FUNCTIONAL DIFFERENTIAL EQUATIONS ABEL-SCHRODER TYPE EQUATIONS FOR MAPS ...

20 Pages·2014·0.66 MB·English

FUNCTIONAL VOLUME 10
DIFFERENTIAL 2003, NO 1-2
EQUATIONS PP. 239-258
ABEL-SCHRODER TYPE EQUATIONS FOR MAPS OF
OPERATOR BALLS
V. A. KHATSKEVICH AND V. A. SENDEROV '
Dedicated to bicentennial of Niels Henrik Abel
Abstract. Suppose that 'Jl and 'Jl are complex Banach spaces of positive dimensions
1 2
=
and JV('Jl;, 'l'l;) is the space of all linear continuous maps from 'Jl; to 'l'l; (i,j 1, 2). Let
F = FA he a linear fractional map of the following form:
where K E JV('Jl1, 'Jl2) and A;; E JV('l'l;, 'Jl;), i,j = 1, 2.
In the paper linear fractional maps F: M -t JV('Jl1, 'Jl2), where their domains M are
open subsets of JV('Jl1, 'Jl2), are studied. The results obtained in this study are used to
find solutions to functional equations. Namely, the Ahel-Schroder equation
is considered, where <p, 'f/;, and f are maps of subsets of the unit ball; the maps 'P and 'f/;
are given, and the map f is unknown. Under various natural restrictions on the spaces
'Jl and 'Jl and on the maps 'P and 'lj;, solutions to the Abel··-8chroder equation are found.
1 2
Even in the case of a one-dimensional space 'Jl and a finite-dimensional Hilbert space
1
'Jl2, the classes of solutions described by the formulas found in the paper are broader than
those previously known.
Keywords: Abel-Schroder equations, linear fractional map, operator ball, indefinite
space, Krein space.
2000 Mathematics Subject Classification: Primary 47B50, 47 A53.
' Braude College, College Campus P.O. Box 78, Karmiel 21982, Israel, e-mail:
[email protected]; 23-2-156, Pyatnitskoye shosse, Moscow, 125430, Russia,
e-mail: senderov@mccme. ru
239
240 V. A. KHATSKEVICH AND V. A. SENDEROV
1. Introduction. The subject of the present paper is now over a cen
tury old and has an extensive literature. The Schroder equations were first
used by G. Koenigs [1] for solving certain functional equations. The gen
eral notion of Abel-Schroder equation was introduced by C. C. Cowen [2].
These equations are related to many other branches of mathematics in dif
ferent ways: they are related to the problem of local change of variables in
the theory of analytic functions; to the theory of composition operators on
functional Banach spaces, in particular, on Hardy and Bergman spaces of
analytic functions on a disk, a polydisk, and a ball; to the problem of frac
tional iteration and the related problem of extending a discrete semigroup to
a continuous one (the so-called Koenigs embedding problem); to the Bottcher
type equations, etc. (see [2-4] and the references therein).
In solving the Koenigs embedding problem for a semigroup of iterations
in a Banach space 'B, the problem arises of how to find the most general
solvability conditions for the Abel-Schroder equation and, on the other hand,
to describe the existence conditions for special (star-like, spiral-like, etc.)
solutions of such equations. In the case where 'B is the space of all continuous
linear operators acting from one Hilbert space 91 to another Hilbert space
1
912, such a problem was solved in [5, 6]. In the present paper we consider
the general case of normed spaces 911 and 912.
Suppose that 91 and 91 are complex normed spaces of positive dimen
1 2
sions and N (91;, 91j) is the space of all linear continuous maps from 91; to
91j (i,j = 1, 2). Let :F =:FA be a linear fractional map of the form
(1.1)
where K E N(911, 912) and A;j E N(91j, 91;), i,j = 1, 2.
We consider linear fractional maps :F: M -! N(91b 91z), where M is
a subset of N(91~> 912) (its own subset for each :F). If in this case M = K
(M = K) is the unit open (closed) ball centered at the origin and if M' is a set
containing Im:F, then we write :FE Hol(K, M') (respectively, Hol(K, M')).
If, in addition, M' = M, then we use the notation Hol(K) and Hol(K).
Besides, we use the definitions and the notation of [7-15] without specially
mentioning this.
We set
A = ( An A12 ) . (1.2)
Az1 Azz
The operator A acts continuously in the space
+
91 = 911 91z, (1.3)
ABEL-SCHRODER TYPE EQUATIONS FOR MAPS 241
i.e., in the direct topological (with continuous projection operators) sum of
spaces 911 and 912.
By P1 and P2 we denote the projection operators onto 911 and 912,
respectively, which correspond to the decomposition (1.3). Throughout this
paper, we assume that the space 91 is endowed with the topology generated
=
by the norm llxll' = viiP1xll2 + 11Pzxll2 vllx1ll2 + llxzll2. This topology
coincides with the initial one; in what follows, the norm 11·11' is also assumed
to be equal to the initial norm: llxll' = llxll·
Endowing a space of the form (1.3) with the Jv-metric Jv(·) given by the
formulaJv(x) = llxdlv-llxzllv (wherex E 91and v > 1) makes it into a space
with indefinite metric or, in other words, into an indefinite space, which in
the case of Hilbert spaces 911 and '.11:1 is called a Krein space. Methods and
results of the theory of such spaces (e.g., see [9, 11, 13, 14]) are used below
to study the properties of linear fractional maps of the form (1.1).
The main subject of our study is the Abel-Schroder equation
fo<p='!j;of. (1.4)
Here the given maps <p and 'lj; and the unknown map f are maps of subsets
of the ball K..
In the classical case of the complex plane IC, a complete solution to (1.4)
is given in [2]. Besides, a deep result of the author of [2] (Theorem 3.2)
makes it possible, under rather general assumptions, to reduce the study of
the analytic maps <p, 1/J, and f to the special case of linear fractional maps.
In [16], Theorem 3.2 of [2] is partially generalized to the case of iC". A
special case of (1.4), i.e., the Schroder equation, was considered in [5, 17, 18]
in the case where 91 is a Krein space (in [17] this space is finite-dimensional).
In [7] an injective solution f E Hol(K,N(911, 912)) to the Schroder equation
is obtained already in the case of a Banach space 91.
The approach to Abel-Schroder equations developed in [5-7, 17, 18]
is to a large extent based on results of the theory of spaces with indefinite
metric, in particular, on the theory of linear fractional maps of operator balls,
which was developed by M. G. Krein and Yu. L. Shmul'yan [19, 20] (see also
[5, 7, 18]) and which was extended to the case of symmetric homogeneous
domains by L. A. Harris [21, 22] (see also [23]), and on the properties of
linear fractional maps of finite-dimensional and infinite-dimensional spaces,
which were established in [17] and in [5, 7, 18], respectively.
For other approaches to solving Schroder type equations see, e.g., the
recent paper [24].
The present paper extends and develops the results obtained in [7]. Some
242 V. A. KHATSKEVICH AND V. A. SENDEROV
of the results obtained in Sections 3 and 5 were briefly announced in [25].
The paper is organized as follows.
Section 2 is of preliminary character. We recall and introduce some
notions and propositions of normed linear space theory, which are necessary
for further exposition. In particular, in this section we introduce the notion
of v-space, which is a generalization of the notion of Clarkson space [26].
In Section 3 we study plus-operators in indefinite spaces 91 = 91 +91
1 2.
Basically, we consider operators whose block matrices A = ( ~~: ~:~ ) are
either of upper triangular form (I) or of lower triangular form (II), i.e., either
A = 0 or A = 0. We study relations between the blocks Aij as well as
21 12
some spectral properties. In doing this, we generalize and strengthen the
corresponding results of [5, 7, 17, 18].
In Section 4 we deal with linear fractional maps. In particular, we es
tablish various conditions for a linear fractional map to be defined on K. or
on iC and to map this ball onto itself.
The results of Sections 2-4 underlie the proofs of the theorems in the
final Section 5, which is the main part of the paper. In this section, under
various natural restrictions on the spaces 911> 91 and on the maps <p, 1/J, we
2
obtain solutions to Schroder type and Abei-Schriider type equations. The
results of this section extend and generalize those obtained in [5, 7, 17, 18].
Moreover, even in the case of a finite-dimensional (indefinite) Krein space,
the formulas of Section 5 determine broader classes of solutions as compared
with those obtained in the papers cited above.
We wish to express our gratitude to J. Arazy for useful discussions of
some results. We are also grateful to C. C. Cowen for kindly placing at our
disposal the materials of his works. The ideas of C. C. Cowen's works, as well
as the ideas of his joint works with B. D. MacCiuer [4, 16, 17], significantly
stimulated us in writing this paper.
2. Strictly convex normed spaces; v-spaces. In Sections 3 and 5
the notions of strict and uniform convexity of a normed space 91 and of its
subspaces 91 and 91 are essentially used. The property of strict convexity
1 2
is characterized by the following theorem, which is a slight modification of
the Ruston Theorem [12].
II · II,
THEOREM 1. For a normed space 91 with the norm the following
assertions are equivalent.
a) The space 91 is strictly convex.
ABEL-SCHRODER TYPE EQUATIONS FOR MAPS 243
b) For any x, y E iJl, where y f 0, the inequality
max{llx + Yll, llx- Yll} > llxll
holds.
c) Any support hyperplane of the unit ball in the space iJ1 is tangent to
this ball at at most one point.
d) Let iJ1 = £-i-9Jt, where .£ and 9Jt are subspaces and the norm of the
projection operator onto.£ is equal to 1, and let x E.£, y E 9Jt \ {0}. Then
llx + Yll > llxll·
DEFINITION 1. We say that the space iJ1 is a v-space if
(llxllv + IIYIIv)l/v :':: max{llx + Yll, llx- Yll} (2.1)
for all x, y E lJl.
It follows from the definition of a v-space that v 2 2. Indeed, by setting
x f 0 andy= ix in (2.1), we obtain 21/v :; 2112. On the other hand, for any
v 2 2, there exist v-spaces. Examples of them are spaces in which the norms
satisfy the Clarkson inequality [26].
PROPOSITION 1. Each v-space is uniformly convex.
Proof. We assume the contrary and consider the following vector se
quences in and Un: lltnll :':: 1, llunll :':: 1, lltn -unll > € > 0, and lltn +unll -> 2
as n -+ oo. Since iJ1 is a v-space, for sufficiently large n we have
This is a contradiction. 0
In what follows, in Theorem 10, Lemma 3, and Theorem 13, the space
lJ1 is assumed to be smooth. The properties of smoothness and uniform
2
smoothness are closely related to the properties of strict and uniform con
vexity, as well as to the property of reflexivity. The proof of the above-listed
assertions is based on these properties. Namely, the following assertions hold.
THEOREM 2 (see [12]). Uniformly convex spaces are reflexive. Unifor·mly
smooth spaces are reflexive.
THEOREM 3 (see [12]). A space iJ1 is uniformly convex if and only if the
space iJl* is uniformly smooth.
THEOREM 4 (see [12]). If a space iJ1 is strictly convex and reflexive, then
iJl* is smooth. If iJ1 is smooth and reflexive, then iJl' is strictly convex.
244 V. A. KHATSKEVICH AND V. A. SENDEROV
3. Plus-operators in indefinite spaces. In what follows, by \)3+ and
\)3++ we, respectively, denote the sets of nonnegative and positive vectors:
\13+ = {x: llx1JI ~ Jlx2JJ},
\13++ = {x: Jlxlll > JJxzll}.
As usual, vectors in \)3+ \ \)3++ are said to be neutral, and any operator A
such that A\)3+ c \)3+ is called a plus-operator.
For the most part, we study operators of classes (I) and (II): A21 = 0
or A12 = 0, respectively. These classes are dual to each other with respect
to taking the adjoint operator. Hence we can prove some assertions for op
erators of one class and then extend them to operators of the other class
automatically. However, the adjoint operator of a plus-operator is not nec
essarily a plus-operator [9]. Therefore, our results are established for both
these classes in parallel.
As is known (see [5, 7, 18]), the properties of the type A\)3++ C \)3++>
which are studied later in Propositions 2-4, are closely related to the prop
erties of the linear fractional maps generated by the plus-operators A. Such
properties of operators play an important role in the present paper.
PROPOSITION 2. Suppose that A is a plus-operator and Au is an injec
tion. Then
A does not annihilate vectors in \13+ \ {0} if and only if JJA!/ A1zxzll < JJxzJI
for any Xz E '.n2 \ {0};
A does not annihilate vectors in \13++ if and only if JJAj11A 1zll :::; 1.
Proof. First, we note that, for any plus-operator A and any vector
x E \)3+, the conditions Ax= 0 and Aux1 + A12x2 = 0 are equivalent.
We assume that x E \)3+ \ {0} and Aux1 + A1zxz = 0. Then x2 # 0.
Otherwise, we would have Aux1 = 0 and, because Au is injective, x1 = 0.
1
Further, A12x2 E ImAu and x1 + Aj1A12xz = 0. Therefore, we obtain
JIA!/ A1zxzll = IJxdl ~ JJxzJJ.
Conversely, let IIA!/A!zxzll ~ JlxzJJ, where Xz E '.nz \ {0}. By setting
x = x1 +x2 = -Aj11A 12xz+xz, we obtain x E \13+ \ {0} and Aux1 +A12X2 =
= -A12xz + A zXz = 0.
1
The second assertion can be proved similarly. D
The following fact is well known: if a plus-operator A in a Krein space
takes a positive vector to a neutral one, then it satisfies the condition ImA C
c \13+· In a Banach space, this is not generally true (see [11, 27]); however,
the following assertion holds.
PROPOSITION 3. Let A be a plus-operator of the form (I), and let An
be a surjection. Suppose that A takes a positive vector x to a neutral one.
ABEL-SCHRODER TYPE EQUATIONS FOR MAPS 245
Then Ax = 0 and ImA = 'Yl1.
Proof. Let us prove the first assertion. We assume that x = x1 + x2,
where x1 E 'Yl1 and xz E 'Ylz, is a positive vector and Ax is a neutral vector,
i.e., IIAnxi +A1zxzll = IIAzzxzll· We denote y = -(Anxi + Aizxz). Since An
is surjective, there exists a vector z E '.)11 such that A z = y. We assume
11
that y of 0. Clearly, in this case the vector u = Ax + Ey is negative for
0 < E < 1. On the other hand, u = A(x + Ez) and, since the functional Jv(·)
is continuous on 'Yl, the vector x + is positive for sufficiently small The
EZ E.
contradiction obtained shows that y = 0, which is equivalent to Ax = 0.
Now we prove the second assertion. Since A is surjective, we have
11
'Yl1 <;;; ImA. Hence it suffices to prove the inclusion ImA <;;; ')11. We assume
that this inclusion does not hold. Then there exists a vector Ay E '.)12 \ { 0}.
In this case, for any number E of 0, we have A(x + Ey) = EAy E q:~ __ . Now,
choosing an E of 0 so that x + EY E q:l++, we obtain a contradiction. The
proof of the proposition is complete. 0
The following proposition can be proved similarly.
PROPOSITION 4. Let A be a plus-operator of the form (II), and let A22
be a surjection. Then Aq:l++ c q:l++·
Let us pass to finding relations and interconnections between the blocks
Aij of the operator block matrix A defining a plus-operator of the form (I)
or (II). We shall need them in the study of the Abel-Schr6der equations (see
Section 5).
THEOREM 5. Let A be a plus-operator of the form (II). Then
a)
IIAd ::; inf IIAnxdl. (3.1)
Xl E 911
llxdl = 1
b) In addition, suppose that ')12 is strictly convex and the vector x1 E 'Yl1
possesses the following property: there exists a vector Xz E 'Yl such that
2
llxzll::; llx111 and IIAnx1ll = IIA22xzll· Then x1 E KerA21·
c) If 'Yl2 is uniformly convex, then
d) Let '.)1 be a complete strictly convex space, and let Azz be an operator
2
adjoint to a compact one. Then inclusion (3.2) holds. In particular, inclusion
(3.2) holds if '.)1 is a reflexive strictly convex space and Azz is a compact
2
operator.
246 V. A. KHATSKEVICH AND V. A.SENDEROV
Proof. a) Suppose that x2 E 912 with llx2ll = 1. Then A(x1 ± x2) E \)3+,
and hence
IIAnx1ll 2: max {IIA21X1 + A22x2ll, IIA21x1- Azzx2!1} 2:
2: max{IIA21xlll, IIA22x2ll}.
b) Assume the contrary: A21x1 f 0. Using Theorem 1, we see that
IIAnx1ll 2: max{IIA21x1 + A22xzll, IIA21X1- A22x2!1} > IIA22x2ll· This is a
contradiction.
c) Assuming the contrary, we can set IIA11x111 = 1 without loss of gen
erality. In this case, for x2 E 912 with llx211 = 1, we have
Since the space 912 is uniformly convex, the last inequality implies
+ + s;
II(A22X2 A21xd (A22X2- A21x1)ll 2(1- J(IIA21x111)),
where J(IIA21x1!1) > 0. So we have obtained
which contradicts the assumption.
d) This assertion follows from the fact that the operator A22 attains its
norm on the unit sphere of the space 912. 0
CoROLLARY 1. Suppose that A is a plus-operator of the form (II),
IIA22II = IIAnll· Then
a) A = ,\ (BEn BO ) , where B is an isometry and B21, B22 are
11
21 22
contractions;
b) if, in addition, 91 is uniformly convex, then -\B = 0;
2 21
c) on the other hand, if 912 is strictly convex and IIA22x2ll = II Au II for
some x2 E 912 with llx2ll = 1, then -\B21 = 0.
The assumptions of strict convexity and uniform convexity in Theorem 5
and in Corollary 1 cannot be omitted.
+
ExAMPLE 1. In the space 91 = 911 912, where 911 = (e1) and 912 =
~ ~ ~
= (e2, e3) is a Chebyshev space, consider the linear operator A= ( ) .
0 0 1
It is clear that A is a plus-operator and 11Aue111 = IIA22e31!; however, we have
A21e1 = e2.
ABEL-SCHRODER TYPE EQUATIONS FOR MAPS 247
COROLLARY 2. Suppose that 'Yt1 is complete, A is a plus-operator of the
form (II), and An is a bijection of 'Yt1 onto itself Then
(3.3)
In the case of a general plus-operator A (neither of the form (I) nor of
the form (II)) with bijective An, inequality (3.3) (and hence (3.1)) does not
hold.
ExAMPLE 2. Suppose that '}1 is a Hilbert space, { eb ez, e3} is its or
thonormal basis, '}11 = (e1, e2), and '}12 = (e.,). We define the J-unitary [9}
= = + = +
operator A by the relations Ae1 e1, Aez ~ez ~e3, and Ae3 ~e2 ~e3.
We have IIA!lll = 1 and IIAnll = ~·
In the case where '}12 is a v-space, inequality (3.1) can be strengthened
significantly.
THEOREM 6. Suppose that '}12 is a v-space and A is a plus-operator of
the form (II). Then
inf IIAz1x1ll" + IIAnll" :::; inf IIAnxdl".
llx,JI=1 llxlll=1
Proof Suppose that x1 E 'Yt1 and Xz E 'Ytz with lh II = llxzll = 1. We
have IIAz1x1ll" + IIAzzxzll":::; (max{IIAz1X1 + AzzXzll, IIAz1X1- AzzXzll})":::;
:::; IIA x1ll", which implies the statement of the theorem. 0
11
Let us return to the general case of an indefinite space 'Yt.
THEOREM 7. Suppose that A is a plus-operator of the form (I), A is a
11
1
bijection of'Yt1 onto itself, the operator Aj1A12 is bounded, and II A!/ Ad I :::; 1.
Then the following inequality holds for all xz E 'Ytz such that llxzll :::; 1:
(3.4)
Proof. Let A12x2 = 0. Then, because A is a plus-operator, for any
x1 E 'Yt1 with llxdl = 1, we have
Let A12x f 0. We consider a vector x1 E 'Yt1 with llx1ll = 1 such that
2
A x1 = aA12xz, where a > 0. We have
11
248
V. A. KHATSKEVICH AND V. A. SENDEROV
and hence
IIAnxJII-IIAJ2X211 = IIAnx1- A12X2II ~ IIAzzXzll (**)
(the last inequality follows from the fact that x1 - x2 E \)3+).
The statement of the theorem follows from relations (* ) and ( **). 0
REMARK 1. Inequality (3.4) cannot be strengthened by replacing IIAnll
in it with II A!/ 11-1
.
+
EXAMPLE 3. In a Krein space ')1 = ')11 ')12, where dim ')11 = 2 and
~ ~ ~
dim 'Jtz = 1, it suffices to consider the linear operator A = ( ) ,
0 0 0
where 1 ~ lal > 1!31 > 0.
REMARK 2. Under the assumptions of Theorem 1, if the opemtor
1
A!/ A12 either is unbounded or satisfies the inequality IIA!1A12ll > 1, then
ImA = <Jt1. This follows fT"Om Propositions 2 and 3. Clearly, in this case
inequality (3.4) does not hold in general.
THEOREM 8. Suppose that A is a plus-operator of the form (I) and An
is a surjection of ')11 onto itself. Then inequality (3.1) holds. But if <Jt1 is
complete and An is a bijection, then inequality (3.3) holds.
Proof. Suppose that x1 E 'Jt1, Xz E 'Jtz, llx1ll = llxzll, and Y1 E 'Jt1,
Any1 = A1zx2. We consider the vector x = ay1 + x1 - ax2, where a= ±1.
We choose a so that llaY1 + xdl 2: llx& In this case we have x E \13-r, and
hence Ax E \)3+, i.e., IIAnx111 ~ IIAzzxzll· Precisely this inequality implies
inequality (3.1). 0
Theorems 7 and 8 are supplemented with the following assertion, which
is an analog of Theorem 5, b).
THEOREM 9. Suppose that ')11 is strictly convex, A is a plus-operator of
the form (I), and A11 is a bijection of ')11 onto itself. Suppose that a vector
x2 E ')12 has the following property: there exists a vector x1 E <Jt1 such that
llx2ll::; llx1ll and IIAnx1ll = IIA2zx& Then Xz E KerA12·
Proof. Assume the contrary: A12x2 # 0. Precisely as above, in the proof
of Theorem 8, we consider a vector y1 E ')11 such that Any1 = A1zx2 and set
x = ay1+ x1- ax2, where a = ±1. Because the space 'Jt1 is strictly convex, we
can choose a number a so that llay1 + x1ll > llx& Thus we have x E \)3++,
and the vector Ax is neutral: IIP1Axll = IIAnx1ll = IIAzzXzll = IIPzAxii
Then it follows from Proposition 3 that A22 = 0. Hence Anx1 = 0, x1 = 0,
and x = 0, which contradicts the assumption. 0
2
REMARK 3. It is easy to see that in the statement of Theorem 9, the
bijectivity of An cannot be replaced by its surjectivity. We show that the
strict convexity of 'Jt1 is also important.