R if and only if Pn(t, s) = 0.
A standard way for describing all ∗-representation of a semilinear relation is to de-
scribe all irreducible ∗-representation up to a unitary equivalence and decompose any
representation into a direct sum or an integral of irreducible ones. Thus in the present
paper we study only irreducible representations.
The structure of irreducible representations crucially depends on the structure of con-
nected components of the graph Γ. Applying the well-known facts about ∗-representations
of graphs, we see that the problem of unitary description of all ∗-representations for pairs
2000 Mathematics Subject Classification. Primary 47A62; Secondary 16G20.
Key words and phrases. Theory ∗-representations, semilinear relations.
168
ABOUT ∗-REPRESENTATIONS OF POLYNOMIAL SEMILINEAR RELATIONS 169
of operators witch satisfy semilinear relations is not ∗-wild (see [10]) if and only if all
connected components of the corresponding graph are of the forms: � , � , � �.
In Section 2 we show that arbitrarily complicated graphs can arise even for polynomial
semilinear relations: for any finite simple non-oriented graph Γ we construct a polynomial
characteristic function such that Γ is its graph (Theorem 2.1).
In Section 3 we list graphs which arise as connected components of characteristic
functions of degree one and two (Theorem 3.1). To do it, we construct a dynamical
system on the set of zeroes of the polynomial such that the connected components are
described in terms of orbits of the dynamical system.
As one could expect, only few of them are ∗-tame, so it is natural to study pairs (A, B)
satisfying semilinear relations with additional conditions. In Section 4 we introduce the
condition of G-orthoscalarity and show that the description of G-orthoscalar pairs sat-
isfying a semilinear relation with a quadratic characteristic function is ∗-tame (Theo-
rem 4.1). The G-orthoscalarity condition is closely related to the orthoscalarity con-
dition in representations of graphs (see [6, 7]). On the other hand, the Fairlie alge-
bra [3, 5, 13] generated by self-adjoint elements a = a∗, b = b∗ and two semilinear rela-
tions [a, [a, b]q]q−1 = −b, [b, [b, a]q]q−1 = −a, (q ∈ T∪R\{0}) gives an important example
of semilinear relation with a quadratic characteristic function and the G-orthoscalarity
condition.
2. Graphs of polynomials
To begin with, we discuss some general properties of polynomial characteristic func-
tions of semilinear relations.
Proposition 1. Let Pn(t, s) =
∑
0≤i+j≤n
aij t is j satisfies the following properties:
• Pn(t, s) is an irreducible polynomial on R2, that is Pn(t, s) cannot be decomposed
into a product of two real polynomials,
• Pn(t, s) = 0 if and only if Pn(s, t) = 0,
• the set V = {(t, s) ∈ R| Pn(t, s) = 0} contains at least one regular point
of Pn(t, s) (a point (t0, s0) ∈ V in which gradPn|(t0,s0) �= 0),
then either Pn(t, s) = Pn(s, t), or Pn(t, s) = c(t − s).
Proof. The proof is based in the following lemma [8].
Lemma 1. Let V be a real algebraic set defined by an irreducible polynomial equa-
tion f(x) = 0, x ∈ Rn, and the set V contain at least one regular point of the polynomial f.
Then any polynomial which vanishes in V is a multiple of the polynomial f.
It follows from the lemma above that under the conditions of the theorem, Pn(t, s) =
cPn(s, t). If c �= 1, then Pn(t, t) = 0 for all t and therefore, by the irreducibility, Pn(t, s) =
c(t − s). �
By using singular polynomials (the set V contains only singular points) we can build
polynomial with any preassigned finite graph.
Theorem 2.1. For any finite simple non-oriented graph Γ there exists a polynomial
Pn(t, s) = Pn(s, t) such that is Γ is a graph of Pn(t, s).
Proof. Let Γ = (V, E) be a given graph with sets of vertices V and edges E,
V = {v1, v2, . . . , vr}, vi ∈ R, r ∈ N, i = 1, r,
E = {e1, e2, . . . , ew} ⊆ (V × V ) ⊆ (R × R), ej = (vj1 , vj2), w ∈ N, j = 1, w.
170 P. V. OMEL’CHENKO
Construct the polynomial Pn(t, s) as follows:
(2.1) Pn(t, s) =
w
∏
i=1
[(t − vi1)
2 + (s − vi2)
2][(t − vi2 )
2 + (s − vi1 )
2]
We see that only elements of the set E are roots of this polynomial. �
Proposition 2. Let Pn(t, s) be a polynomial characteristic function of a semilinear re-
lation of degree n, Γ be the corresponding graph, then the valency of any vertex of Γ is
less or equal to n, if for any x0 ∈ R Pn(x0, s) �≡ 0.
Proof. If we fix t = t0 ∈ R, then Pn(t0, s) is a polynomial of degree n on R. Then Pn(t0, s)
has no more than n real roots, that is, any vertex t0 of the graph is connected with not
more than n vertex. �
Remark 1. If there exist x0 ∈ R such that Pn(x0, s) ≡ 0, then the valency of its vertex
is equal to ℵ1.
3. Graphs of polynomials of degrees one and two
3.1. Polynomials of degree one.
Proposition 3. Symmetric and antisymmetric polynomials of degree one has the follow-
ing connected components of its graph:
(1) For P−
1 (t, s) := t − s,
�
t
, t ∈ R;
(2) For P+
1 (t, s) := t + s − a1, a1 ∈ R,
�
a1
2
, � �
t+
a1
2
−t+
a1
2
, t ∈ R\ {0} .
Corollary 1. The problem of describing all irreducible pairs of self-adjoint operators
(A, B) up to unitary equivalence satisfying the semilinear relations
AB − BA = h1(A), AB + BA − a1B = h2(A),
a1 ∈ R, h1(t), h2(t) are Borel function on R.
is ∗-tame (for a complete description, see [10]).
Consider the algebraic curve
(3.1) P2(t, s) := a0 + 2a1(t + s) + a2(t
2 + s2) + 2a11ts = 0, a0, a1, a2, a11 ∈ R.
Consider two different cases.
3.2. Polynomials without squares. First, let a2 = 0. Making a shift of variables,
t → t − a1/a11, s → s − a1/a11, and normalizing by setting a11 �= 0, we can assume
that a1 = 0 and a11 = 1, and have the equation: P2(t, s) := ts = a0.
Two different cases arise:
(1) if a0 �= 0, then any connected component of the graph of P2(t, s) is one of the
following:
�
±√
a0
(if a0 > 0), �
0
, � �
t a0/t
, t ∈ R\
{
±√
a0, 0
}
.
ABOUT ∗-REPRESENTATIONS OF POLYNOMIAL SEMILINEAR RELATIONS 171
Corollary 2. The problem of describing all irreducible pairs of self-adjoint op-
erators (A, B) up to unitary equivalence satisfying the semilinear relation
a0B + ABA = h(A),
h(t) is Borel function on R, a0 ∈ R, a0 �= 0, is ∗-tame (for a full description
see [10]).
(2) if a0 = 0, then the whole graph P2(t, s) is connected, any point is connected to
zero.
Corollary 3. The problem of describing all irreducible pairs of self-adjoint op-
erators (A, B) up to unitary equivalence satisfying the semilinear relation
ABA = h(A), h(t) is a Borel function on R,
is ∗-wild.
3.3. Polynomials with square members. Now let a2 �= 0. Assume that a2 = 1, then
we have the curve
(3.2) P2(t, s) := a0 + 2a1(t + s) + (t2 + s2) + 2a11ts = 0.
To describe the graph corresponding to (3.2), we introduce a dynamical system on the
set V = {(t, s) ∈ R2| P2(t, s) = 0}. For any (t, s) ∈ V construct two maps:
(3.3) f1(t, s) := (s, t), f2(t, s) := (t, ϕ(t, s)), where ϕ(t, s) : V → R.
Since P2(t, s) = 0 iff P2(s, t) = 0, it follows that ϕ(t, ϕ(t, s)) = s and f1f1(t, s) = (t, s),
f2f2(t, s) = (t, s) for any (t, s) ∈ V .
Consider the following map
(3.4) F (t, s) := f1f2(t, s) = (ϕ(t, s), t).
Proposition 4. The dynamical system generated by the maps f1, f2 is completely defined
by the dynamical system generated by the map F .
Proof. For any (t, s) ∈ R2 and n ∈ N0, this map has the following properties:
(f1f2)
n(t, s) = Fn(t, s), f1(f2f1)
n(t, s) = Fn(s, t),
(f2f1)
n(t, s) = f1F
n(s, t), f2(f1f2)
n(t, s) = f1F
n+1(t, s).
Then the trajectory of any point (t0, s0) ∈ R2 with respect to the action of the maps f1, f2
can be uniquely recovered from the trajectory of points (t0, s0), (s0, t0) ∈ R2, with respect
to the action of the map F as follows:
Trf1,f2
(t0, s0) = {Fn(t0, s0), f1F
n(t0, s0), F
n(s0, t0), f1F
n(s0, t0), n = 0, 1, . . .}. �
The proposition above implies the following fact.
Proposition 5. Let Fn(t, s) := (tn, sn), n ∈ N0. If Γ = (V, E) is the graph of the
polynomial P2(t, s), then the set of its vertices is V = V1 ∪ V2 and the set of its edges
is E1 ∪ E2, where V1 = {sn, n ∈ N0}, E1 = {(sn, sn+1), n ∈ N0}, with starting points
(s0, s1) = (λ, µ) and V2 = {sn, n ∈ N0}, E2 = {(sn, sn+1), n ∈ N0}, with starting points
(s0, s1) = (µ, λ), (λ, µ) ∈ R2, such that P2(λ, µ) = 0.
Now we apply these facts to describe connected components of the graph of a quadratic
characteristic function. Consider the dynamical system (V , N0, F ), where
(3.5) V = {(t, s) ∈ R2 | a0 + 2a1(t + s) + (t2 + s2) + 2a11ts = 0, a0, a1, a11 ∈ R}
F (t, s) = (−2a11t − s − 2a1, t), for any (t, s) ∈ V .
Let Fn(t, s) = (tn, sn), then it follows from (3.5) that
{
tn+1 = −2a11tn − sn − 2a1,
sn+1 = tn
⇒
{
tn+1 = −2a11tn − sn − 2a1,
tn+1 = sn+2
172 P. V. OMEL’CHENKO
and we have the difference equation
(3.6) sn+2 + 2a11sn+1 + sn = −2a1, n ∈ N0.
Solutions of this equation are the following.
• The hyperbolic case (a2
11 > 1),
sn =
s1 − s0λ1
2
√
a2
11 − 1
λn
2 − s1 − s0λ2
2
√
a2
11 − 1
λn
1 − a1
a11 + 1
, λ1, 2 = −a11 ∓
√
a2
11 − 1.
• The parabolic case (a2
11 = 1),
if a11 = −1, then sn = s0 + (s1 − s0)n − a1n
2,
if a11 = 1, then sn = (−1)n(s0 − (s1 + s0)n) − a1
2
.
• The elliptic case (a2
11 < 1),
(3.7) sn = s0 cos(n ψ) +
s1 + s0a11
√
1 − a2
11
sin(n ψ) − a1
a11 + 1
, ψ = arccos(−a11).
It is easy to prove that a dynamical system of hyperbolic type has no cycles. A
dynamical system of parabolic type has cycles if and only if a11 = −1 and a0 = a1 = 0
or a11 = 1 and a0 = a2
1.
Proposition 6. The map (3.7) has cycles if and only if ψ = πl/k, where l ∈ Z, k ∈ N.
If l/k is an irreducible fraction then the length of the cycle is equal to k if l is even and
2k if l is odd. If the map (3.7) does not have cycles then the set {(sn+1, sn), n ∈ N0} is
dense in the set V for any (s0, s1) ∈ V .
Proof. Find the angle ψ such that there exist m1, m2 ∈ N0 for which
(3.8) s0 cos(m1 ψ) +
s1 + s0a11
√
1 − a2
11
sin(m1 ψ) = s0 cos(m2 ψ) +
s1 + s0a11
√
1 − a2
11
sin(m2 ψ).
There exist such an angle α that the equality (3.8) can be rewritten as follows:
cos(α) cos(m1 ψ) + sin(α) sin(m1 ψ) = cos(α) cos(m2 ψ) + sin(α) sin(m2 ψ),
that is, ψ = 2πl
m2−m1
with some l ∈ Z. Suppose that m1 = 0, m2 = 2k; we obtain that if
l/k is an irreducible fraction then the length of the cycle is equal to k if l is even and 2k
if l is odd. If the map (3.7) does not have cycles then ψ = pπ, p ∈ (−1/π, 1/π)\(Q ∪ πQ)
and the map (3.7) is adjoint to the irrational rotation of the unit circle. Thus we have
that the set {(sn+1, sn), n ∈ N0} is dense in the set V for any (s0, s1) ∈ V . �
Summing up the above, we obtain a theorem about graphs of polynomials of degree
two.
Theorem 3.1. Any connected component of P2(t, s) = a0+2a1(t+s)+(t2 +s2)+2a11ts,
a0, a1, a11 ∈ R is one of the following.
(1) The hyperbolic case (a2
11 > 1).
For a11a0 > 0 the connected components are
� � � � . . . , . . . � � � . . . ;
for a11a0 < 0 the connected components are
. . . � � � . . ..
ABOUT ∗-REPRESENTATIONS OF POLYNOMIAL SEMILINEAR RELATIONS 173
(2) The parabolic case (a2
11 = 1).
For a11 = −1 and a1 = a0 = 0 the connected components are �
t
, t ∈ R;
for a11 = −1 and a1a0 �= 0 the connected components are
� , � � � � . . . , � � � . . . , . . . � � � . . . ;
for a11 = 1 and a2
1 �= a0 the connected components are
. . . � � � . . . .
(3) The elliptic case (a2
11 < 1).
For a0 > 0 the connected components are �;
for a0 = 0 the connected components are �, �
0
;
for a0 < 0 the connected components are the following: if arccos(−a11) = lπ
k ,
l
k ∈ Q (rational ellipse),
� , . . .� � � � , . . .� � � � , . . .� � � � , l is odd,
� , . . .� � � � , . . .� � � � , l is even;
if arccos(−a11) = pπ, p ∈ (− 1
π , 1
π ) \ (Q ∪ πQ) (irrational ellipse),
� , � � � � . . . , � � � . . . , . . . � � � . . . .
Remark 2. Irreducibility of pairs of the operators (A, B) that satisfy semilinear rela-
tion (1.1) implies that the spectrum of the operator A is discrete in any case with an
exception of the case of an irrational ellipse. Below we will show that if the characteristic
function of a semilinear relation is an irrational ellipse then there exists an irreducible
representations with continuous spectrum of the operator A.
Remark 3. Irreducibility of the pair (A, B) implies that in the hyperbolic and the par-
abolic (except when a11 = −1 and a0 = a1 = 0 or a11 = 1 and a0 = a2
1) cases the
operator A is unbounded, and in the elliptic case the operator A is bounded.
Following [14] we define unbounded representations of the polynomial semilinear re-
lation (1.1). We will say that a pair of symmetric operators A, B satisfies the semilinear
relation (1.1) if there exist a dense set K ⊂ H such that:
• K is invariant with respect to A, B, EA(∆), ∆ ∈ B(R),
• K consists of bounded vectors for A, K ⊂ HB(A) ⊂ D(A),
• relation (1.1) holds on K.
Then the following holds.
Corollary 4. The problem of describing all (bounded and unbounded) irreducible pairs of
self-adjoint operators (A, B) up to a unitary equivalence satisfying the semilinear relation
a0B + 2a1(AB + BA) + A2B + BA2 + 2a11ABA = h(A),
for any Borel on R function h(t) is ∗-wild.
174 P. V. OMEL’CHENKO
4. Representations of semilinear relations with condition of
G-orthoscalarity
It follows from Corollary 4 that it is natural to study pairs of operators (A, B) sat-
isfying semilinear relations with additional conditions. Here we give a condition of G-
orthoscalarity such that under this condition the unitary classification of representations
of semilinear relations with any quadratic characteristic function can be done.
Let H be a Hilbert space, (A, B) be a pair of linear self-adjoint operators in H.
Let G0(t), G1(t), G2(t) be a Borel function on R. We will say that a pair of opera-
tors (A, B) satisfy a semilinear relation with the condition of G-orthoscalarity, if they
satisfy the relations
(4.1)
m
∑
i=1
f i(A)B g i(A) = 0,
G1(A)B2 + 2BG2(A)B + B2G1(A) = G0(A),
where f i(t), g i(t), i = 1, m.
Following Remark 3 we can consider representations (A, B) of semilinear relations with
the condition of G-orthoscalarity (4.1) by unbounded operators A, B, assuming that the
relations hold on the domain K described in Remark 3.
Theorem 4.1. The problem of describing all irreducible pairs of self-adjoint (bounded or
unbounded) operators (A, B) in a Hilbert space H up to a unitary equivalence satisfying
(4.2)
{
A2B + BA2 + 2a11ABA + 2a1(AB + BA) + a0B = 0,
G1(A)B2 + 2BG2(A)B + B2G1(A) = G0(A)
is ∗-tame for any Borel functions G0(t), G1(t), G2(t), the numbers a0, a1, a11 ∈ R; if
a11 = − cos(pπ), p ∈ (−1/π, 1/π) \ (Q ∪ πQ) then we additionally assume that σ(A) is
discrete.
Proof. The proof of this theorem is based on following lemma [1] (here we give an equiv-
alent formulation of this lemma).
Lemma 2. Let (A, B) be a pair of linear self-adjoint operators in a Hilbert space H which
satisfies the semilinear relation (1.1) with the graph Ak ( ...� � � �). Let the spectrum σ(A)
of the operator A be discrete. If, with respect to the decomposition H = Hλ1
⊕ Hλ2
⊕
. . . ⊕ Hλk
, where σ(A) = {λ1, λ2, . . . , λk}, Hλi
is the eigenspace corresponding to the
eigenvalue λi, i = 1, k, k ∈ N ∪ {∞}, the matrix blocks of the operator B satisfy the
following relations
(4.3)
p2B21B12 + q2B23B32 = r2IH2
,
p3B32B23 + q3B34B43 = r3IH3
,
· · · · · · · · · · · · · · · · · · · · · · · · · · · · · ·
pk−1Bk−1 k−2Bk−2 k−1 + qk−1Bk−1 kBk k−1 = rk−1IHk−1
,
pi, qi, ri ∈ R, qi �= 0, Bij : Hj → Hi, i, j = 2, . . . , (k − 1),
then, in irreducible representation of its semilinear relation, dimHi = 1, i = 1, k.
It follows from Theorem 3.1 and the first relation above that under the condition of
the theorem the spectrum of the operator A in any irreducible representation is discrete.
Any connected component of the graph of the semilinear relation contains the graph Ak,
thus the operator B can be written as the corresponding block matrix. The second
relation (G-orthoscalarity) gives conditions on the blocks of the operator B which contain
the conditions 4.3. A routine analysis of the additional relations on the blocks of the
operator B gives us the result. �
ABOUT ∗-REPRESENTATIONS OF POLYNOMIAL SEMILINEAR RELATIONS 175
An example of a semilinear relation with G-orthoscalarity condition is given by the
Fairlie algebra [3, 13]. Consider the following ∗-algebra:
Aq,µ := C〈a = a∗, b = b∗| [a, [a, b]q]q−1 = sign(µ)b, [b, [a, b]q]q−1 = sign(µ)a〉,
where [x, y]q := xy − qyx, q ∈ R ∪ T, µ ∈ R.
Irreducible ∗-representations of Aq µ are pairs of self-adjoint operators (A, B) which
satisfy the relations
(4.4)
A2B + BA2 + 2a11ABA + a0B = 0,
B2A + BA2 + 2a11BAB + a0A = 0, a0 ∈ R, a11 = −q + q−1
2
.
Note that, if a0 = 1, then this algebra is the Fairlie algebra [3], and if a0 = a11 = 1
then it is the universal enveloping algebra of the Lie algebra so(3), if a0 = a11 = −1 then
it is a graded analogue of the Lie algebra so(3) [4, 10].
Let A and B be operators of a ∗-representation of the ∗-algebra Aq, µ. Suppose that
the spectrum σ(A) of the operator A is discrete. A description of all irreducible ∗-
representations up to unitary equivalence of the ∗-algebra Aq, µ can be obtained as ∗-
representations of the first relation of (4.4) with the condition of G-orthoscalarity (for
G = 0, G0(t) = −a0t, G1(t) = t, G2(t) = a11t. For a description of ∗-representations
of Aq, µ, see [13].
Consider the semilinear relation (4.4) with
arccos(−a11) = pπ, p ∈
(
− 1
π
,
1
π
)
\ (Q ∪ πQ) .
From Theorem 3.1 it follows that any orbit of the corresponding dynamical system is
dense in the set of zeroes of the characteristic function of the semilinear relation. There
is no measurable section which contains one point from each orbit of the dynamical
system. We show that there exists an irreducible ∗-representation of (4.4) with continuous
spectrum of the operator A, σ(A) =
[
− 1
i sh(−pπ)
,
1
i sh(−pπ)
]
.
Example 1. Let H = L2
([
− 1
i sh(−pπ)
,
1
i sh(−pπ)
]
, dx
)
. Consider operators in H :
Af(x) = xf(x), Bf(x) = b1(x)f(F−1
1 (x)) + b2(x)f(F−1
2 (x)),
where F1,2(x) = −a11x ±
√
x2(a2
11 − 1) + 1, b1(x), b2(x) ∈ H.
The pair (A,B) satisfies (4.4) and is irreducible.
Acknowledgments. The author is sincerely grateful to Professor Yu. S. Samoilenko,
his adviser V. L. Ostrovskyi, and to V. V. Fedorenko for a permanent attention to this
work and useful discussions.
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Institute of Mathematics National Academy of Sciences of Ukraine, 3 Tereshchenkivska
Str., Kiyv, 252601, Ukraine
E-mail address: omelchenko@imath.kiev.ua
Received 21/04/2009
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