Definitizability of normal operators on Krein spaces and their functional calculus

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by  Michael Kaltenbäck

Definitizability of normal operators on Krein
spaces and their functional calculus
Michael Kaltenba¨ck1
6
1
Abstract: We discuss a new concept of definitizability of a normal
0
2 operator on Krein spaces. For this new concept we develop a functional
calculus φ7→φ(N) which is the proper analogue of φ7→R φdE in the
n
Hilbert space situation.
a
J
Mathematics Subject Classification (2010): 47A60, 47B50, 47B15.
5
1
Keywords: Krein space, definitizable operators, normal operators, spectral theorem

]
A
1 Introduction
F
.
h
AboundedlinearoperatorN onaKreinspace( ,[.,.])isnormal,ifN commutes
t K
a with its Krein space adjoint N+. If we write N as A+iB with the selfadjoint
m realpartA:=ReN := N+N+ andtheselfadjointimaginarypartB :=ImN :=
2
[ N−N+, then N is normal if and only if AB = BA. In [K] we called a normal
2i
1 N definitizable wheneverAandB werebothdefinitizable inthe classicalsense,
v i.e.thereexistso-calleddefinitizingpolynomialsp(z),q(z) R[z] 0 suchthat
3 ∈ \{ }
[p(A)x,x] 0 and [q(B)x,x] 0 for all x .
7 ≥ ≥ ∈K
For such definitizable operators in [K] we could build a functional calculus
8
3 in analogy to the functional calculus φ φdE mapping the -algebra of
7→ ∗
0 bounded and measurable functions on σ(N) to B( ) in the Hilbert space case.
1. The functional calculus in [K] can also be sReen aHs a generalization of Heinz
0 Langersspectraltheoremondefinitizable selfadjointoperatorsonKreinspaces;
6 see [L], [KP]. Unfortunately, there are unsatisfactory phenomenons with this
1
concept of definitizability in [K]. For example, it is not clear, whether for a
:
v bijective, normal definitizable N also N−1 definitizable.
i Inthepresentpaperwechooseamoregeneralconceptofdefinitizability. We
X
shallsaythatanormalN onaKreinspace isdefinitizableif[p(A,B)u,u] 0
ar for all u for some, so-called definitizinKg, p(x,y) C[x,y] 0 with r≥eal
∈ K ∈ \{ }
coefficients. ThenwestudytheidealI generatedbyalldefinitizingpolynomials
with real coefficients in C[x,y], and assume that I is large in the sense that it
is zero-dimensional, i.e. dimC[x,y]/I < . By the way, if N is definitizable in
∞
the sense of [K], then I is always zero-dimensional.
Using results from algebraic geometry, under the assumption that I is zero-
dimensional, the variety V(I) = a C2 : f(a) = 0 for all f I is a finite
set. We split this subset of C2 up{as∈ ∈ }
V(I)=(V(I) R2)˙(V(I) R2),
∩ ∪ \
and interpret VR(I) :=V(I) R2 in the following as a subset of C by consider
∩
the first entry as the real and the second entry as the imaginary part.
1Thisworkwassupported byajointprojectofthe AustrianScience Fund(FWF, I1536–
N25)andtheRussianFoundationforBasicResearch(RFBR,13-01-91002-ANF).
1
By the ascending chain condition the ideal I is generated by real defini-
tizing polynomials p ,...,p . With the help of the positive semidefinite scalar
1 m
m
products[p (A,B).,.], j =1,...,mand [p (A,B).,.]weconstructHilbert
j k=1 k
spaces ,j =1,...,mand togetherwithboundedandinjectiveT :
j j j
and T H: . We consiHder Θ : (TPT+)′ (T+T )′ and Θ : (THT+)→′ K
H → K j j j → j j →
(T+T)′ by Θ (C) := (T T )−1(C) and Θ(C) := (T T)−1(C), as studied
j j j
in [KP]. Here (T T+)′,(TT×+)′ B( ) and (T+T )′ B×( ),(T+T)′ B( )
j j ⊆ K j j ⊆ Hj ⊆ H
denote the commutant of the respective operators.
Theproperfamily offunctionssuitablefortheaimedfunctionalcalculus
N
F
are functions defined on
σ(Θ(N)) VR(I) ˙(V(I) R2).
∪ ∪ \
Moreover, the function(cid:0)s φ N assume(cid:1)values in C on σ(Θ(N)) VR(I) and
∈ F \
valuesinacertainfinite dimensional -algebras (z)atz VR(I)and ((ξ,η))
at (ξ,η) V(I) R2. On σ(Θ(N))∗ VR(I) weAassume φ∈to be bounBded and
∈ \ \
measurable. Finally, φ satisfies a growth regularity condition at all w
N
∈ F
pointsfromVR(I)whicharenotisolatedinσ(Θ(N)) VR(I). Vaguelyspeaking,
∪
this growthregularityconditionmeans thataroundw the function φ admits an
approximation by a Taylor polynomial, which is determined by φ(w) (w).
Anypolynomials(x,y) C[x,y]canbeseenasafunctions ina∈naAtural
N N
∈ ∈F
way.
For each φ we will see that there exists p C[x,y] and bounded,
N
measurable f1,.∈..F,fm : σ(Θ(N)) VR(I) C with f∈j(z) = 0 for z VR(I)
∪ → ∈
such that
φ(z)=p (z)+ f (z)(p ) (z) (1.1)
N j j N
j
X
for all z σ(Θ(N)) VR(I), and that φ((ξ,η)) = pN((ξ,η)) for all (ξ,η)
V(I) R2∈. We then d∪efine ∈
\
m
φ(N):=p(A,B)+ T f dET+,
k k k
k=1 Zσ(Θk(N))
X
andshow thatthis operatordoes notdepend onthe actualdecomposition(1.1)
and that φ φ(N) is indeed a -homomorphismsatisfying φ(N)=s(A,B) for
7→ ∗
φ=s .
N
2 Multiple embeddings
In the present section ( ,[.,.]) will be a Krein space and ( ,(.,.)),
K H
( ,(.,.)), j = 1,...,m, will denote Hilbert spaces. Moreover, let T : ,
j
H H →K
T : and R : bounded, linear and injective mappings such
j j j j
thatHTR→=KT . By T+H: → H and T+ : we denote the respective
j j K → H j K → Hj
Krein space adjoints.
IfDisanoperatoronaKreinspace,thenweshalldenotebyD′ thecommu-
tant of D, i.e. the algebraof all operatorscommuting with D. For a selfadjoint
D this commutant is a -algebra with respect to forming adjoint operators.
For j = 1,...,m we∗shall denote by Θ : (T T+)′ ( B( )) (T+T )′ (
j j j ⊆ K → j j ⊆
B( )), and by Θ : (TT+)′ ( B( )) (T+T)′ ( B( )) the -algebra
j
H ⊆ K → ⊆ H ∗
2
homomorphisms mapping the identity operator to the identity operator as in
Theorem 5.8 from [KP] corresponding to the mappings T and T:
j
Θ (C )=(T T )−1(C )=T−1C T , C (T T+)′,
j j j × j j j j j j ∈ j j
Θ(C)=(T T)−1(C)=T−1CT, C (TT+)′. (2.1)
× ∈
We can apply Theorem 5.8 in [KP] also to the bounded linear, injective R :
j
, and denote the corresponding -algebra homomorphisms by Γ :
j j
H → H ∗
(R R∗)′ ( B( )) (R∗R )′ ( B( )):
j j ⊆ H → j j ⊆ Hj
Γ (D)=(R R )−1(D)=R−1DR , D (R R∗)′.
j j × j j j ∈ j j
For the following note that due to (ranT+)[⊥] = kerT = 0 the range of T+
{ }
is dense in .
H
2.1 Lemma. For j = 1,...,m we have Θ((T T+)′ (TT+)′) (R R∗)′
j j ∩ ⊆ j j ∩
(T+T)′, where in fact
Θ(C)R R∗ =R Θ (C)R∗ =R R∗Θ(C), C (T T+)′ (TT+)′. (2.2)
j j j j j j j ∈ j j ∩
Moreover,
Θ (C)=Γ Θ(C), C (T T+)′ (TT+)′. (2.3)
j j ◦ ∈ j j ∩
Proof. AccordingtoTheorem5.8in[KP]wehaveΘ (C)T+ =T+C andT+C =
j j j
Θ(C)T+ for C (T T+)′ (TT+)′. Therefore,
∈ j j ∩
T(R Θ (C)R∗)T+ =T Θ (C)T+ =T T+C
j j j j j j j j
=TR R∗T+C =T(R R∗Θ(C))T+.
j j j j
kerT = 0 and the density of ranT+ yield R Θ (C)R∗ =R R∗Θ(C). Apply-
{ } j j j j j
ing this equation to C+ and taking adjoints yields R Θ (C)R∗ = Θ(C)R R∗.
j j j j j
In particular, Θ(C) (R R∗)′. Therefore, we can apply Γ to Θ(C) and get
∈ j j j
Γ Θ(C)=R−1T−1CTR =T−1CT =Θ (C).
j ◦ j j j j j
❑
For the following assertionnote that by (2.3) and by the fact that Γ is a -
j
∗
algebrahomomorphismmapping the identity operatorto the identity operator,
for j =1,...,m we have
σ(Θ(C)) σ(Θ (C)) for all C (T T+)′ (TT+)′. (2.4)
⊆ j ∈ j j ∩
2.2 Corollary. For a j 1,...,m let N B( ) be normal such that N
(T T+)′ (TT+)′. Then Θ∈({N) is a no}rmal op∈eratoKr in the Hilbert space , an∈d
j j ∩ H
Θ (N) is a normal operator in the Hilbert space . Denoting by E (E ) the
j j j
H
spectral measure of Θ(N) (Θ (N)), we have E(∆) (R R∗)′ (T+T)′ and
j ∈ j j ∩
Γ (E(∆))=E (∆),
j j
for all Borel subsets ∆ of C, where E (∆) (R∗R )′ (T+T )′.
j ∈ j j ∩ j j
3
Moreover, hdE (R R∗)′ (T+T)′ and
∈ j j ∩
R
Γ hdE = hdE
j j
(cid:18)Z (cid:19) Z
for any bounded and measurable h : σ(Θ(N)) C, where hdE (R∗R )′
→ j ∈ j j ∩
(T+T )′.
j j R
Proof. The normality of Θ(N) and Θ (N) is clear, since Θ and Θ are -
j j
∗
homomorphisms. From Lemma 2.1 we know that Θ(N) (R R∗)′ (T+T)′.
∈ j j ∩
According to the well known properties of Θ(N)’s spectral measure we obtain
E(∆) (R R∗)′ (T+T)′ and,in turn, hdE (R R∗)′ (T+T)′. Inparticu-
∈ j j ∩ ∈ j j ∩
lar,Γ canbe appliedto E(∆) and hdE. Similarly,Θ (N) (T+T )′ implies
j R j ∈ j j
E (∆), hdE (T+T )′ for a bounded and measurable h.
j j ∈ j j R
Recall from Theorem 5.8 in [KP] that Γ (D)R∗x = R∗D for D (R R∗)′.
R j j j ∈ j j
Hence, for x and y we have
j
∈H ∈H
(Γ (E(∆))R∗x,y)=(R∗E(∆)x,y)=(E(∆)x,R y)
j j j j
and, in turn,
s(z,z¯)d(Γ (E)R∗x,y)= s(z,z¯)d(Ex,R y)=(s(Θ(N),Θ(N)∗)x,R y)
j j j j
C C
Z Z
=(R∗s(Θ(N),Θ(N)∗)x,y)=(Γ s(Θ(N),Θ(N)∗) R∗x,y)
j j j
for any s(z,w) C[z,w]. By (2.3) and the fact, tha(cid:0)t Γ is a -homo(cid:1)morphism,
j
∈ ∗
we have Γ (s(Θ(N),Θ(N)∗))=s(Θ (N),Θ (N)∗). Consequently,
j j j
s(z,z¯)d(Γ (E)R∗x,y)= s(z,z¯)d(E R∗x,y).
j j j j
C C
Z Z
Since E(C K)=0 and E (C K)=0 for a certain compact K C and since
j
the set of a\ll s(z,z¯), s C[z,w\], is densely contained in C(K), w⊆e obtain from
∈
the uniqueness assertion in the Riesz Representation Theorem
(Γ (E(∆))R∗x,y)=(E (∆)R∗x,y) for all x , y ,
j j j j ∈H ∈Hj
for all Borel subsets ∆ of C. Due to the density of ranR∗ in we even have
j Hj
(Γ (E(∆))z,y)=(E (∆)z,y) for all y,z , and in turn Γ (E(∆))=E (∆).
j j j j j
∈H
Since Γ maps into (R∗R )′, we have E (∆) (R∗R )′. This yields hdE
j j j j ∈ j j j ∈
(R∗R )′ for any bounded and measurable h.
jIfjh : σ(Θ(N)) C is bounded and measurable, then by (2.4R) also its
→
restriction to σ(Θ (N)) = σ(Γ Θ(N)) is bounded and measurable. Due to
j j
◦
E (∆)R∗ =Γ (E(∆))R∗ =R∗E(∆), for x and y we have
j j j j j ∈H ∈Hj
(Γ hdE R∗x,y)=(R∗ hdE x,y)=( hdE x,R y)
j j j j
(cid:18)Z (cid:19) (cid:18)Z (cid:19) (cid:18)Z (cid:19)
= hd(Ex,R y)= hd(E R∗x,y)=( hdE R∗x,y).
j j j j j
Z Z (cid:18)Z (cid:19)
The density of ranR∗ yields Γ hdE = hdE . ❑
j j j
(cid:0)R (cid:1) R
4
Recall from Lemma 5.11 in [KP] the mappings (j =1,...,m)
Ξ :B( ) B( ), Ξ (D )=T D T+,
j Hj → K j j j j j
and Ξ:B( ) B( ) with Ξ(D)=TDT+. By (j =1,...,m)
H → K
Λ :B( ) B( ), Λ (D )=R D R∗,
j Hj → H j j j j j
we shall denote the corresponding mappings outgoing from the mappings R :
j
. Due to T =TR we have Ξ =Ξ Λ .
j j j j j
H →H ◦
According to Lemma 5.11 in [KP], Λ Γ (D) = DR R∗ for D (R R∗)′.
j ◦ j j j ∈ j j
Hence, using the notation from Corollary 2.2
Ξ ( hdE )=Ξ Λ Γ hdE =Ξ(R R∗ hdE). (2.5)
j j j ◦ j j j
Z (cid:16) (cid:18)Z (cid:19)(cid:17) Z
2.3Lemma. Assumethatfor j 1,...,m theoperator T T+ commuteswith
∈{ } j j
TT+ on . Then the operators R R∗, T+T commute on and R∗R , T+T
K j j H j j j j
commute on . Moreover,
j
H
Θ(T T+)=R R∗T+T =T+TR R∗. (2.6)
j j j j j j
Proof. If T T+ and TT+ commute on , then
j j K
T(T+TR R∗)T+ =TT+T T+ =T T+TT+ =T(R R∗T+T )T+.
j j j j j j j j
EmployingT’sinjectivityandthedensityofranT+,weseethatR R∗ andT+T
j j
commute. From this we derive
T+T R∗R =R∗(T+TR R∗)R =R∗(R R∗T+T)R =R∗R T+T .
j j j j j j j j j j j j j j j j
(2.6) follows from
T−1T T+T =T−1TR R∗T+T =R R∗T+T .
j j j j j j
❑
3 Definitizability
In [K] we said that a normal N B( ) is definitizable, if its real part A :=
N+N+ and its imaginary part B∈:= NK−N∗ are definitizable in the sense that
2 2i
there exist real polynomials p,q R[z] 0 such that [p(A)v,v] 0 and
∈ \ { } ≥
[q(B)v,v] 0 for all v . In the present note we will relax this condition.
≥ ∈K
3.1 Definition. For a normal N B( ) we call p(x,y) C[x,y] 0 a
∈ K ∈ \ { }
definitizing polynomial for N, if
[p(A,B)v,v] 0 for all v . (3.1)
≥ ∈K
whereA= N+N+ andB = N−N+. Ifsuchadefinitizingp C[x,y] 0 exists,
2 2i ∈ \{ }
then we call N definitizable normal. ♦
5
Clearly, we could also write p as a polynomial of the variables N and N+.
ButbecauseofA=A+ andB =B+,writingpasapolynomialofthe variables
A and B has some notational advantages.
3.2 Remark. According to (3.1) the operator p(A,B) B( ) must be selfad-
∈ K
joint; i.e. p(A,B)+ =p#(A,B), where p#(x,y)=p(x,y). Hence, q := pj+p#j is
2
real, i.e. q(x,y) R[x,y] 0 , and satisfies q(A,B) = p(A,B). Thus, we can
∈ \{ }
assume that a definitizing polynomial is real. ♦
In the present section we assume that p (x,y) R[x,y] 0 , j =1,...,m,
j
∈ \{ }
are real, definitizing polynomial for N.
3.3 Proposition. With the above assumptions and notation there exist Hilbert
spaces ( ,(.,.)), ( ,(.,.)), j = 1,...,m and bounded linear and injective op-
j
H H
erators T : , T : , such that
j j
H→K H →K
m m
T T+ =p (A,B), and TT+ = T T+ = p (A,B).
j j j k k k
k=1 k=1
X X
Proof. Let ( ,(.,.)) be the Hilbert space completion of /kerp (A,B) with
j j
H K
respect to [p (A,B).,.] and let T : be the adjoint of the factor
j j j
mapping x x+kerp (A,B) of inHto →.KSince T+ has dense range, T
7→ j K Hj j j
must be injective. Similarly, let ( ,(.,.)) be the Hilbert space completion of
m H m
/(ker p (A,B))withrespectto[ p (A,B) .,.]andletT :
K k=1 k k=1 k H→K
be the injective adjoint of the factor mapping of into .
FromP[TT+x,y] = (T+x,T+y) =(cid:0)P(x,y) =K [ (cid:1)Hm p (A,B) x,y] and
k=1 k
[T T+x,y]=(T+x,T+y)=(x,y)=[p (A,B)x,y] for all x,y we conclude
j j j j j (cid:0)P ∈K (cid:1)
m
T T+ =p (A,B) and TT+ = p (A,B),
j j j k
k=1
X
wheretheoperatorsT T+ =p (A,B),j =1,...,m,pairwisecommute,because
j j j
A and B do. ❑
Proposition3.3 in particular yields
m
TT+ = T T+ (3.2)
k k
k=1
X
Since for x and j 1,...,m we have
∈K ∈{ }
m m
(T+x,T+x)=[TT+x,x]= [T T+x,x]= (T+x,T+x) (T+x,T+x),
k k k k ≥ j j
k=1 k=1
X X
one easily concludes that T+x T+x constitutes a well-defined, contractive
7→ j
linear mapping from ranT+ onto ranT+. By (ranT+)⊥ = kerT = 0 and
j { }
(ranT+)⊥ = kerT = 0 these ranges are dense in the Hilbert spaces and
j j { } H
. Hence, there is a unique bounded linearcontinuationofT+x T+xto ,
Hj 7→ j H
which has dense range in .
j
H
6
Denoting by R the adjoint mapping of this continuation we clearly have
j
T =TR and kerR kerT = 0 . From (3.2) we conclude
j j j j
⊆ { }
m m
T(I )T+ =TT+ = TR R+T+ =T( R R+)T+.
H k k k k
k=1 k=1
X X
kerT = 0 and the density of ranT+ yield m R R∗ =I .
{ } k=1 k k H
3.4 Lemma. With the above notations and Passumptions for j =1,...,m there
existinjectivecontractionsR : suchthatT =TR and m R R∗ =
j Hj →H j j k=1 k k
I . Moreover, we have
H
P
N,N+ ′ = A,B ′ (T T+)′ (TT+)′ (3.3)
{ } { } ⊆ k k ⊆
k=1,...,m
\
for all j 1,...,m . Finally,
∈{ }
m
p (Θ(A),Θ(B))=R R∗ p (Θ(A),Θ(B))
j j j k
k=1
(cid:0)X (cid:1) (3.4)
m
= p (Θ(A),Θ(B)) R R∗,
k j j
k=1
(cid:0)X (cid:1)
and for any u C[x,y]
∈
p (A,B)u(A,B)=Ξ u(Θ (A),Θ (B)) =Ξ R R∗u(Θ(A),Θ(B)) , (3.5)
j j j j j j
whereΘ:(TT+)′ ( B((cid:0))) (T+T)′ ( (cid:1)B( ))(cid:0)isas in (2.1)andΞ:(cid:1)B( )
⊆ K → ⊆ H H →
B( ) with Ξ(D)=TDT+.
K
Proof. The first part was shown above, and (3.3) is clear from Proposition 3.3.
From (2.6) and Theorem 5.8 in [KP] we get
p (Θ(A),Θ(B))=Θ(p (A,B))=Θ(T T+)=R R∗T+T =R R∗Θ(TT+)
j j j j j j j j
m m
=R R∗Θ( p (A,B))=R R∗ p (Θ(A),Θ(B)) ,
j j k j j k
k=1 k=1
X (cid:0)X (cid:1)
where R R∗ commutes with T+T = m p (Θ(A),Θ(B)) by Lemma 2.3. Fi-
j j k=1 k
nally, (3.5) follows from (see Lemma 5.11 in [KP])
P
p (A,B)u(A,B)=Ξ Θ (u(A,B)) =Ξ Λ Γ Θ(u(A,B))
j j j j j
◦ ◦
=Ξ (cid:0)RjRj∗u(Θ(A),(cid:1)Θ(B)) . (cid:0) (cid:1)
❑
(cid:0) (cid:1)
By (3.3) we can apply Corollary 2.2 in the present situation. In particular,
Θ(N) is a normal operator on the Hilbert space . Condition (3.1) for p =
H
p , j =1,...,m, implies certain spectral properties of Θ(N).
j
3.5 Lemma. With the above assumptions and notation for j 1,...,m we
∈ { }
have
m
z C: p (Rez,Imz) > R R∗ p (Rez,Imz) ρ(Θ(N)).
{ ∈ | j | k j jk·| k |}⊆
k=1
X
In particular, the zeros of m p (Rez,Imz) in C are contained in ρ(Θ(N))
z C:p (Rez,Imz)=0 fko=r1alkl j =1,...,m . ∪
j
{ ∈ P }
7
Proof. Let n N and set
∈
m
1
∆ := z C: p (Rez,Imz)2 > + R R∗ 2 p (Rez,Imz)2 .
n { ∈ | j | n k j jk ·| k | }
k=1
X
For x E(∆ )( ), where E denotes Θ(N)’s special measure, we then have
n
∈ H
p (Θ(A),Θ(B))x 2 = p (Reζ,Imζ)2d(E(ζ)x,x)
j j
k k | | ≥
Z∆n
m
1
d(E(ζ)x,x)+ R R∗ 2 p (Reζ,Imζ)2d(E(ζ)x,x)
n k j jk | k |
Z∆n Z∆n k=1
X
m
1
x 2+ R R∗ p (Θ(A),Θ(B)) x 2.
≥ nk k k j j k k
k=1
(cid:0)X (cid:1)
By (3.4) this inequality can only hold for x = 0. Since ∆ is open, by the
n
Spectral Theorem for normal operators on Hilbert spaces we have ∆ ρ(N).
n
⊆
The asserted inclusion now follows from
m
z C: p (Rez,Imz) > R R∗ p (Rez,Imz) = ∆ .
{ ∈ | j | k j jk·| k |} n
k=1 n∈N
X [
❑
InthefollowingletI theideal p ,...,p generatedbytherealdefinitizing
1 m
polynomials p ,...,p in the rinhg C[x,y]. Ti he variety V(I) is the set of all
1 m
common zeros a = (a ,a ) C2 of all p I. Clearly, V(I) coincides with the
1 2
set of all a C2 such that p∈1(a1,a2)= ∈=pm(a1,a2)=0. VR(I) is the set of
all a R2,∈which belong to V(I). It is c·o·n·venientfor our purposes, to consider
VR(I)∈as a subset of C:
VR(I):= z C:f(Rez,Imz)=0 for all f I (3.6)
{ ∈ ∈ }
= z C:p (Rez,Imz)=0 for all k 1,...,m .
k
{ ∈ ∈{ }}
3.6 Corollary. Let E denote the special measure of Θ(N). Then we have
p (Rez,Imz)
RjRj∗E(C\VR(I))=E(C\VR(I))RjRj∗ =ZC\VR(I) mk=j1pk(Rez,Imz)dE(z).
Proof. First note that the integral on the righPt hand side exists as a
bounded operator, because by Lemma 3.5 we have p (Rez,Imz) R R∗
m p (Rez,Imz) for z σ(Θ(N)). The firstequ|ajlity is known|f≤rokmCj orjokl-·
| k=1 k | ∈
lary 2.2.
P
Concerning the second equality, note that both sides vanish on the range of
E(VR(I)). Its orthogonalcomplement :=ranE(C VR(I)) is invariant under
Q \
m m
p (Rez,Imz) dE(z)= p (Θ(A),Θ(B)).
k k
Z k=1 k=1
(cid:0)X (cid:1) X
8
By Lemma 3.5 the restriction of this operator to is injective, and
Q
hence, has dense range in . If x belongs to this dense range, i.e. x =
m Q
p (Θ(A),Θ(B)) y with y , then
k=1 k ∈Q
(cid:0)P p (Rez,(cid:1)Imz)
j
dE(z)x= p (Rez,Imz)dE(z)y
m j
ZC\VR(I) k=1pk(Rez,Imz) ZC\VR(I)
m
P=p (Θ(A),Θ(B))y =R R∗ p (Θ(A),Θ(B)) y =R R∗x.
j j j k j j
k=1
(cid:0)X (cid:1)
By a density argument the second asserted equality of the present corollary
holds true on and in turn on . ❑
Q H
3.7Remark. InProposition3.3thecasethatp (A,B)=0forsomej,orevenfor
j
all j, is not excluded, and yields = 0 , T =0 and R =0 (in Lemma 3.4),
j j j
H { }
oreven = 0 andT =0. Alsotheremainingresultsholdtrue,ifweinterpret
ρ(R) asHC an{d}σ(R) as for the only possible linear operator R = (0 0) on
∅ 7→
the vector space 0 . ♦
{ }
4 An Abstract Functional Calculus
In this section let be again a Krein space, N B( ) be a definitizable
normal operator. LKet I be the ideal in C[x,y], whi∈ch isKgenerated by all real
definitizing polynomials. By the ascending chain condition for the ring C[x,y]
(see for example [CLO1], Theorem 7, Chapter 2, 5) I is generated by finitely
§
many realdefinitizing polynomials p ,...,p , i.e. I = p ,...,p . We employ
1 m 1 m
h i
the samenotionasinthe previoussectionsforthese polynomialsp ,...,p . In
1 m
particular, E (E) denotes the spectral measure of Θ (N) on (Θ(N) on ).
j j j
We also make the convention that for p C[x,y] and z HC we write pH(z)
∈ ∈
short for p(Rez,Imz).
4.1 Lemma. For a bounded and measurable f : σ(Θ(N)) C and j
→ ∈
1,...,m we have
{ }
Ξ fdE =
j j
Zσ(Θj(N)) !
p
Ξ f j dE+R R∗ fdE .
m p j j
Zσ(Θ(N))\VR(I) l=1 l Zσ(Θ(N))∩VR(I) !
P
Proof. By (2.5) the left hand side coincides with
Ξ R R∗ fdE+R R∗ fdE .
j j j j
Zσ(Θ(N))\VR(I) Zσ(Θ(N))∩VR(I) !
fdE = E(C VR(I)) fdE together with Corol-
σ(Θ(N))\VR(I) \ σ(Θ(N))\VR(I)
lary 3.6 prove the equality. ❑
R R
9
4.2 Lemma. Let f,g : σ(Θ(N)) C be bounded and measurable, and let
r C[x,y]. For j,k 1,...,m we→then have
∈ ∈{ }
r(A,B)Ξ ( fdE )=Ξ ( fdE )r(A,B) (4.1)
j j j j
Zσ(Θj(N)) Zσ(Θj(N))
=Ξ ( rfdE ),
j j
Zσ(Θj(N))
and
Ξ fdE Ξ gdE (4.2)
j j k k
Zσ(Θj(N)) ! Zσ(Θk(N)) !
p p
j k
=Ξ fg dE
m
Zσ(Θ(N)) l=1pl !
P
=Ξ fgp dE =Ξ fgp dE .
j k j k j k
Zσ(Θj(N)) ! Zσ(Θk(N)) !
Proof. By Lemma 5.11 in [KP] we have
r(A,B)Ξ (D)=Ξ (Θ(r(A,B))D)=Ξ (r(Θ (A),Θ (B))D),
j j j j j
Ξ (D)r(A,B)=Ξ (DΘ (r(A,B)))=Ξ (Dr(Θ (A),Θ (B)))
j j j j j j
for D (T+T)′. For D = fdE this implies (4.1).
∈ σ(Θj(N)) j
According to (2.5) the expression in (4.2) coincides with
R
Ξ R R∗ fdE Ξ R R∗ gdE .
j j k k
Zσ(Θ(N)) ! Zσ(Θ(N)) !
By Lemma 5.11 and Theorem 5.8 in [KP], we also know that Ξ(D )Ξ(D ) =
1 2
Ξ(T+TD D )=Ξ(Θ(TT+)D D ), where (see Proposition 3.3 and (3.6))
1 2 1 2
m m m
Θ(TT+)= pl(Θ(A),Θ(B))= pldE =( pldE)E(C VR(I)).
\
l=1 Z l=1 Z l=1
X X X
Therefore, by Corollary 3.6 and the fact, that E(C VR(I)) commutes with
\
fdE, (4.2) can be written as
σ(Θ(N))
R
m
p p
j k
Ξ ( p dE)( dE)( fdE)( dE)( gdE) =
l m p m p
Z l=1 Z l=1 l Z Z l=1 l Z !
X
P P
p p
j k
Ξ fg dE .
m p
Zσ(Θ(N)) l=1 l !
TheremainingequalitiesfollowfromLemPma4.1sincetherespectiveintegrands
vanish on VR(I). ❑
4.3 Lemma. For a bounded and measurable f : σ(Θ(N)) C and j
→ ∈
1,...,m the operator Ξ fdE belongs to N,N+ ′′.
{ } j σ(Θj(N)) j { }
(cid:16)R (cid:17)
10