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Unitary operators with discrete spectrum induced by measure preserving transformations
Vol. 4 (1973)
REPORTS
UNITARY INDUCED
ON MATHEMATICAL
OPERATORS
BY MEASURE
No. 3
PHYSICS
WITH DISCRETE PRESERVING
SPECTRUM
TRANSFORMATIONS
J. KWIATKOWSKI Institute
of Mathematics,
N. Copernicus
(Received
The purpose
there exists two systems
such that A = G
v
S
subset
a dynamical
A is the set of all eigenvalues there exist
A on the circle
of T and n(l)
with discrete
K and an integer-valued spectrum
is the multiplicity
{Gi}, EN and
and
{S,}, EM (E< \. \
00
for
l.ES,
LEG{}
for
AEG\S,
(X, F,
function h’,, ,
function
_u, T) such that
of T if and onIy if
A?
n(l)= {ifN;
u G,, IEN
27, 1972)
system
of subgroups
; where G=
Toruli, Poland
of this paper is to prove the following theorem.
THEOREM.Given a countable n(L) on’A,
March
University,
S=
U
S,.
JEM
Introduction Any dynamical system (X, 9, ,D, T) determines a corresponding unitary operator U, on the space Lz(X, p) defined by (U,f) (x) =f(Tx) for f l L’(X, ,u)and x E X. It is known that if U is a unitary operator on Lz(X, ,u), where (X, 9, cc) is a non-atomic Lebesgue space, then there exists an automorphism T on the space X such that U,= U if and only if Uu.g)= U cf>. U(g), w h eneverf, g, f.g belong to L2(X, ,D) (see [l], $4). We say that a unitary operator U on an abstract separable Hilbert.space H is induced by an automorphism if there exists a dynamical system (X, 9, ,u, T) such that the operators U and U, are unitarily equivalent. J. R. Choksi [I] found necessary and sufficient conditions for a unitary operator to be inducible by an automorphism, in terms of an auxiliary spectral measure. The purpose of this paper is to characterize those unitary operators with discrete spectrum on a separable Hilbert space which are induced by automorphism, in terms of the spectral measures of those operators. Suppose U is a unitary operator with discrete spectrum on L2(X, ,D), where (X, 9, ,u) is a non-atomic Lebesgue space. A necessary and sufficient condition for the existence of an automorphism T on X, such that the operators U and U, are unitarily equivalent, is given for this particular case. 12031
J. KWIATKOWSKI
204 Part 1 1.1.
Let U be a unitary
operator
on a separable
Hilbert
space H and let
be the sequence of spectral types of this operator (see [4]). We shall denote by K the circle group. For each positive integer i, gi is a normalized measure on K. The type Oi (i> 1) is called the i-th spectral type of the operator U, the type a1 is called the maximal spectral type of U. We recall the following facts: (a) lfE is the spectral measure of the operator U, then there exists a vector h, E H such that Z1 = phO and for each h E H we have ph<<&,, where ph(A)=(E(A) h, h) for h E H, A being any Borelian subset of K. (b) For A c K we obtain E(A) #O if ~~ (A)>O, have o1 ({A}) > 0 ifs i is an eigenvalue of U. DEFINITION 1. We say that a unitary operator if its maximal spectral type is purely atomic.
whence by ([3], p. 57, Lemma
3) we
U is an operator with discrete spectrum
THEOREM 1. A unitary operator U on a separable Hilbert space H is an operator with discrete spectrum if and only if there exists a basis of H consisting of eigenvectors of U. Proo$ Necessity. Let A = {%i}i EN, (g
so that E(A) is the projection
of H on the subspace
0 Hi, isN
where the symbol 0 denotes the direct sum of Hi (i 6 IV). Finally, alently, there exists a basis of H consisting of eigenvectors of U. Suficiency.
We have H= @ Hi. It is easy to show that if h= c hi, where hi is the proieN
isN
jection
of 12on Hi, then for A c K we have E(A)h = c
(E(A) h, h)=
@Hi= H, or equivicN
C Ilhil12.
hi, where Nd = (ic N; E-iEA>. Hence,
ieN*
It follows from (a) that if 11, is a vector such that its projections on all subspaces (i E N) are non-zero vectors, then Pho is the maximal spectral type of U. The measure pho is purely atomic with atoms {iijicN. This completes the proof.
Hi
1.2. Consider a unitary operator U with discrete spectrum on a separable Hilbert space Hand let A be the set of all eigenvalues of this operator. Let k(n) (J_E A) denote the dimension of the eigenspace of 2.
UNITARY
OPERATORS
WITH DISCRETE
205
SPECTRUM
Denote
If A1 # 0,
then
k;= In the above
manner
k(/l)=k;).
A,=(n~/i;
min k(l); rlEA\A~
k(IZ)=k;}.
we can find k;
the integers satisfying
A,={&QI;
k; = min k(A) ; l.PA we put
Ainn,=
for i#j
A,={AEA;
k(l)=k;)
the conditions
(*>
. ..
and for
/>l.
then we write A, =A\ 6 A,. If lJ ci,zn, I=1 1=1 The following cases are possible: (c)
The sequence (*) is$nite. Let s be the number of elements of this sequence.
(d)
The sequence (*) is infinite.
Put
/cl = kl + 1, k, = 1, A, _ 1= U Ai for
13 1. Let pI be a normalized,
purely
atomic
i>l
measure with the set of all atoms Now, we prove
A, (I=O, 1, . ..).
THEOREM2. If(d) or (c) holds and k,= co, then FL is the j-th spectral type of the operator U, where k,
Proof: We assume case (d). Let H, (SEA) be the eigenspace of A and let {h~}j,, be a basis of HA. It follows from Theorem 1 that H= @ HA. For any positive integer j, 2.6‘4 let Hj be the space spanned by (hf}AeAl, where I is an integer such that k, < j< k,+ 1. It is easy to show that H= follows:
6 Hj. We define a transformation
Vj from Hj into
L’(pJ
as
j=l
h)
(4
=
@
9
@/JPl(4
3
where Iz E Hi and 2. E A,. It may be easily verified that Vj is the transformation which maps Hj onto L2 (pi) and (Vjh, Vjg)=(h, g), whenever h, g E Hi, so that Hi and L’(p,) are isomorphic. Let KpI denote a unitary operator on L2(pl) defined by (K,,f)
(A) =A .f(A)
for
f~ L’(pJ
and
For h E Hj we have Uh=
c (h, h;)XJh;= leA,
1 (h, h;)+h; lEAI
il E A,.
J. KWIATKOWSKI
206 and ((vjU)h)(I)=(~(rrh))(I)=(h,
h:).A/Jpl(A),
where
IZEA~.
Simultaneously,
Thus,
Vj. U=Ke;
Vj. Therefore,
both operators
UIHj and
K,_ are unitarily
equivalent.
Hence, UlHj is a cyclic operator and its spectral type is m. Because H= 6 Hj = G ;I’& IH) j=l I=0 j=kr the jth spectral type of U is equal to m. and &>>;ii,>>&>>..., This completes the proof for the case (d). For (c) the proof is similar. 1.3. Suppose U is a unitary operator with discrete spectrum and let Z,>>Z,>>... be the sequence of its spectral types. By ri we denote the set of all atoms of the measure pi. We have rI~r2~r3~..., and rI is the set of all eigenvalues of U. We put A = rI and n(il)=max(k&l;
AE~,>,
for AEA (if I_Erk for k=l,2, . . . . then n(A) = co). The function n(A) is called the multiplicity function of U. Since r,= (i&E A; n(A)ak), we see that the pair (A, n(1)) completely determines the sequence of spectral types of U. In the sequel the pair (A, n(A)) will be called the spectrum of the operator U. From Theorem 2 we get THEOREM3. If U is a unitary operator with discrete spectrum space H, then n(2) = k(l) for each eigenvalue i of U.
Part
on a separable
Hilbert
2
In this part we give a characterization of the pairs (A, n(L)), where A is a subset of K (2
measure p and let T be an on X.
DEFINITION2. The dynamical system (X, 9, ,u, T) is said to be a system with discrete spectrum if U, is an operator with discrete spectrum. The pair (A, n(2)) of U,, which was defined above, will be called the spectrum of this system. LEMMA 1.
If f is an eigenfunction
of U,, then there exist real numbers r, R satisfying the condition R > r > 0 and a subset A of X invariant under T such that p(A) > 0 and r < 1f(x)/
f. There
UNITARY
Putting
B= z
OPERATORS
WITH DISCRETE
207
SPECTRUM
TkB’, we have p(B) = 0 and
k=-m
f(T”x)=I”j-(x)
(1)
f Ifl’+ = llf/’ >O , it follows that there X\B exists a set A’cX\ B such that ,u(A’)> 0 and If(x)1 > llfl I2 for x E A’. Let AA={xEA’; n
and n=O, +l,
+2, . . . Since
which p((A&) >O (this is possible because A’ = 6 Ah) and put A = ‘6 n=O k=-co AcX\ B. Since IiLl= 1, we get from (1)
Ilfj/
for
TkA:, . We have
XEA.
Simultaneously, the set A is invariant under Tand ,a(A)>O. We admit r= Ilfjl, R=no+ 1, so that the lemma is proved. LEMMA 2. If A is an eigenvalue of the operator U,, then for any irtteger n, 1” is also an eigenvalue of U,.
Pcoof: By Lemma 1 we can choose real numbers R, r satisfying R>r>O invariant set A such that ,V(A) > 0 and r < 1f (x)1
{
f”(x) o
for
XEA,
for
x$A.
J lgn12&>0 and Urg,=i”*g,
for n=O, fl,
and an
5-2, . . . The proof of
X
the lemma is thus complete. 2.2. Let (X, 9, p, T) be a dynamical system, where (X, 9, ,u) is a Lebesgue space. Put X=(x E x; Tx=x}. Consider the algebra J$ of all invariant subsets of X \Xi and let Al, A2, . . . be a sequence of all atoms of &‘. We may assume that Ai n Aj = 0 for i#j. Put X, = U Ai, X, = X\ (X, u Xi). The sets Xi, A’, and X, are invariant, they are pairwise disjoint and
X=XiVX,VX,.
The decomposition LEMMA
3.
(2)
above is unique modulo null sets ([2], p. 309).
If p(X,> >O and J. is an eigenvalue of the operator U, on L2(XS, p), then
k(A)=oo. Proof: Let f E L2(XS, ,u) and Ur f =1 .f. According to Lemma 1 there exist real numbers r, R satisfying R>r >O and an invariant set A0 cX, such that ,u(A,) >O and r< If(x)1
,u(Aj)>O
and
p(Ao)>p(A1)>p(Az)>‘?
J. KWIATKOWSKI
208 Put
n= 1,2, . . . We have fn~-L%x~,
F),
j lfnl’&>O x.
and
U,f,=&
fi , fi , . . . are pairwise
At the same time, the functions lemma is proved.
(n=l,
orthogonal,
2, . ..).
so that k(A)= 0~). The
THEOREM4. Given a countable subset A of the circle K and an integer-valuedfunction n(A) on A, the pair (A, n(A)) can be the spectrum of a dynamical system with discrete spectrum if and only if there exist two systems of subgroups {Gi)ieN and (Sj}jEM (Ij< K,-,, I%< K,) of the circle such that A= GUS and
00 n(A)= 1 {ieN;
AEGJ
,for
AES,
for
AEG-S,
where G= U Gi, S= U Sj. isN
jEM
Proof: Necessity. Let (X, F-, ,u, T) be a dynamical system with discrete spectrum and let X= Xi u X, u X, be the decomposition of X the same as that in (2). Suppose p(X,) > 0, p(X,) > 0 and ,u(X,) > 0. The dynamical systems (X,,, F”^,, p,, T) and (X,, gs, ps, T), where .FO=gjxU,
Ps=F/Xs,
cl,(*)=~(*)~~(U
f4*)=,0Ic1(X3,
are with discrete spectra (see [2], p. 309). Let S denote the set of all eigenvalues of U, on Lz(X,, p). According to Lemma 2 we have S= U Sj, where Sj is a cyclic group and i%< No. Let X,= U Ai (g< EC,) be the ieN
jsM
of all the atoms of the algebra d, i.e. the dynamical systems with discrete spectra (Ai, pi, pi, T) (i E N) are ergodic. Denote by G, the countable subgroup of K which forms the set of all eigenvalues of U, on LZ(Ai, pi) or, equivalently, on L’(Ai, p). Since L’(X,, ,u)= @ L’(Ai, ,u), therefore union
ieN
G= U Gi is the set of all eigenvalues
of UT on L’(X,,
,u). Let k,(i),
k,(I) and ki(n) be the
ieN
dimensions the eigenspace of 1 in L2(X,, ,B), L’(X,, ,u) or L’(X,, ,u) for 2 E G, 1 E S or 1= 1, respectively. We have k(l)=k,(l)+k,(l)+ki(l) and k,(l)
for
IES\G,
k(l) = I k,(A) + k,(l)
for
IESnG,
1k,(4
for
IEG\S
UNITARY
for I# 1. Because
OPERATORS
p(X,>> 0 and S=
WITH DISCRETE
SPECTRUM
U Sj (the Sj’s are groups),
209
therefore
1 E S and, by
jeht
3, k,(A)=co
Lemma
for A ES,
so that
k(l)=co.
Since L2(X,~)=Lz(X,,~))oLz(X,,~)
@L2(Xi, ,u), we see that _4= GUS is the set of all eigenvalues taneously,
of UT on Lz(X, ,u). Simul-
k,(i) = (i E N, E.E Gi) (A E G) by the fact that if 2 is an eigenvalue
LZ(Ai, ,u), then 1, is simple ((Ai pi,
k(l)=n(‘)= In the remaining is similar.
cases
pi, T) is ergodic).
I
FEN
(p(X,) =0
; A&)
Finally,
from
for
1ES,
for
I~EG\S.
or p(X,) = 0 or p(XJ = 0) the
of Ur on
Theorem
3 we obtain
proof
of necessity
Suficiency. Let (Xi, Fi, pi, Ti) (i E N) be an ergodic dynamical system with discrete spectrum for which G, is the set of all eigenvalues of Ur, whenever Gi#(l). Let Xi=(xJ, ~i((Xi))=l, Ti(XJ=Xi, if G,=(l). B e1ow we show (Lemma 4) that for each j E M there exists a dynamical system ( Yj, ~j, Vj, Vi) with discrete spectrum such that Sj is the set of all eigenvalues of lJ Vj and the multiplicity function nj(A) = co for /z E Sj. We may assume that the sets Xi (i E N) and Yj (je M) are pairwise disjoint. PutX= UXiuU YjandTx=TixifxEXiorTx=I/jxifxEYj. ieN
jeM
Let F denote the o-field of subsets of X defined as follows: A E 9 if and only if AnX, EFiandAnYjE%?jforeveryiENandjEM. For AEF we put ,u(A)= C ai*pi(AnXJ+ C bj*vj(AnYj), where ai>O(iEN), icN
bj>O(jEM)and
c ieN
a,+
c
b,=l.
jsM
Since
L’(X,,U)=
@ L’(Xi,~))o iEN
jeM
@ L2(Yj,p)and jeM
~(.)lXi=ai.~i(.), ,~(.)]Y~=b~.v~(.) (in N, jEM), we see that the dynamical system (X, 9, ,u, T) is a system with discrete spectrum. It is easily shown that the pair (A, n(2)) is the spectrum of (X, P-, p(, T). This completes the proof. LEMMA
4.
Let G be a countable subgroup of the circle group. There exists a dynamical
system with discrete spectrum such that G is the set of all eigenvalues of this system and n(I)=
oo for i E G.
Proof:
Let Y be a countable
(?= K,) Abelian
group.
Let X= GQ Y. By 8 we denote
the character group of X. 8 is a separable, Abelian compact topological group. Let b be the homomorphism from X into the circle K such that if x E X, then a(x) = x,, where xe is the projection of x on G. We have fi E 8. Denote by 9 the class of Borelian subsets of X, by p a normalized Haar measure and let T;;k) = il .x for x E 2. It is known that X is the character group of 8 and that X, as a subset of the space L*(8, p), is the basis of this space. At the same time, for each x E XcL*(8, p), x is the eigenfunction of U, with a(x) as the eigenvalue. We see that (8, 9, ,u, T;) is a dynamical system with discrete spectrum and its set of eigenvalues is {a(x); x E X} = G.
210
J. KWIATKOWSKI
Let k(g) be the dimension of an eigenspace of g (g E G). Since a(g)=g, we have ___.k(g)={xeX; a(x)=g}={xEX; a(x)=il(g))=(xEX; iz(xg_l)=l} __ i =(x~X;xg-‘~ker8}=keri2=Y=t\‘,. The multiplicity
function
n(n) =k(A) (2 E G), and therefore
IZ(;~)= co. The lemma is proved.
THEOREM 5. Let (X, 9, ,u) be a non-atomic Lebesgue space, let U be a unitary nontrivial operator with discrete spectrum on L2(X, n), and let (A, n(2)) be the spectrum of U. Then there exists an automorphism T of X such that the operators U and UT are unitarily equivalent if and only if there exist two systems of subgroups {Gili EN, {Sj} j sM (I\i< KO , $< No) of the circle group and ci = h’, (i E N) such that A = GUS and
CL) n(A)= i (iEN=} where G= U ieN
Gj, S= U
for
ilES,
for
AEG\S,
Sj.
ieM
Proof: The proof of this theorem is just the same as that of Theorem 4. Since the measure p is non-atomic, the measures ,U~(i E N) are also non-atomic, hence it follows that the groups G, are infinites. At the same time, if Gi (i E N) are infinites, then according to the proof of sufficiency of Theorem 4 we see that there exists a dynamical system (Y, 93, II, V) with discrete spectrum such that the measure v is non-atomic and (A, n(A)) is the spectrum of this system. Since the measures ,U and v are non-atomic, there exists an isomorphism R: (X, 9, n) + (Y, B, v). We put T= R- 1 VR. Thus, T is an automorphism on X and the operators U, and U, are unitarily equivalent. Since the operators U and U, are unitarily equivalent (they have the same spectrum (A, n(I))), the operators U and UT are unitarily equivalent. This completes the proof. REFERENCES [l] [2] [3] [4]
Choksi, J. R., Journ,. Math. Mech. 16.1 (1966), 83. -, Ill. Journ. Math.‘92 (1965), 307. Dunford, N., J. T. Schwartz, Linear operators, Vol. 2, Moscow, 1965 (in Russian). Plesner, A. L., The spectral theory of linear operators, Moscow, 1965 (in Russian).
REPORTS
UNITARY INDUCED
ON MATHEMATICAL
OPERATORS
BY MEASURE
No. 3
PHYSICS
WITH DISCRETE PRESERVING
SPECTRUM
TRANSFORMATIONS
J. KWIATKOWSKI Institute
of Mathematics,
N. Copernicus
(Received
The purpose
there exists two systems
such that A = G
v
S
subset
a dynamical
A is the set of all eigenvalues there exist
A on the circle
of T and n(l)
with discrete
K and an integer-valued spectrum
is the multiplicity
{Gi}, EN and
and
{S,}, EM (E< \. \
00
for
l.ES,
LEG{}
for
AEG\S,
(X, F,
function h’,, ,
function
_u, T) such that
of T if and onIy if
A?
n(l)= {ifN;
u G,, IEN
27, 1972)
system
of subgroups
; where G=
Toruli, Poland
of this paper is to prove the following theorem.
THEOREM.Given a countable n(L) on’A,
March
University,
S=
U
S,.
JEM
Introduction Any dynamical system (X, 9, ,D, T) determines a corresponding unitary operator U, on the space Lz(X, p) defined by (U,f) (x) =f(Tx) for f l L’(X, ,u)and x E X. It is known that if U is a unitary operator on Lz(X, ,u), where (X, 9, cc) is a non-atomic Lebesgue space, then there exists an automorphism T on the space X such that U,= U if and only if Uu.g)= U cf>. U(g), w h eneverf, g, f.g belong to L2(X, ,D) (see [l], $4). We say that a unitary operator U on an abstract separable Hilbert.space H is induced by an automorphism if there exists a dynamical system (X, 9, ,u, T) such that the operators U and U, are unitarily equivalent. J. R. Choksi [I] found necessary and sufficient conditions for a unitary operator to be inducible by an automorphism, in terms of an auxiliary spectral measure. The purpose of this paper is to characterize those unitary operators with discrete spectrum on a separable Hilbert space which are induced by automorphism, in terms of the spectral measures of those operators. Suppose U is a unitary operator with discrete spectrum on L2(X, ,D), where (X, 9, ,u) is a non-atomic Lebesgue space. A necessary and sufficient condition for the existence of an automorphism T on X, such that the operators U and U, are unitarily equivalent, is given for this particular case. 12031
J. KWIATKOWSKI
204 Part 1 1.1.
Let U be a unitary
operator
on a separable
Hilbert
space H and let
be the sequence of spectral types of this operator (see [4]). We shall denote by K the circle group. For each positive integer i, gi is a normalized measure on K. The type Oi (i> 1) is called the i-th spectral type of the operator U, the type a1 is called the maximal spectral type of U. We recall the following facts: (a) lfE is the spectral measure of the operator U, then there exists a vector h, E H such that Z1 = phO and for each h E H we have ph<<&,, where ph(A)=(E(A) h, h) for h E H, A being any Borelian subset of K. (b) For A c K we obtain E(A) #O if ~~ (A)>O, have o1 ({A}) > 0 ifs i is an eigenvalue of U. DEFINITION 1. We say that a unitary operator if its maximal spectral type is purely atomic.
whence by ([3], p. 57, Lemma
3) we
U is an operator with discrete spectrum
THEOREM 1. A unitary operator U on a separable Hilbert space H is an operator with discrete spectrum if and only if there exists a basis of H consisting of eigenvectors of U. Proo$ Necessity. Let A = {%i}i EN, (g
so that E(A) is the projection
of H on the subspace
0 Hi, isN
where the symbol 0 denotes the direct sum of Hi (i 6 IV). Finally, alently, there exists a basis of H consisting of eigenvectors of U. Suficiency.
We have H= @ Hi. It is easy to show that if h= c hi, where hi is the proieN
isN
jection
of 12on Hi, then for A c K we have E(A)h = c
(E(A) h, h)=
@Hi= H, or equivicN
C Ilhil12.
hi, where Nd = (ic N; E-iEA>. Hence,
ieN*
It follows from (a) that if 11, is a vector such that its projections on all subspaces (i E N) are non-zero vectors, then Pho is the maximal spectral type of U. The measure pho is purely atomic with atoms {iijicN. This completes the proof.
Hi
1.2. Consider a unitary operator U with discrete spectrum on a separable Hilbert space Hand let A be the set of all eigenvalues of this operator. Let k(n) (J_E A) denote the dimension of the eigenspace of 2.
UNITARY
OPERATORS
WITH DISCRETE
205
SPECTRUM
Denote
If A1 # 0,
then
k;= In the above
manner
k(/l)=k;).
A,=(n~/i;
min k(l); rlEA\A~
k(IZ)=k;}.
we can find k;
the integers satisfying
A,={&QI;
k; = min k(A) ; l.PA we put
Ainn,=
for i#j
A,={AEA;
k(l)=k;)
the conditions
(*>
. ..
and for
/>l.
then we write A, =A\ 6 A,. If lJ ci,zn, I=1 1=1 The following cases are possible: (c)
The sequence (*) is$nite. Let s be the number of elements of this sequence.
(d)
The sequence (*) is infinite.
Put
/cl = kl + 1, k, = 1, A, _ 1= U Ai for
13 1. Let pI be a normalized,
purely
atomic
i>l
measure with the set of all atoms Now, we prove
A, (I=O, 1, . ..).
THEOREM2. If(d) or (c) holds and k,= co, then FL is the j-th spectral type of the operator U, where k,
Proof: We assume case (d). Let H, (SEA) be the eigenspace of A and let {h~}j,, be a basis of HA. It follows from Theorem 1 that H= @ HA. For any positive integer j, 2.6‘4 let Hj be the space spanned by (hf}AeAl, where I is an integer such that k, < j< k,+ 1. It is easy to show that H= follows:
6 Hj. We define a transformation
Vj from Hj into
L’(pJ
as
j=l
h)
(4
=
@
9
@/JPl(4
3
where Iz E Hi and 2. E A,. It may be easily verified that Vj is the transformation which maps Hj onto L2 (pi) and (Vjh, Vjg)=(h, g), whenever h, g E Hi, so that Hi and L’(p,) are isomorphic. Let KpI denote a unitary operator on L2(pl) defined by (K,,f)
(A) =A .f(A)
for
f~ L’(pJ
and
For h E Hj we have Uh=
c (h, h;)XJh;= leA,
1 (h, h;)+h; lEAI
il E A,.
J. KWIATKOWSKI
206 and ((vjU)h)(I)=(~(rrh))(I)=(h,
h:).A/Jpl(A),
where
IZEA~.
Simultaneously,
Thus,
Vj. U=Ke;
Vj. Therefore,
both operators
UIHj and
K,_ are unitarily
equivalent.
Hence, UlHj is a cyclic operator and its spectral type is m. Because H= 6 Hj = G ;I’& IH) j=l I=0 j=kr the jth spectral type of U is equal to m. and &>>;ii,>>&>>..., This completes the proof for the case (d). For (c) the proof is similar. 1.3. Suppose U is a unitary operator with discrete spectrum and let Z,>>Z,>>... be the sequence of its spectral types. By ri we denote the set of all atoms of the measure pi. We have rI~r2~r3~..., and rI is the set of all eigenvalues of U. We put A = rI and n(il)=max(k&l;
AE~,>,
for AEA (if I_Erk for k=l,2, . . . . then n(A) = co). The function n(A) is called the multiplicity function of U. Since r,= (i&E A; n(A)ak), we see that the pair (A, n(1)) completely determines the sequence of spectral types of U. In the sequel the pair (A, n(A)) will be called the spectrum of the operator U. From Theorem 2 we get THEOREM3. If U is a unitary operator with discrete spectrum space H, then n(2) = k(l) for each eigenvalue i of U.
Part
on a separable
Hilbert
2
In this part we give a characterization of the pairs (A, n(L)), where A is a subset of K (2
measure p and let T be an on X.
DEFINITION2. The dynamical system (X, 9, ,u, T) is said to be a system with discrete spectrum if U, is an operator with discrete spectrum. The pair (A, n(2)) of U,, which was defined above, will be called the spectrum of this system. LEMMA 1.
If f is an eigenfunction
of U,, then there exist real numbers r, R satisfying the condition R > r > 0 and a subset A of X invariant under T such that p(A) > 0 and r < 1f(x)/
f. There
UNITARY
Putting
B= z
OPERATORS
WITH DISCRETE
207
SPECTRUM
TkB’, we have p(B) = 0 and
k=-m
f(T”x)=I”j-(x)
(1)
f Ifl’+ = llf/’ >O , it follows that there X\B exists a set A’cX\ B such that ,u(A’)> 0 and If(x)1 > llfl I2 for x E A’. Let AA={xEA’; n
and n=O, +l,
+2, . . . Since
which p((A&) >O (this is possible because A’ = 6 Ah) and put A = ‘6 n=O k=-co AcX\ B. Since IiLl= 1, we get from (1)
Ilfj/
for
TkA:, . We have
XEA.
Simultaneously, the set A is invariant under Tand ,a(A)>O. We admit r= Ilfjl, R=no+ 1, so that the lemma is proved. LEMMA 2. If A is an eigenvalue of the operator U,, then for any irtteger n, 1” is also an eigenvalue of U,.
Pcoof: By Lemma 1 we can choose real numbers R, r satisfying R>r>O invariant set A such that ,V(A) > 0 and r < 1f (x)1
{
f”(x) o
for
XEA,
for
x$A.
J lgn12&>0 and Urg,=i”*g,
for n=O, fl,
and an
5-2, . . . The proof of
X
the lemma is thus complete. 2.2. Let (X, 9, p, T) be a dynamical system, where (X, 9, ,u) is a Lebesgue space. Put X=(x E x; Tx=x}. Consider the algebra J$ of all invariant subsets of X \Xi and let Al, A2, . . . be a sequence of all atoms of &‘. We may assume that Ai n Aj = 0 for i#j. Put X, = U Ai, X, = X\ (X, u Xi). The sets Xi, A’, and X, are invariant, they are pairwise disjoint and
X=XiVX,VX,.
The decomposition LEMMA
3.
(2)
above is unique modulo null sets ([2], p. 309).
If p(X,> >O and J. is an eigenvalue of the operator U, on L2(XS, p), then
k(A)=oo. Proof: Let f E L2(XS, ,u) and Ur f =1 .f. According to Lemma 1 there exist real numbers r, R satisfying R>r >O and an invariant set A0 cX, such that ,u(A,) >O and r< If(x)1
,u(Aj)>O
and
p(Ao)>p(A1)>p(Az)>‘?
J. KWIATKOWSKI
208 Put
n= 1,2, . . . We have fn~-L%x~,
F),
j lfnl’&>O x.
and
U,f,=&
fi , fi , . . . are pairwise
At the same time, the functions lemma is proved.
(n=l,
orthogonal,
2, . ..).
so that k(A)= 0~). The
THEOREM4. Given a countable subset A of the circle K and an integer-valuedfunction n(A) on A, the pair (A, n(A)) can be the spectrum of a dynamical system with discrete spectrum if and only if there exist two systems of subgroups {Gi)ieN and (Sj}jEM (Ij< K,-,, I%< K,) of the circle such that A= GUS and
00 n(A)= 1 {ieN;
AEGJ
,for
AES,
for
AEG-S,
where G= U Gi, S= U Sj. isN
jEM
Proof: Necessity. Let (X, F-, ,u, T) be a dynamical system with discrete spectrum and let X= Xi u X, u X, be the decomposition of X the same as that in (2). Suppose p(X,) > 0, p(X,) > 0 and ,u(X,) > 0. The dynamical systems (X,,, F”^,, p,, T) and (X,, gs, ps, T), where .FO=gjxU,
Ps=F/Xs,
cl,(*)=~(*)~~(U
f4*)=,0Ic1(X3,
are with discrete spectra (see [2], p. 309). Let S denote the set of all eigenvalues of U, on Lz(X,, p). According to Lemma 2 we have S= U Sj, where Sj is a cyclic group and i%< No. Let X,= U Ai (g< EC,) be the ieN
jsM
of all the atoms of the algebra d, i.e. the dynamical systems with discrete spectra (Ai, pi, pi, T) (i E N) are ergodic. Denote by G, the countable subgroup of K which forms the set of all eigenvalues of U, on LZ(Ai, pi) or, equivalently, on L’(Ai, p). Since L’(X,, ,u)= @ L’(Ai, ,u), therefore union
ieN
G= U Gi is the set of all eigenvalues
of UT on L’(X,,
,u). Let k,(i),
k,(I) and ki(n) be the
ieN
dimensions the eigenspace of 1 in L2(X,, ,B), L’(X,, ,u) or L’(X,, ,u) for 2 E G, 1 E S or 1= 1, respectively. We have k(l)=k,(l)+k,(l)+ki(l) and k,(l)
for
IES\G,
k(l) = I k,(A) + k,(l)
for
IESnG,
1k,(4
for
IEG\S
UNITARY
for I# 1. Because
OPERATORS
p(X,>> 0 and S=
WITH DISCRETE
SPECTRUM
U Sj (the Sj’s are groups),
209
therefore
1 E S and, by
jeht
3, k,(A)=co
Lemma
for A ES,
so that
k(l)=co.
Since L2(X,~)=Lz(X,,~))oLz(X,,~)
@L2(Xi, ,u), we see that _4= GUS is the set of all eigenvalues taneously,
of UT on Lz(X, ,u). Simul-
k,(i) = (i E N, E.E Gi) (A E G) by the fact that if 2 is an eigenvalue
LZ(Ai, ,u), then 1, is simple ((Ai pi,
k(l)=n(‘)= In the remaining is similar.
cases
pi, T) is ergodic).
I
FEN
(p(X,) =0
; A&)
Finally,
from
for
1ES,
for
I~EG\S.
or p(X,) = 0 or p(XJ = 0) the
of Ur on
Theorem
3 we obtain
proof
of necessity
Suficiency. Let (Xi, Fi, pi, Ti) (i E N) be an ergodic dynamical system with discrete spectrum for which G, is the set of all eigenvalues of Ur, whenever Gi#(l). Let Xi=(xJ, ~i((Xi))=l, Ti(XJ=Xi, if G,=(l). B e1ow we show (Lemma 4) that for each j E M there exists a dynamical system ( Yj, ~j, Vj, Vi) with discrete spectrum such that Sj is the set of all eigenvalues of lJ Vj and the multiplicity function nj(A) = co for /z E Sj. We may assume that the sets Xi (i E N) and Yj (je M) are pairwise disjoint. PutX= UXiuU YjandTx=TixifxEXiorTx=I/jxifxEYj. ieN
jeM
Let F denote the o-field of subsets of X defined as follows: A E 9 if and only if AnX, EFiandAnYjE%?jforeveryiENandjEM. For AEF we put ,u(A)= C ai*pi(AnXJ+ C bj*vj(AnYj), where ai>O(iEN), icN
bj>O(jEM)and
c ieN
a,+
c
b,=l.
jsM
Since
L’(X,,U)=
@ L’(Xi,~))o iEN
jeM
@ L2(Yj,p)and jeM
~(.)lXi=ai.~i(.), ,~(.)]Y~=b~.v~(.) (in N, jEM), we see that the dynamical system (X, 9, ,u, T) is a system with discrete spectrum. It is easily shown that the pair (A, n(2)) is the spectrum of (X, P-, p(, T). This completes the proof. LEMMA
4.
Let G be a countable subgroup of the circle group. There exists a dynamical
system with discrete spectrum such that G is the set of all eigenvalues of this system and n(I)=
oo for i E G.
Proof:
Let Y be a countable
(?= K,) Abelian
group.
Let X= GQ Y. By 8 we denote
the character group of X. 8 is a separable, Abelian compact topological group. Let b be the homomorphism from X into the circle K such that if x E X, then a(x) = x,, where xe is the projection of x on G. We have fi E 8. Denote by 9 the class of Borelian subsets of X, by p a normalized Haar measure and let T;;k) = il .x for x E 2. It is known that X is the character group of 8 and that X, as a subset of the space L*(8, p), is the basis of this space. At the same time, for each x E XcL*(8, p), x is the eigenfunction of U, with a(x) as the eigenvalue. We see that (8, 9, ,u, T;) is a dynamical system with discrete spectrum and its set of eigenvalues is {a(x); x E X} = G.
210
J. KWIATKOWSKI
Let k(g) be the dimension of an eigenspace of g (g E G). Since a(g)=g, we have ___.k(g)={xeX; a(x)=g}={xEX; a(x)=il(g))=(xEX; iz(xg_l)=l} __ i =(x~X;xg-‘~ker8}=keri2=Y=t\‘,. The multiplicity
function
n(n) =k(A) (2 E G), and therefore
IZ(;~)= co. The lemma is proved.
THEOREM 5. Let (X, 9, ,u) be a non-atomic Lebesgue space, let U be a unitary nontrivial operator with discrete spectrum on L2(X, n), and let (A, n(2)) be the spectrum of U. Then there exists an automorphism T of X such that the operators U and UT are unitarily equivalent if and only if there exist two systems of subgroups {Gili EN, {Sj} j sM (I\i< KO , $< No) of the circle group and ci = h’, (i E N) such that A = GUS and
CL) n(A)= i (iEN=} where G= U ieN
Gj, S= U
for
ilES,
for
AEG\S,
Sj.
ieM
Proof: The proof of this theorem is just the same as that of Theorem 4. Since the measure p is non-atomic, the measures ,U~(i E N) are also non-atomic, hence it follows that the groups G, are infinites. At the same time, if Gi (i E N) are infinites, then according to the proof of sufficiency of Theorem 4 we see that there exists a dynamical system (Y, 93, II, V) with discrete spectrum such that the measure v is non-atomic and (A, n(A)) is the spectrum of this system. Since the measures ,U and v are non-atomic, there exists an isomorphism R: (X, 9, n) + (Y, B, v). We put T= R- 1 VR. Thus, T is an automorphism on X and the operators U, and U, are unitarily equivalent. Since the operators U and U, are unitarily equivalent (they have the same spectrum (A, n(I))), the operators U and UT are unitarily equivalent. This completes the proof. REFERENCES [l] [2] [3] [4]
Choksi, J. R., Journ,. Math. Mech. 16.1 (1966), 83. -, Ill. Journ. Math.‘92 (1965), 307. Dunford, N., J. T. Schwartz, Linear operators, Vol. 2, Moscow, 1965 (in Russian). Plesner, A. L., The spectral theory of linear operators, Moscow, 1965 (in Russian).