Advances in Pure Mathematics, 2013, 3, 29-33
http://dx.doi.org/10.4236/apm.2013.31006 Published Online January 2013 (http://www.scirp.org/journal/apm)
Factorization of Operators in Krein Spaces and
Linear-Fractional Relations of Operator Balls
Victor Anatoly Khatskevich1, Valery Anatoly Senderov2
1Department of Mathematics, ORT Braude Academic College, Karmiel, Israel
2Pyatnitskoe Highway, 23-2-156, Moscow, Russia
Received July 22, 2012; revised September 18, 2012; accepted September 29, 2012
ABSTRACT
We consider plus-operators in Krein spaces and generated operator linear fractional relations of the following form:
   
:andare contractions between Hilbert spaceand
F
KQABKQCDKK QXY

.
We study some special type of factorization for plus-operators T, among them the following one: T = BU, where B is a
lower triangular plus-operator, U is a J-unitary operator. We apply the above factorization to the study of basical prop-
erties of relations (1), in particular, convexity and compactness of their images with respect to the weak operator topol-
ogy. Obtained results we apply to the known Koenigs embedding problem, the Krein-Phillips problem of existing of
invariant semidefinite subspaces for some families of plus-operators and to some other fields.
Keywords: Krein Space; Linear Fractional Relation; Plus-Operator; Factorization
1. Introduction Since that time the invariant subspace problem has
become one of the central problems in the theory of spaces
with indefinite metric (see [6]), and the spaces of the
class axiomatized by M. G. Krein have been called Krein
spaces.
During World War II, one of the burning problems was
the firing accuracy of the new rocket weapon called “Ka-
tyusha”.
In 1943, S. L. Sobolev showed how to control the fir-
ing accuracy by using an eigenvector with a special ei-
genvalue of a linear operator A that is self-adjoint with
respect to an indefinite metric and whose canonical ex-
pansion contains a single negative square (Sobolev’s
paper was published in an administrative military journal
and appeared in the public press only in 1960 [1]). In
1944, L. S. Pontryagin [2] generalized Sobolev’s result
as follows: he proved that, for an operator A that is self-
adjoint with respect to an indefinite metric and has a fi-
nite number of negative squares, there exists a
Factorizations of different types are considered for op-
erators in Krein spaces. The fundamental paper [7] deals
with the so-called “J-polar decomposition of operators”
which is an analog and a natural generalization of the
usual polar decomposition of operators in Hilbert spaces.
Another type of decompositions is the factorization of
operators of the form
(1)
A
UB

or

-
dimensional invariant subspace such that the spectrum of
the restriction to this subspace satisfies a special condi-
tion.
CU
A
, (2)

where B is an upper triangular operator, C is a lower
block triangular operator, and U and are J-unitary
operators.
U

Lin ,2,CeiH
In 1950, M. G. Krein successfully used a fixed point
principle to prove the equivalence between the existence
of a specific invariant subspace of an operator A and the
fact that a linear-fractional mapping FA has a fixed point;
he also obtained several new results in this field.
It follows from the results presented below that each
strict plus-operator (the definition is given below) admits
factorization (1). We show that this is not the case for
factorization (2).

Example. Let 1i,

Lin eH21 ,
and let 1ii
In 1959-1960, R. S. Phillips published several papers
[3-5], where he showed how to solve some systems of
differential equations by using extensions of invariant
subspaces of several operator families.
A
ee


for iN


10aUe 
. We also assume that A =
BU is factorization (2), 1


12
BUa Be. Then H1 AHH. On the other hand, . There-

C
opyright © 2013 SciRes. APM

V. A. KHATSKEVICH, V. A. SENDEROV
30
fore, we have and 0 0aAa 

. But a significant generalization of the factorization
theory (to arbitrary linear operators) opens new ways for
studying the relations. Let us consider several results.
(By Lin we denote the linear span of a set, and by
, its closure).
LinC

Factorizations (1) and (2) are rather useful tools for
studying both the operators in spaces with indefinite
metric and the so-called linear-fractional relations of op-
erator balls. Several results contained in the present paper
have not yet been published; the others were published
recently and are practically unknown. But, to make our
presentation clear and consistent, we begin with one of
the classical schemes.
The most of them were obtained for bistrict plus-op-
erators in [15].
Let us introduce the definitions and notation used in
the present paper.
For more details, see [6].
2. Basic Results
Let be the open unit ball of the space


1
,HH
A

 
1
11 12
2
,
where 1 and 2 are Hilbert spaces, and let be a
linear-fractional mapping of the form
H H
 
21 22A
K
AAKAAK

,
where ij,

i
1,HHA,1,2ij

, 2
,1
ij ij
AA

, and
, and
K11 12
A
A


K is invertible.
Since the mappings satisfy the “chain rule”
121 2
AAAA , factorizations (1) and (2) lead to the
decomposition of the linear-fractional mapping A into
an automorphism U of the ball and a linear-fractional
mapping


B

 
A
fixing the origin or into an affine linear-
fractional mapping and an automorphism .
C U
These facts have interesting applications. As an exam-
ple, we mention the well-known problem, namely, the
problem of the geometric and topological properties of
some operator sets (see, for example, [8-14]).
The further development of this theory led to the
problem of describing the structure of operator sets also
in the case where is a multivalued generating func-
tion.

Definition. The formula


11 122122
A
AK KAAK

K,for,1,2,ij
,
where K, , and
determines a linear-fractional relation (l.f.r.)
A


,

ijj i
AHH
K
K  in . Note that if 11 12 
A
AK

is
invertible for all , then becomes a linear
fractional mapping defined earlier.
K
1
H
A

There does not exist any direct generalization of the
mapping factorization theory to the case of linear-frac-
tional relations. The point is that, even for very simple
generating operators, the basic (“chain”) equality may
not hold in the case of relations.
Example. Let , where ,

ni eLiCi


i
e is
an orthonormal basis; 1ii
s
ee

; and
.
0
00
u 

10Ke
I
0



00
s
v
Clearly, uv are operators K such that

. On
the other hand, we have .

0
uIm 
H
12
Let be a Krein space:

HH H
H
 
H
,
where 1 and 2
H are nontrivial complex Hilbert spaces;
1 and 2 are the orthogonal projection operators
corresponding to this decomposition of : ii

HH
1, 2i, where

and 12
I

 xH. For any , we set
ii
x
x

1, 2, where i

; the indefinite metric


,


H
in
the space is given by the formula




1122
,, ,
x
yxy xy .
Here 12
x
xx12 yyy, H

, , and







0SS



is the
(usual) inner product.
A set S is said to be nondegenerate if ,
where


:,0 for all Sy yzzS
 H.
By

P and

P

we denote the sets of all nonnega-
tive and all nonpositive vectors, respectively:



:, 0,xxx
 PH




:, 0.xxx
 PH
:P To each lineal (linear manifold)  there cor-
responds a contraction


12
:KH so that


 
1111
,xKxx  .

The operator

K

is called the angular operator of
the lineal .
A lineal

P


is said to be uniformly positive if
2
,
x
xd x

for all x and some


0.d This is equivalent to
the condition


K1

e uniformly negative lineal
is defined similarly.
. Th
:
If the lineal

P is maximal (with respect to
inclusion), then 11

H

, and hence K

. From
now on, is the closed unit ball of the space



1
,HH1 H
H
2
of bounded linear operators acting from
into and is its interior:
21KK 

AH
Im AH

.
A linear operator A with domain D and
range (image) is called a plus-operator if


AA


PDP.
The main objects of study in the present paper are the

V. A. KHATSKEVICH, V. A. SENDEROV 31
plus-operators , where denotes the
set of all bounded linear operators with

H:AA

H

A

DH

.
A plus-operator A is said to be strict if



,1 , 0AAxAx
c
inf
xx

,
and bistrict if, along with A, the operator
A
that is the
adjoint of A with respect to the indefinite metric


,

:


,c ,
A
xy
cxAy
 ,xy

, is also strict. In this case,
H
A
JA J

, where .
12
If A is a bistrict operator, then
J

c


A
A 
V
 .
An operator is said to be J-expansive if

H




, ,Vx Vxxx for all xH
c
V

,
and J-bi-expansive if both operators V and are J-
expansive.
Each strict plus-operator is collinear to a J-expansive
operator, and each bistrict plus-operator is collinear to a
J-bi-expansive operator.
Each plus-operator A determines a linear-fractional re-
lation on the ball
A:
 

11 12

21 22
:
A
K
KAAK

  KAAK

0V  
.
An operator V is said to be focusing if there is a con-
stant such that


2
,Vx VxVVx  for all x P

H
Im
.
An operator U is said to be J-unitary if
and
UH




, ,Ux Uxxx for all xH
V
2
.
Each J-unitary operator V is determined by the fol-
lowing three parameters: and operators 1 and
unitary in and H, respectively, by the relation
2
V1 H
 
 
11
22
12
11
22
12
I
VI
V
IV


  


V
IV












AH
12
.
Theorem 1. Let . Then th e following asser-
tions are equivalent:
1) 11 111211 11
A
AAA



0ddA
AH
dAA

, where ;
2) The lineal is uniformly positive;
1
3) 12 11
A
AK

K for some


0B
;
4) A admits the factorization
A = BU,
where U is J-unitary and 12

.
Now we outline the proof of Theorem 1.
1) 2) Let
1
y
Ax

11
xH
yP
, where . Then
, which implies




1,
2
11 111 1
,,
y
ydAxAx

d yy

 



A H
1
H

.
2) 3) It suffices to consider the angular operator
of the lineal 1  and continue it to the entire
space preserving the norm.

3) 4) Let us consider the operator

H

12
,HH

V
whose block-matrix in the basis has the form


11
22
11
22
IKKKIKK
V
KI KKI KK










 

1
BAV
12 0,BABU
.
K is strictly contractive, as follows from (c). The op-
erator V (and hence the operator V) is J-unitary
(straightforward calculated). We set and obtain
1 UV



, where .

A

UB 4) 1) We have  BHH
U
1
AH
, where 11
and the J-unitary operator takes uniformly positive
lineals to uniformly positive ones, which implies that the
lineal is uniformly positive, and hence assertion 1)
follows.
It is natural to compare the statements of Theorem 1
with the following proposition.
Theorem 2. The plus-operator A is exactly bistrict if


11
0
A
 11 , that is, 0 is a regular point of
A
,
1
11 121AA
1
21 111AA

, and .


Lemma. Assume that the lineal
A
H
11 11
11 11
is nonnegative
under the conditions of Theorem 2. Then A is a focusing
strict plus-operator.
Proof of the lemma. It is easy to show that
A
AZ
AWAWA Z



,
where 1W

and 1Z

(cf. [16], Proposition 2.3).
Thus, for x

P

, we have





22
11 11
22
1
111
2
2
22 2
23
,1
1
1,
2
Ax AxWAxAZx
cAx Zx
cx Zx
cZxcx






 
 





0c0c0

AH


where , and c.
1 23
On the other hand, for some , the inequality
2
,
A
xAxcA x

0cA
x, where , exactly holds for all


P

if A is a focusing strict plus operator. This
can be easily proved using ([6], Theorem 2.4.11).
Proof of Theorem 2. It follows from the lemma and ([6],
Corollary 2.4.5) that A is a strict plus-operator. Since


011
A
 , this implies that the plus-operator A is bis-
trict ([6], Theorem 2.4.17). 
We illustrate the new methods and approaches listed
above by an obvious example.
Namely, we use the method of operator factorization
in indefinite spaces that genetically originates from T. Ya.

V. A. KHATSKEVICH, V. A. SENDEROV
32
Azizov’s work [15]. This method allows us to prove that,
for any strict plus-operator A satisfying the condition
1112 is a definite operator: D ≥ 0 or D ≤ 0,
the set A is convex and compact in the weak op-
erator topology (w.o.t.).
11 12
A AA


Im 
 

DA
To prove this, we need some auxiliary assertions con-
cerning the case of a strict plus-operator A with an arbi-
trary operator D.
Lemma 3. Let A be a strict plus operator, and let
1. Then and A
H

 are uniformly negative
subspaces.
Proof. Let . Then x 

A
 
,,xAxH

,0xH
111
. Thus,


Ker 1
x
PA

PA
,
where 1 is a strict plus operator. Because of ([6],
Proposition 2.4.14), the subspace 1 is uni-
formly negative. The end of the proof follows from the
relation
PA

Ker
J


(where ).
12
Now we prove that, in the case of an arbitrary strict
plus-operator A, the lineal contains a “sufficiently
large” positive lineal. If 1, then
J 
AH
dim 
1
H
A

AH

1
dH
dim H
H

1
dim
is a strict
plus-operator, which implies that the subspace 1 is
positive and . Now we assume that
.
1
Hdim imA
1
We have the following theorem.
H
Theorem 4. Let 1, and let A be a strict
plus-operator. Then, there exists an infinite-dimensional
positive lineal in .
1
To prove this theorem, it suffices to prove the follow-
ing assertion.
A
Lemma. Assume that, under the conditions of Theo-
rem 4, 1 is some maximal (with respect to inclusion)
positive lineal contained in . Then
1
AH

.
Proof of the lemma. Let 1
A

H, and let 1 dim

.
Then


1, where . On the other hand,
using Lemma 3, it is easy to prove that

 

P






1
H= . Indeed



 
11
 





  .
Hence 1 is the maximum positive subspace, and this is
a contradiction.
Further, we shall need the following proposition, which
can also be proved by using Lemma 3.
Proposition. If A is a strict plus-operator, then
1
A H is nondegenerate; if





P 
 , where

, then  and   are uniformly positive and
uniformly negative subspaces, respectively.

Proof. The nondegeneracy of 






 
 H

1
A H
follows from the
fact that is (uniformly) negative. Further, since




 , the definite lineals in paren-
theses are closed and uniformly definite.
Corollary. If there exists a finite-dimensional maximal
negative lineal in , then the lineal 



0DAH
Im 

is uniformly positive.
We return to the case of a strict plus-operator A with a
definite operator D. It follows from Theorem 4 that
. Further, it follows from the proposition that the
lineal is uniformly positive.
1
Thus, to complete the proof, it suffices to prove the
following assertion.
Theorem 5. Assume that a plus-operator A satisfies
the conditions of Theorem 1. Then the set A is
convex and compact in the weak operator topology
(w.o.t.) (see the definition in [17]).
Proof (cf. [18]). Since U is a linear-fractional map-
ping of the ball 
ABU
onto itself, we have

,
see ([19], Proposition 4.20). Because of the affine rela-
tion
B
 and the equality

U, this readily im-
plies that the set is convex.
 
Im A
It remains to prove that the set

B
 is compact in
w.o.t.
Since the ball 
1

21 22
is a compact set in w.o.t., its image
is also a compact set under the continuous mapping
K
BBK


  . Hence 1 is a closed subset of the
space


,HH12 . Therefore, the complete preimage


B
  of the set 1 under the continuous mapping
11
K
KB



 of the ball is also closed. Thus, since 

is a compact set,


Im Im
B is also a compact set. The
proof of the theorem is complete. 
We consider another application, namely, we show
how both the factorization of operators and the linear-
fractional relations can be used to prove the existence of
invariant subspaces.
Theorem 6. (cf. [20]. 1) Under the conditions of Theo-
rem 1, 1211
A
A.
2) Under the conditions of Theorem 1, assume in addi-
tion that


1
122221 1112
AA AAA


,




1
122221 1112
that is,
A
AAAA


MA

is a compact operator. Then
there exists an such that and .

To prove this theorem, we need the following auxiliary
lemmas.
Lemma 7. The norms generated by different decom-
positions of the form

12
Proof. This lemma readily follows from the Banach
theorem. 



HH H are equivalent.
Lemma 8. ([6], Remark 3.2.4) For the existence of the
decomposition


12
, where the matrix of the
operator A has the property 12



HHH A

, it is necessary
and sufficient that there exist a uniform contraction




21
,QHH
222112 11
BQAQAQA AQ
such that



.
Lemma 9. ([6], Proposition 3.3.4) A subspace


M
with an angular operator K is exactly invariant under the

V. A. KHATSKEVICH, V. A. SENDEROV
Copyright © 2013 Sci APM
33

Res.
action of a strict plus-operator A if
K
AK
Im
.
Proof of Theorem 6. 1) It follows from Theorem 1 that
the factorization (2) and hence the inclusion
12 11
A

2 1
,HH

11
11 12
AA
exist, and hence .
1
11 12
AA

Im
A
2) By setting , we obtain
1
11 12
QAA


11 12
AA

22 2122 21
BQA AQAA



 .
It follows from Lemma 7 that the operator A is
bounded in the norm generated by the decomposition in
Lemma 8. This, as usual, permits considering the linear-
fractional relation A
, where 
A


12
,

HH
is the matrix of the
operator A in the basis . It easily follows from
the inclusion 12

A

 that the linear-fractional relation
A
 is a closed (multivalued) mapping. Thus, we are
under the conditions of Glicksberg’s theorem [6], which
implies that A
 has at least one fixed point in the ball




11X (here and

12

HHX1 

is the norm
generated by the new decomposition).
By Lemma 9, this implies that the operator A has a
maximal nonnegative invariant subspace.
3. Conclusions
We note that statement 2) of Theorem 6 holds for any
operator with the property
Im A

P

M
. (3)
Indeed, for any such that , we
have .
Im A
A
In particular, each nonstrict plus-operator in a Krein
space satisfies condition (3).
We note that the method of mapping factorization
constantly finds new applications. In particular, in [18],
precisely this method permits obtaining new conditions
under which the given operator is a plus-operator.
REFERENCES
[1] S. L. Sobolev, “On the Motion of a Symmetric Top with a
Cavity Filled with a Liquid,” Zh. Prikl. Mekhan. i Tekhn.
Fiz., No. 3, 1960, pp. 20-55.
[2] L. S. Pontryagin, “Hermitian Operators in Spaces with
Indefinite Metric,” Izvestiya Akademii Nauk SSSR, Seriya
Matematicheskaya, Vol. 8, 1944, pp. 243-280.
[3] R. S. Phillips, “Dissipative Operators and Hyperbolic
Systems of Partial Differential Equations,” Transactions
of the American Mathematical Society, Vol. 90, No. 2,
1959, pp. 193-254.
doi:10.1090/S0002-9947-1959-0104919-1
[4] R. S. Phillips, “Dissipative Operators and Parabolic Par-
tial Differential Equations,” Communications on Pure
and Applied Mathematics, Vol. 12, No. 2, 1959, pp. 249-
276. doi:10.1002/cpa.3160120206
[5] R. S. Phillips, “The Extensions of Dual Subspaces In-
variant under an Algebra,” Proceedings of the Interna-
tional Symposium on Linear Spaces, Jerusalem, 5-12 July
1960, pp. 366-398.
[6] T. Ya. Azizov and I. S. Iokhvidov, “Foundations of The-
ory of Linear Operators in Spaces with Indefinite Metric,”
Nauka, Moscow, 1986.
[7] M. G. Krein and Yu. L. Shmul’yan, “J-Polar Representa-
tion of Plus Operators,” Materialy Issledovaniya, Vol. 1,
No. 2, 1966, pp. 172-210.
[8] V. A. Khatskevich, “Some Global Properties of Fractional-
Linear Transformations,” Operator Theory, Vol. 73, 1994,
pp. 355-361.
[9] V. A. Khatskevich and V. S. Shulman, “Operator Frac-
tional-Linear Transformations: Convexity and Compact-
ness of Image, Application,” Studia Mathematica, Vol.
116, No. 2, 1995, pp. 189-195.
[10] V. A. Khatskevich and L. Zelenko, “Indefinite Metrics
and Dichotomy of Solutions for Linear Differential Equa-
tions in Hilbert Spaces,” Chinese Journal of Mathematics,
Vol. 24, No. 2, 1996, pp. 99-112.
[11] V. A. Khatskevich and L. Zelenko, “The Fractional-Lin-
ear Transformations of the Operator Ball and Dichotomy
of Solutions to Evaluation Equations,” Contemporary Ma-
thematics, Vol. 204, 1997, pp. 149-154.
doi:10.1090/conm/204/02628
[12] V. A. Khatskevich, “Generalized Fractional Linear Trans-
formations: Convexity and Compactness of the Image and
the Pre-Image; Applications,” Studia Mathematica, Vol.
137, No. 2, 1999, pp. 169-175.
[13] V. A. Khatskevich and L. Zelenko, “Bistrict Plus-Opera-
tors in Krein Spaces and Dichotomous Behavior of Irre-
versible Dynamical Systems,” Operator Theory: Advances
and Applications, Vol. 118, 2000, pp. 191-203.
[14] V. Khatskevich and V. A. Senderov, “On Convexity,
Compactness, and Non-Emptiness of Images and Pre-
images of Operator Linear-Fractional Relations,” Doklady
Akademii Nauk, Vol. 69, No. 3, 2004, pp. 409-411.
[15] T. Ya. Azizov, “On Extension of Invariant Dual Pairs,”
Ukrainian Mathematical Journal, Vol. 41, No. 7, 1989,
pp. 958-961. doi:10.1007/BF01060700
[16] V. Khatskevich, V. Senderov and V. Shulman, “On Op-
erator Matrices Generating Linear Fractional Maps of Op-
erator Balls,” Contemporary Mathematics, Vol. 364, 2004,
pp. 93-102. doi:10.1090/conm/364/06679
[17] N. Dunford and J. Schwartz, “Linear Operators,” Wiley,
New York, 1958.
[18] V. Khatskevich and V. A. Senderov, “On Operator Sets
Generated by Plus-Operators,” Vestnik Voronezhskogo
Gosudarstvennogo Universiteta, Seriya Fizika, Matematika,
No. 2, 2010, pp. 170-174.
[19] V. Khatskevich, M. Ostrovskii and V. Shulman, “Linear
Fractional Relations for Hilbert Space Operators,” Mathe-
matische Nachrichten, Vol. 279, No. 8, 2006, pp. 875-
890. doi:10.1002/mana.200310400
[20] T. Azizov and V. Khatskevich, “A Theorem on Existence
of Invariant Subspaces for J-Bi-Expansive Operators,”
Operator Theory: Advances and Applications, Vol. 198,
2009, pp. 41-48.