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Tides in this series Improperly posed boundary value problems A Carasso and A P Stone
2 Lie algebras generated by finite dimensional ideals IN Stewart
3 Bifurcation problems in nonlinear elasticity
15 Unsolved problems concerning lattice points JHammer 16 Edge-colourings of graphs S Fiorini and R J Wilson
17 Nonlinear analysis and mechanics: Heriot-Watt
RWDickey
Symposium Volume I RJKnops
4 Partial differential equations in the complex domain DLColton
18 Actions of finite abelian groups C Kosniowski
5 Quasilinear hyperbolic systems and waves A Jeffrey
19 Closed graph theorems and webbed spaces MOe Wilde
6 Solution of boundary value problems by the
20 Singular perturbation techniques applied to
method of integral operators DLColton
7 Taylor expansions and catastrophes T Poston and I N Stewart
8 Function theoretic methods in differential equations R P Gilbert and R J Weinacht 9 Differential topology with a view to applications D R J Chillingworth
10 Characteristic classes of foliations HVPittie 11
Stochastic integration and generalized martingales AUKussmaul
12 Zeta-functions: An introduction to algebraic geometry AD Thomas 13
Explicit a priori inequalities with applications to boundary value problems V G Sigillito
14 Nonlinear diffusion WE Fitzgibbon III and H F Walker
integro-differential equations H Grabmiiller
21
Retarded functional differential equations: A global point ofview SEA Mohammed
22 Multiparameter spectral theory in Hilbert space BDSieeman
23 Recent applications of generalized inverses MZNashed
24 Mathematical modelling techniques RAris
25 Singular points of smooth mappings CGGibson
26 Nonlinear evolution equations solvable by the spectral transform. FCalogero
27 Nonlinear analysis and mechanics: Heriot-Watt Symposium Volume II RJKnops
28 Constructive functional analysis OS Bridges
B D Sleeman University ofDundee
Multiparameter spectral theory in Hilbert space
Pitman LONDON· SAN FRANCISCO· MELBOURNE
PITMAN PUBLISHING LIMITED 39 Parker Street, London WC2B 5PB FEARON-PITMAN PUBLISHERS INC. 6 Davis Drive, Belmont, California 94002, USA
Associated Companies Copp Clark Ltd, Toronto Pitman Publishing New Zealand Ltd, Wellington Pitman Publishing Pty Ltd, Melbourne First published 1978 AMS Subject Classifications: (main) 47A50, 47BI5, 47B25 (subsidiary) 47E05, 34B25, 46CIO British Library Cataloguing in Publication Data Sleeman, Brian D Multiparameter spectral theory in Hilbert space. - (Research notes in mathematics; no. 22). I. Hilbert space 2. Linear operators 3. Spectral theory (Mathematics) I. Title II. Series 515'.73 QA322.4 78-40060 ISBN 0-273-08414-3
this book arose out of a series of lectures given at the University of Tennessee at Knoxville in the spring of 1977.
It is a .pleasure to acknow-
ledge the hospitality of the Department of Mathematics at the University of Tennessee during 1976-77 when the author was a visiting Professor there. The purpose of this book is to bring to a wide audience an up-to-date account of the developments in multiparameter spectral theory in Hilbert space.
Chapter one is introductory and is· intended to give a background
and motivation for the material contained in subsequent chapters.
Chapter
two sets down the basic concepts and ideas required for a proper understanding of the theory developed in chapters three, four and five.
It is
mainly concerned with the concept of tensor products of Hilbert spaces and the spectral properties of linear operators in such spaces.
Most of the
theorems contained in this chapter are given without proof but nevertheless adequate references are included in which complete proofs may be found.
In
chapter three multiparameter spe·ctral theory is developed for the case of bounded operators and this is generalised to include unbounded operators in chapters four and five.
Chapter six deals with a certain abstract relation
arising in multiparameter spectral theory and is analogous to the integral equations and relations well known in the study of boundary value problems for ordinary differential equations.
Chapters seven and eight exploit the
theory in application to coupled operator systems and to polynomial bundles. Finally chapter nine reviews the material of the previous chapters, points out open problems and indicates paths of new investigations.
OVer the years the author bas benefited from collaboration and guidance from a number of colleagues.
In particular it is a pleasure to acknowledge
the guidance and stimulation from my colleague and former teacher Felix Arscott who first aroused my interest in multiparameter spectral theory. I also wish to acknowledge the influence and collaboration of Patrick Browne (who read the entire manuscript and made a number of suggestions for improving certain sections), Anders KBllstrBm and Gary Roach whose contributions are significant in much of the theory developed here.
I would also
like to thank Julie my wife for her sustained encouragement during the writing of this book.
She not only prepared the index and list of
references but also contributed to the style and layout of the work. Finally I would like to express my appreciation to Mrs Norah Thompson who so skilfully typed the entire manuscript.
Contents
CHAPTER 1
AN INTRODUCTION References
CHAPTER 2
TENSOR PRODUCTS OF HILBERT SPACES
2.1
The algebraic tensor product
2.2
Hilbert tensor product of Hilbert spaces
2.3
Tensor products of linear operators
2.4
Functions of several commuting self-adjoint operators
2.5
Solvability of a linear operator system Notes and References
CHAPTER 3
MULTIPARAMETER SPECTRAL THEORY FOR BOUNDED OPERATORS
3.1
Introduction
3.2
Multiparameter spectral theory
3.3
Eigenvalues
3.4
The case of compact operators Notes and References
CHAPTER 4
MULTIPARAMETER SPECTRAL THEORY FOR UNBOUNDED OPERATORS (The right definite case)
4.1
Introduction
4.2
Commuting self-adjoint operators
4.3
Multiparameter spectral theory
4.4
The Compact case
4.5
CHAPTER 5
An application to ordinary differential equations
58
Notes and References
60
MULTIPARAMETER SPECTRAL THEORY FOR UNBOUNDED OPERATORS (The left definite case)
5.1
Introduction
62
5.2
An eigenvalue problem
64
5.3
The factorisation of W
66
5.4
An application to ordinary differential equations
73
5.5
A comparison of the definiteness conditions
77
Notes and References
80
CHAPTER 6
II!
AN ABSTRACT RELATION
6.1
Introduction
81
6.2
The problem
81
6.3
Some applications to ordinary differential equations
87
References
90
CHAPTER 7
COUPLED OPERATOR
SYSTE~ffi
7.1
Introduction
91
7.2
Direct sums of Hilbert spaces
92
7.3
Reduction of strongly coupled systems
94
7.4
Spectral theory for weakly coupled systems
96
Notes and References
98
CHAPTER 8
SPECTRAL THEORY OF OPERATOR BUNDLES
8.1
Introduction
8.2
The Fundamental reformulation
99
101
8.3
Two-parameter spectral theory
8.4
Concerning eigenvalues
8.5
The case of unbounded operators
8.6
An application to ordinary differential equations Notes and References
CHAPTER 9
OPEN PROBLEMS
9.1
Solvability of linear operator systems
9.2
Multiparameter spectral theory for bounded operators
9.3
Multiparameter spectral theory for unbounded operators
9.4
The abstract relation
9.5
Applications References
INDEX
1 An introduction
Multiparameter spectral theory like its one parameter counterpart, spectral theory of linear operators, which is the subject of a vast and active literature, has its roots in the classic problem of solving boundary value problems for partial differential equations via the method of separation of variables.
In the standard case the separation technique leads to the
study of systems of ordinary differential equations coupled via spectral parameters (i.e. separation constants) in only a non-essential manner.
For
example the problem of vibration of a rectangular membrane with fixed boundary leads to a pair of Sturm-Liouville eigenvalue problems for ordinary differential equations which are separate not only as regards their independent variables but also in regard to the spectral parameters as well.
The
same problem posed for the circular membrane leads to only mild parametric coupling.
This is a kind of triangular situation.
The parameter in the
angular equation must be adjusted for periodicity and the resulting values substituted in the radial equation leading to the study of various Bessel functions.
The multiparameter situation arises in full if we pursue this
class of problems a little further.
Take for example the vibration problem
of an elliptic membrane with clamped boundary. use elliptic coordinates.
It is appropriate here to
Application of the separation of variables method;
leads to the study of eigenvalue problems for a pair of ordinary differential equations both of which contain the same two spectral parameters. This is then a genuine two-parameter eigenvalue problem.
The ordinary
differential equations which arise are Mathieu equations whose solutions are 1
expressible in terms of Mathieu functions.
Other problems of this type give
rise to two or three parameter eigenvalue problems and their resolution lies to a large extent in the properties of the "higher" special functions of mathematical physics, e.g. Lame functions, spheroidal wave functions, paraboloidal wave functions, ellipsoidal wave functions etc,
We refer to
the encyclopaedic work of Erdelyi et al [12J and also the book of Arscott [1] for an account of these functions.
Many of these special functions
possess as yet unrevealed secrets even though they have been studied vigorously over the past fifty or so years.
It is perhaps not so surprising
then that multiparameter spectral theory per se has been rather neglected over the years despite the fact that it arose almost as long ago as the classic work of Sturm and Liouville regarding
one~parameter
eigenvalue
problems particularly oscillation theory. In its most general setting the multiparameter eigenvalue problem for ordinary differential equations may be formulated in the following manner, Consider the finite system of ordinary, second order, linear, formally selfadjoint differential equations in then-parameters, A1 ,.,,,An'
+
0 s xr s 1,
fE
a
s•l rs
(x )A
r
s
- q (x
r
r
>}
n
~
2,
(1. 1)
yr = 0,
r • l, ••• ,n with ars(xr)' qr(xr) continuous and real valued
functions defined on the interval 0 s xr s 1.
By writing A for (A 1 , ••• ,An ) . ~
we may formulate an eigenvalue problem for (1.1) by demanding that
~be
chosen so that.all the equations of (1,1) have non-trivial solutions with each satisfying the homogeneous boundary conditions
2
cos ary r (0) - sin a r
dy (0) r dx r
0,
0 s a
r
n-1 e Hn
(1)
e ... e
g
(1\)
)A where the n
..... notation means that the corresponding index in { 1,2, ••• ,n} is omitted. In subsequent chapters we shall have occasion to make use of the following theorems The tensor product space H1 e ... 8 Hn of separable Hilbert
Theorem 2.2: spaces Hi'
f • 1, ••• ,n is separable.
Furthermore if {elk) : i e Uk}
(Uk·an index set) is an orthormal basis in ~· then (1)
(n)
{ ei1 8 • • • 8 ein : i 1 e u1 , ••• , in e Un} is an orthonormal basis in
e
,.,
H ...
n
~
~:
Of
is separable then there is a countable orthonormal basis
{e~k) 1
i £ Uk} in Hk.
Consequently the set
T,. {e~l) 8 ••• 8 e.(n) 1 11 n is countable.
il £ ul'
••• J
i
n
£ u } n
1rrAA
Also T is an orthonormal system since
(2.12) That Tis a basis in H1 8 .•• 8 Hn follows from the observation that the
~ = ( e (k) , e 2(k) 11'near hull ~k 1 H_, and -K
so~
1
8
a
•.. 8
~
a
n
· dense 1n · .•• ) spanne d by {e (k) 1 , e 2(k) , ••• } 1s
which is contained in the linear space T
spanned by Tis dense in H1 8 ... 8 Hn. Theorem (2.2) admits the following extension Theorem 2.3:
If {e. : i £ U} is an orthonormal basis in the separable 1
Hilbert space H1 and
{e~
., j £ V} is, for each value of the index i an orth1J onormal basis in the separable Hilbert space H2 , then T ={e. 8 e~.,i£U,j £V} 1 1J is an orthonormal basis in H1 8 H2 •
Proof:
That T is an orthonormal system follows from (2.10) with n = 2.
let f be any element of H1 8 H2 •
Now
Since {ei 8 eik : i £ U, k £ V} is by
theorem (2.2) a basis in H1 8 H2 then for any given E > 0 there is a g £ H1 8 H2 of the form
Consequently T the linear space spanned by T is dense in H1 8 H2 , i.e.
T: 2.3
H1 8 H 2 •
TENSOR PRODUCTS OF LINEAR OPERATORS
Definition 2. 1 Let A1 , ... , A ben bounded linear operators in the Hilbert spaces n H1 , _. .. , Hn respectively.
The tensor product A1 8 ••• 8 An of these operators
is that bounded linear operator on H1 8 •.• 8 Hn acts on f 1 8 .•. 8 fn, fiE Hi'
i • l, ••• ,n, in the following way. (2.13)
The relation (2.11) determines the operator A1 8 ..• 8 An on all elements of the form : 1 8 ••• 8 fn and so, due to the presupposed linearity of A1 8 ••• 8 An' on the linear manifold spanned by all such vectors.
14
Since this
linear manifold is dense in H1 8 ••• 8 Hn and A1 8 ... 8 An defined by (2.12) is bounded, it has a unique extension to the whole of H1 8 .•• 8 Hn.
Hence
the definition is consistent. In a similar fashion the bounded linear operator Ak on Hk induces a linear operator
Ak·r
on H1 8 ••• 8 Hn in the following way; if
(2.14) This operator is then extended by linearity and continuity to the whole of H1 8 ••• 8 Hn as above. If the operators Ai' i = 1, ••• , n, are n unbounded linear operators defined on dense domains D(Ai)' i
= 1,
••. , n, in Hi' i
= 1,
•.• , n, then
A1 8 ••• 8 An can be defined uniquely via (2.11) at least on the algebraic However if the A., i = 1, .•• , n, 1
are self-adjoint unbounded operators then A1 8 ..• 8 An, defined in this way may not be self-adjoint on such a domain. To overcome this difficulty we proceed in the following way: Let E.(A) be the resolution of the identity for the self-adjoint operator 1
8 H n
We then define
A!. 1
=
[A -co
the spectral measure
A·~1 via the resolution of the identity E·~1 (A) that is
-r
dE. (A). 1
(2.15)
15
2.4
FUNCTIONS OF SEVERAL COMMUTING SELF-ADJOINT OPERATORS.
A fundamental tool in the multiparameter spectral theory developed in this book is the theory of several commuting self-adjoint operators.
In this
section we describe those aspects of the theory particularly relevent to develo~ent.
the subsequent
Full details of the theory of functions of
several commuting operators are to be found in the book of Prugovecki [5.1. Let E.(·) denote the resolution of the identity for the self-adjoint 1
operator A. and let B. c R be a.Borel set. 1
1
Definition 2.2 Two self-adjoint operators A. and A. are said to commute if their respective J
1
resolutions of the identity E.(B.) and E.(B.) commute. 1
i.e.
J
1
J
E.(B.)E.(B.) = E.(B.)E.(B.) 1
J
1
J
J
J
1
(2.16)
1
for all B., B. Borel subsets of R. 1
J
In the particular case when A. and A. are bounded self-adjoint operators 1
J
defined on the whole of the Hilbert space H then it may be shown that (2.16) is a necessary and sufficient condition for A. and A. to commute. 1
Theorem 2.4:
Let E.(B), i 1
J
1, ••• ,n, with B a Borel set, be the resolutions
of identity for n commuting self-adjoint operators A., i 1
= 1,
•.•• n, in H.
Define (2.17) where Bi c R, i = 1, •••• n
are Borel sets.
Thus E(•) defines a spectral
measure on the Borel subsets of Rn.
.. .'
A ) is a bounded Borel measurable function on Rn. i.e • n
IFCA 1 , .•.• An)l ~ M for all A1 , •••• An£ R, then there exists a unique
16
bounded linear operator A conveniently denoted by F(A 1 ••• ,An) such that (f,Ag) =
f
F(A 1 , ... , An)d(E(A 1 , ... , An)f,g)
(2.18)
Rn for all f,g e: H. The following two theorems may be used to prove that F(A1 , ••. ,An) is self-adjoint when F(A 1 , .•• , An) is real. Theorem 2.5:
Let A1 , ••• ,An be a commuting set of self-adjoint operators.
If F(A 1, ••• , An) is a complex-valued, bounded, Borel measurable function on Rn and F*(A 1 , .•.• An) is its complex conjugate, then F*(A 1 , .••• An) is the adjoint of F(A1 , ••• • An). Theorem 2.6: If F(A 1, ..• , An) is a real bounded Borel measurable function on Rn and A1 , • • • J An are n commuting self-adjoint operators then A= F(A1, ••. ,An) is self-adjoint and its spectral measure EA(B) satisfies
for all Borel sets B in R. The next theorems extend theorems 2.4 and 2.5 to the case when F(A 1 , ••• , An) is any Borel measurable function on Rn and not necessarily bounded. Theorem 2.7:
Let A1, ••• , An ben commuting self-adjoint operators in H,
and E(B), B a Borel set in Rn, be the spectral measure defined by (2.15). If F(A), A e: Rn is a Borel measurable function, then there exists a unique linear operator A such that (g,Af) =
I
(2.19)
F(A)d(g,E(A)f)
:Rn
for all g e: Hand where D(A), the
do&~in
of A, is given by 17
I
D(A) .. { f :
jF(~) j 2dll E(~)fjj 2
and
s • s 11 s 22
-
s 12 s 21
positive definite.
From the first equation
and When this is inserted into the second equation we find
and by commutativity this reduces to
and so
This completes the proof of theorem 2.9. Notes:
The treatment of algebraic and Hilbert tensor products presented in
sections 2.1 -2.3 is taken largely from the book of Prugovecki [5J as is section 2.4 on the theory of functions of several commuting self-adjoint operators.
A more detailed account of tensor product spaces is to be found
in the classic paper of Murray and Von-Neumann [4] or in the book by Schatten [6]. 26
Section 2.5 on the solvability of linear operator systems
is based on the paper of Klllstr8m and Sleeman [2] wherein it is shown that the system (2.25) is also solvable under the slightly weaker hypothesis that
I (Su, u) I ~
cllull 2 , u
E:
H.
As a by-product of the analysis in [2] it
turns out that the elements of the determinant S enjoy certain commutativity relations.
For example it may be shown that when n • 2,
Actually only the first of these is proved in [2] on the basis that positive definite.
s 11
is
However this requirement is not necessary as is proved
in [3] wherein the complete set of commutativity relations for general n is given. References 1
P. R. Halmos,·
2
A.Killstr8m and
A Hilbert space problem book: Van Nostrand, New York, London, Toro~to (1961).
&n Sleeman,
Solvability of a linear operator system.
3. Math. Anal. Applies. 55 (1976) 785-793.
3
A. Klllstr8m and B. D. Sleeman, Multiparameter spectral theory. Arkiv. £8r Matematik 15 (1977) 93-99
4
F. 3. Murray and 3. Von Neumann, . On rings of operators. (2) 37 (1936) 116-229.
5
E. Prugove~ki,
Quant1DD mechanics in Hilbert space. Press, New York (1971).
6
R. Schatten;
A theory-of cross-apaces. Ann. of Math. studies 26, Princton University Press. Princeton (1950).
Ann. of Math. Academic
27
3 Multiparameter spectral theory for boimded operators 3.1
INTRODUCTION n
Let H1 , ••• • Hn be separable -Hilbert spaces and let H • tensor product. j
= 1,
Hi be their
8
i=l
In each space H1• we assume we have operators A.• S ••• 1
1J
•••• n enjoying the property
(i)
A. • S •• : H. +H. • 1
1J
1
1
i,j • 1, •••• n are Hermitian and continuous.
In addition we shall require a certain "definiteness" condition which may be described as follows: H with fi
€
Let f • f 1 8 •.• 8 fn be a decomposed element of
Hi' i • 1, •••• nand let a 0 , a 1 , .•.• an be a given set of real
numbers not all zero.
Then the operators l1i : H + H,
i • 0, ••• • n, may be
defined by the equation n
Af •
l a.l1.f • det i•O 1 1
al • ... • .. • .. •. • • an
(3.1)
-Ann f
sn lfn
• •• • •• ••• •••
snnf n
where the determinant is to be expanded formally using the tensor product. For example
28
A0 f . • 8
sn lfn
••••••••
snn f n
where a runs through all permutations of { 1 1 2 1 according as a is even or odd.
n} and e:a is +1 or -1
This defines A0 f for decomposable f
and we extend the definition to arbitrary f The operators A., 1
••• •
€
H
€
H by linearity and continuity.
i • 1, •••• n are obtained in a sfmilar fashion.
The definiteness condition referred to above can now be stated as (ii)
A : H + H is positive definite, that is (Af,f)
0!::
cllfll 2
for some constant C > 0 and all f in Hand
11·11
€
H.
the corresponding norm. 8 f
(Af, f)
(3.2)
n
Here(.,.) denotes the inner product Note that for a decomposable element
in H we have
• det
• • • • • • • • • • • • • • • • • •
-(A f
n n• f n ) n
(S 1 f ,f ) n n n n
. ........ . '
(S
Cl
n
f ,f )
nn n
n n
(3.3)
29
where ( •, ·).1 ( 11·11·) denotes the inner pr~duct (norm) in H., i .. 1, ... , n. 1 1 The system of operators {A. ,S •• }, 1
(i)
i,j • 1, .•. , n having the properties
1J
(ii) form the basis for multiparameter spectral theory developed in
this and the next two chapters of this book. Each of the operators A., S .. : H.+ H., 1
1J
1
i = 1, ... , n, induces a corre-
1
sponding operator in H as described in chapter 2. §2.3.
The induced
operators will be denoted by At1., St ..• 1J
The theory to be developed i~ this chapter is based on the solvability of certain systems of linear operator equations. elements f.
€
1
H,
Let f
€
H be given; we seek
i • 0,1, ••• , n satisfying the system of equations
n
r a. f. i=O 1
- f,
1
(3.4)
t
r
t
n
- Ai f 0 + S •• f. • O, j•l 1J J
i • 1, ••• , n.
We have seen (chapter 2. §2.5) that the system (3.4) subject to condition (ii) is uniquely solvable for any f
€
H, and the solution is given by
Cramer's rule, that is -1
fi .. A
!2!!=
6i f,
because of condition (ii) A-l exists as a bounded operator.
The operators ri
r.1
-1
•A
azefundamen~l
3. 2
(3.5)
i "' 0,1, · ••• , n.
6., 1
H+H
i - 0,1,
....
n are defined by
i•O,l, ... , n,
to the multiparameter spectral theory to be developed.
MULTIPARAMETER
SPECTRAL
THEORY
Rather than use the inner product(·,·) in H generated by the inner products (.,.).1 in H., we shall use the inner.product given by (A·,·) which will be 1 30
denoted by[·,·].
The normainduced by these inner products are equivalent
and so topological concepts such as continuity of operators and convergence of sequences may be discussed unambiguously without reference to a particulat inner product. used.
Algebraic concepts however may depend on the inner product ~
For L
H + H we denote by L
that is, for all f,g
the'adjoint of L with respect to[.,.],
H we have
£
tf: . = [f,L gJ.
[Lf,gJ
(3.6)
lbeorem 3 .1:
r.#1
=
r., 1
i = 0,1, ••• , n.
First we observe that the adjoint ~~ of ~. with respect to the
Proof:
1
inner product
(.,.)equals~., 1
Hermitian operators A., S ..• 1
[ri f, gJ = (AA
-1
1J
~if, g)
1
since these operators are formed from the Thus for f,g £Hand i
c
0,1, ••• , n, we have
.. (f,~i g)
-1 • (f, AA ~. g) 1
.. (A f,
• [£,
r.1
g)
r.1 gJ,
and the result is proved.
Lemma 3.1:
Let Ai: Hi+ Hi'
i = 1, ••• , n be continuous linear operators.
Then n n
i=l ~:
t • Ker(A.) 1
Ker(A.) • 1
This result has been established by Atkinson [1. Thm 4.7.2]
case that the spaces Hi'
i = 1, ••• , n are all finite dimensional.
in the A
similar argument using orthonormal bases in the spaces Hi shows it to be true in the present setting.
31
We now establish a fundamental result. The operators r., i - 0,1, •••• n are pairwise commutative.
Theorem 3.2:
1
Let f
~:
(3.4).
H be arbitrary and let f., i
€
1
= 0,1,
••• , n solve the system
Then from [1. Thm 6.4.2 p 106J we see that I!.. f. 1
I!.. f.,
J
J
1
i, j_ •._ 0,1, ••• , n.
Note however that the theorem referred to applied to the case of the spaces Hi being dimensional.
Its extension to the case of countable dimension is
straight forward. -1
Since f.1 • A
I!.. f, 1
i • 0,1, ... , n, we have
or l!..r.f•l!..r.f. 1
J
J
1
An application of A-1 on the left throughout this equation establishes the result. As a Corollary we have Corollary 3.1: n
r a.r. i-o 1
• I,
1
t Ai r0 +
t r S •. r. j•l 1J J n
•
O,
i • 1, ••• , n.
Working with the inner product [ • , • ] in H the opera tors r.1 , i • 0, 1, ... , n form a family of n + 1 commuting Hermitian operators. Let a(r.) denote the 1
spectrum of ri and a0 • XOSiSn a(ri)' the Cartesian product of the a(r.), i • 0,1, ••• , n. 1
Then since a(r.) is a non-empty compact subset ofR 1
it follows that a0 is a non-empty compact subset of Rn+l. 32
Let E.(·) denote the resolution of the identity for the operator 1 let Mi
£
= 0,1,
R be a Borel set, i
••• , n.
r.1
and
We then define
n
E(M x M1 x ••• x M) • IT E.(M.). 0 n i•O 1 1 commute since the operators
r.1
Notice that the projections E.(•) will 1
commute.
Thus in this way we obtain a
spectral measure E(·) on the Borel subsets of Rn+l which vanish outside a0 • Thus for each f,g outside a0 •
H, [E(•)f,g] is a complex valued Borel measure vanishing
£
Measures of the form [E(•)f,f] will be non-negative finite
Borel measures vanishing outside a0 . The spectrum a of the system { Ai' Sij} may be defined as the support of the operator valued measure E(•), that is, a is the smallest closed set outside of which E(·) vanishes or alternatively a is the smallest closed set with the property E(M) • E(M n a) for all Borel sets M c Rn+l. compact subset of Rn+l and if A rectangles M with A
€
M, E(M)
~
€
Thus a is a
a then for all non-degenerate closed
0.
Thus the measures [E(M)f,g], f,g
£
H
actually vanish outside a. We are now in a position to state a fundamental result namely the Parseval equality and eigenvector expansion. Theorem 3.3: (i)
(ii)
Let f e H. (Af,f) •
f •
t
Then
f [E(dA)f,f] a
•
f
a
(E(dA)f,Af).
E(dA)f,
where this integral converges in the norm of H. ~:
The result is a simple application of Theorem 2.4 on choosing
33
§3.3 EIGENVALUES We now turn to a discussion of the eigenvalues of the system {A., S .. } • An 1
1J
"homogeneous" eigenvalue is defined to be an n + 1-tuple of complex numbers A • (A0 ,A1 , ••• , An) for which there exists a non-zero decomposable element u
c
u 1 8 ... 8 un
H such that
£
n
I
1
(3.7)
.. 1,
a.A.
i•O
1
and n
-A0 A.u. + I A. S .. u. • 0, 1 1 j•l J 1J 1 Theorem 3.4:
i • 1, ... , n.
Let A= (A0 , ••• , An)
be an eigenvalue for the system (3.7).
Then if A is positive definite on H each A., 1
Proof:
i • 0, 1, ••• , n is real.
If u • u 1 8 ... 8 un is an eigenvector corresponding to the eigen-
value A we have -A0 (A.u., u.). 1 1 1 1 and n
-X0(u.,A.u.). 1 1 1 1
+
I
I(u.,S •. u.). • 0, 1
j•l
1J 1
1
and since A., S •. are Hermitian we have 1
-(A0 -
1J
X0 HA.u.,u.). 1
1
1
1
n
+
I (A. j•l J
- I'.)(s .. u.,u.) ... o. J
1J 1
1
(3.8)
1
:furthermore n
I i-o
a.(A. 1
1
I.) • o.
(3.9)
1
It now follows from equations (3.8)(3.9) and the positive definiteness of A that A.1 • 34
I., 1
i • 0,1, ••• , n thus proving the result.
If A £ a is such that
theorem 3.5:
i« A})
~ 0 1 then A is an eigeuvalue.
Conversely if A is an eigeuvalue then A £ a and E« A})
g
£
E« A})H1 g rJ 0.
Then since the operators E.« A.}) 1 1
commute E.«A.})g = g for each i. 1 1 Corollary 3.1 we deduce that r.g 1 n
i•o.l •....• n
1
From ordinary spectral theory and
= A.g. 1
i • 0 1 11
••••
nand that
n
r a.r.g • i•O 1
~ 0.
• i•O L a.A.g ,
g
1
1
1
t n t t - Ai A0 g + l A. S •• g • (-Ai r 0 + j•l J
1J
t
n
l j•l
S •• r. )g • 0. 1J J
Hence by lemma 3.1 we have n
n 0 ~ g
n Ker(-Ai A0 +
€
i•l
.
A. s .. ) t r j•l J 1J
n
n
Ker(-Ai A0 +
8
i•l
r A.s .. >. j•l J 1J
where n
r
i•O
a.1 A. • 1. . 1
thus there must be a non-zero decomposable element u • u 1 8 ... 8 un
£
H
such that n
-A0 A. u. + L A. S •• u. • 0 1 1 1 j•l J 1J 1
i • 11
••• •
n.
This shows that A • (A0 .A1 ••••• An) is an homogeneous eigenvalue. Conversely if A is an homogeneous eigenvalue with non-zero decomposable eigenvector u • u 1 8 ... 8 un• we have
r
iii()
a. A. • 1 1
1
35
and
t n - Ao Ai u + I
j•l
t A. s •• u • 0. J
1J .
i
= 1,
• • • • n.
Then from the proof of theorem 3.2 we have, for i • 0,1, ••• , n, r. u • A.u. 1
1
E({ A})u "' u and the result follows. If, as is usual, we adopt
thct~'inhomogeneous"
concept of eigenvalue, i.e.
an n·tuple of complex numbers p • (p 1 , ..• , pn) for which there exists a 8 v
non-zero decomposable element v • v 1 8
n
such that
n
-A.v. 1
1
+I j•l
p.S •. v.•O, J
1J 1
i • l , ••• ,n,
then we can obtain results similar to Theorems (3.3- 3.5) above. it is necessary for AO o
t
~
0.
That is we require
a • a(A- 1s>,
where A is defined by (3.1) and S • det(s!.). 1J
if f
€
To do this
Now 0
€
a(A- 1s) if and only
HA(co) where HA (co) • { f
H I Sf • 0}.
€
Consequently if we a* • {A £ a
d~fine
I
A0 •
o},
then for the inhomogeneous concept of spectrum we have in analogy with theorem 3.3.· Theorem 3.6: (i)
Let f
(Af,f) •
€
f
H 8 HA(co).
a-a*
36
Then
(E(dA)f, A£),
(ii)
f
f •
E(dA)f.
a-a*
ll:.,i
THE CASE OF COMPACT OPERATORS
In this section we shall investigate the nature of the spectrum a, under the additional assumption that each of the operators Ai: Hi+ Hi' • compact. 1s
F or A'
•
i
= 1,
••• , n
('AO' ••• ,An ' ) E lln+l d e f'1ne operators S i (') A :Hi+ Hi
by n
n
r
r
S.(A) • -A0 A. + A. S .• , a. A. 1 1 j•l J 1J i=O 1 1
= 1.
i • 1, ••• , n.
(3.10)
Theorem 3.7: (i)
A
E
a if and only if
S.(A), 1 (ii)
0 is in the spectrum of each
i • 1, ••• , n,
A is an eigenvalue if and only if 0 is an eigenvalue of each S.(A). 1
First
Proof:
note that (ii) is immediate from the definition of
we
homogeneous eigenvalue. A.1
E
a(r.) and 1
we
In the proof of (i) first suppose A
find a sequence fm
ll. •• i • 1, 1
~
0
.... ••• J
n and ).
~
0
n, is non-zero,
then it is impossible for 0 to be in the continuous spectrum of each of Si(A),
i • 1, •.•• n.
Proof:
Suppose that each of the operators Ai : Hi+ Hi"
i • 11
••••
is compact and that 0 is in the continuous spectrum of each Si().). we can find sequences f~ E H., 1
1
II f~ IIi
n Then
• 1 such that f~ tends weakly to
zero and S.().)f~ tends strongly to zero. 1
In
1
case (i) we have >.0 • 1 and so A.S .. f~+O, 1 1J 1 Then on putting ).1
fB = ~
i • 1, ... , n.
8 ••• 8 f: we have, assuming for simplicity that
'I o. (Afm, fm) - (110 fm, fm) •
<suC:·~>l (Snl ~~
•
f:)
• • • • •• • • • • • •• • • • • • • • • • • • •
<slnf~.~>l
1 ·························
1
~
n
< j•l I >..s .fD.fD> J nJ n n n 39
Thus (A0 fm,fM> + 0 which contradicts the fact that in this case A0 is positive definite.
This proves the theorem in this case.
In case (ii) we
have al ......... • .. •. an
<sn 1f'l,n fm) . • • • • • <s fm fA> n n nn n' n n as m + •.
+ 0
!bat is (AfM,fm) >. 1
~
0 which again is a contradiction.
+
In case (iii) suppose
0 then
(Afm,fm)
1
Alal . • • • • .. • • • • • • an
- >.1
>.1 (Sllf~,c:)l • • • • • (Slnf~,f~)l
.
>. (S 1
.
fm fm) (s· fm ..m) nl n' n n • • • • • nn n' rn n
n
1
I a. A•••••••••••••••• an • l 1 1
·x:-1
J•
fm f'A> n n' n n
-(A
Thus
40
cAfM;fM>
+
0 which once again contradicts the positive definiteness
of A.
This completes the proof.
As immediate Corollaries to theorem 3.8 we have Corollary 3.2:
H., i • 1, ••• , n is Suppose each of the operators A.1 : H.+ 1 1
If A • (A0 , ••• , An) is a non-zero point in a then at least one of
compact.
the equations n
-A0 A. f. + 1
1
r AJ. S .J. f j=l 1
1•
-
o
•
i • 1, ••• , n,
with n
r
j-o
a. A. • 1, has a non-trivial solution. J
J
Corollary 3. 3: is compact. ~:
Suppose each of the operators A. : H. +H., 1
1
1
i • 11
• •• 1
D
Then 0 .: a and (A0 0, ••• , 0) .: a provided a 0A0 • 1.
The theory described in this chapter is a generalisation of some
work of Browne [2] and includes that of Klllstr8m and Sleeman [3].
References 1 F. V. Atkinson,
Multiparameter eigenvalues problems Vol 1, Matrices and compact operators. Academic Press, New York (1972).
2 P• .J. Browne,
Multiparameter spect~al theory. Math • .J. 24 (1974) 249-257.
Indiana Univ.
3 A. Klllstr8m and B. D. Sleeman, Multiparameter spectral theory. f8r Matematik 15 (1977) 93-99.
Arkiv
41
4 Multiparameter spectml theory for unbounded opemtors (The right defmite case) 4.1
INTRODUCTION
In this and the following chapter we consider extensions of the spectral theory given in chapter 3 to the case when the operators Ai'
i = 1, •.. , n
are unbounded. As before H1 , ••. , Hn are separable Hilbert spaces and H = H1 8 ... 8 Hn is their tensor product.
In each space Hi'
i = 1, ••• , n, we have
operators A., S .. , i,j • 1, ••• 1 n enjoying the properties. 1
(i)
1J
s ..
H. + H. are Hermitian.
A.
D(A.)
1
1J
(ii)
1
1
1
c
H. +H. are self-adjoint. 1
1
Each of the operators S •. , A. induce operators S!., A! in H as described 1J
in section 2.3 of chapter 2.
1
1J
In particular if W.(A) is the resolution of 1
the identity for A. then we define A! • 1
1
1
f
co -co
A dw! (A). 1
Denote by D the dense subspace of H given by D •
(See 2.13).
n t n D(A.). 1 i•l
On D define
operators !:., t. 0 , t. 1 , ••• , t.n in precisely the same way as in (§3.1, chapter 3). However instead of assuming that t. : D + H is positive definite we assume t:.0 E S • det{S •• } is positive definite on H in the sense of hypothesis 2.1. 1J
This ensures that S-l : H + H exists as a bounded operator. In the main we shall not use the inner product(·,·) in H generated by the inner products (·,·)i in Hi' but rather the inner product given by (!:.0 •,.) wbic~
in chapter 3. 42
will be denoted by[·,·].
This follows the procedure adopted
lbe norms induced by these inner products are equivalent so
that topological concepts such as continuity of operators and convergence of sequences may be treated without reference to a particular inner product. However algebraic concepts may depend on the inner product used.
If
A : D(A) c H + H is a densely defined linear operator, we denote by A• the adjoint of A with respect to [•,•].
Similarly A* will be used to denote the
adjoint of A with respect to (.,.). As in the spectral theory developed in chapter 3 a fundamental role is played by certain operators constructed from the operators A, A0,
... '
An •
Here such operators are defined by
r.1 4.2
d~f
D c H + H,
i = 1, ..• , n.
(4.1)
COMMUTING SELF-ADJOINT OPERATORS
Leuma 4.1: i.e.
-1
= A0 A.1
The operato:mr. : D c H + H,
for all f,g
1
£
i = 1, ••• , n are [·,·]-symmetric,
D
[r 1•• f,g] - [f,r 1.g],
i
= 1,
•••• n.
The proof of this result is identical to the proof of Theorem 3.1 and may be omitted. For subsequent development we introduce the following terminology.
Let
B1 , ••• , Bn be bounded Borel subsets of Rand let B • B1 x ••• x Bn c Rn be their Cartesian product.
Recalling that W.(·), 1
i • 1, ••• , n is the
resolution of the identity for A. we define, as in chapter 2 section 2.4, 1
the spectral measure W(B)
= wi j•l J
st1J .. r J. f That D
n n
c
j=l
-
and for all f e D.
o,
i
= 1,
•.. , n. n
Now suppose f e
D(r.) is obvious. J
n D(r.), then
j•l
J
t n t -1 n t r S .. r. f • - r S .. A0 r Ak AOk. f j•l 1l J j=l 1J k•l J n
..
lim
B~n
n t -1 r S •• A0 A.(B)f • j•l 1J J
Noting that the array (4.2) satisfies the conditions of section 3.1 chapter 3 and using Corollary 3.1 of Theorem 3.2 we have
r
t t n t -1 A.W.(B)f + S .• A AJ.(B)f • O, 1 1 j•l 1J 0
i • 1, .•• , n,
and so
--
st1J .. r.J f Thus
f E
.,..·..
lim A.W!(B)f,
~n
1 1
i • 1, ••• , n.
D(A!) • D and J
s"f. r. f 1J J This proves the lemma.
•
o,
i - 1, ••• , n.
The calculations used in the above proof also show
that
49
f
E
D(r.) ==>lim r. (B)f B4.n
J
J
exists and equals r. f • J
We can now usefully summarize all the above results in Theorem 4.1:
The operators r..
j • 1, •••• n are given by
J
n t D(r.)• n D(A:t:.0k.) •{feHilim r.(B)exists}. J k•l -""It J a.,n J
(i)
for f e D(r.),
(ii)
J
n I r. f • - t:.-1 0 l k•l
·-
Akt t:.0 k.l
• lim
a..n
r .(B)f. l
These operators are [·,·]-self-adjoint and, further f
E
D(r.) • D and for J
D.
n
I s!. r.f j•l 1J J
•
o,
i . 1, •• • 1
D.
Our aim now is to follow the ideas of chapter 3 section 3.2 , to arrive at a spectral theory and associated Parseval equality.
in order Thus
having established the self-adjointness of the operators ri. i . 1, •••• n, it remains to show that they also commute. Let Ei(A),
i • 1, ••• , n be the·resolution of the identity for ri.
According to definition 2.2 the operators ri commute if E.(B.)E.(B.) • E.(B.)E.(B.), 1
1
J
J
J
J
1
1
(4.6)
i,j • 1, ••• , n,
for all Borel subsets B., B. of•.
Note that E.(•) is an orthogonal
projectionwi_th respect to[·,·].
In order to prove (4.6)
1
J
1
we
proceed as
follows: Let E. B(•) denote the resolution of the identity for the [.,.J-self1,
-1
adjoint operator r i (B) • t:. 0 t:.i (B).
so
Prom chapter 3, Theorem 3.2
we
kiunr
that the operators r.(B),
i = 1, ••• , n are pairwise commutative and so
1
E. B(B.)E. B(B.) • E. B(B.)E. B(B.), 1
i,j
1
0
= 1,
... '
Jo
J
Jo
J
1
(4. 7)
1
0
Let B + :Rn through a
n, for all Borel subsets B., B. of R. 1
J
sequence of bounded Borel sets and take any real number a which is not an eigenvalue of r 1••
Then from a theorem of Rellich [6, Ex.38, p. 1263, 8, p.
369] it follows that E. B{ (-oo,a)} +E. { (-oo,a)} strongly. 1,
Consequently if
1
a and a are not eigenvalues of any of the operators r •• 1
i - 1, •••• n we
have E.{ (-oo,a)}E.{ (-oo,a)} J
1
i,j • 1, ••• , n. From
= E.{J (-oo,a)}E.{1 (-oo,a)},
the fact that spectral projections are strongly right
continuous we conclude that E.{ (-oo,a]}E.{ (-oo,B]} = E.{ (-oo,BJ}E.{ (-oo,a]}, J
1
i,j
= 1,
J
••• , n, for all a,a
E
1
:R.
The extension of this result to arbitrary
Borel sets B., B. c:R follows from the strong a-additivity of the spectral 1
measures.
This proves (4.6) and we have
Theorem 4.2: 4. 3
J
Tbe Operator r., .
MULTI-PABAMBTIR
1
i
= 1,
•••• n are pairwise commutative.
SPECTRAL THEORY
We begin, as we did in chapter 3, by formulating a definition of the spectrum of the multi-parameter system{A.,S •. }. 1
1J
Recalling that E.(•) is the 1
resolution of the identity for r., then as in the statement of theorem 2.4 1
we can define a spectral measure E(•) on Borel subsets of Rn by (4.8) for all Borel subsets B.1 of :R,
i • 1, ••• , n.
51
Furthermore E(·) defined in this way vanishes outside a, the Cartesian product of the spectra, a(r.>. of r 1•• 1
i . 1, •••• n.
n
!he set a• x a(ri) is defined to be the i•l
Definition 4.1:
sp~ctrum
of the
multi parameter system { A•• S •• } • 1
1J
We are now in a position to.state the main result of this section. Theorem 4.3:
Let f [f,f] •
(i)
(ii)
I
f -
a
H.
€
Then
fa
[E(d>.)f,f]
·-J
a
(E(d>.)f,60 f)
E(d>.)f,
where this integral converges in the norm of H. The proof of this result follows directly from theory 3.7 on setting F(>.) • 1. We now establish a result which will be of use subsequently. Let A. : D(A.) c H. +H. 1
Lemma 4.5:
1
1
1
1
i • 1, •••• n be self-adjoint operators.
Then n n
Ker(A.)
t
n
• 8 Ker(A.). 1 i•l
1
i•l
Since A., A!1 are self-adjoint and consequently closed it fol.lows that 1
,!!.2.2!:
Ker(A.) is a closed subspace of H.1 and that Ker(At) is a closed subspace of 1 1 H.
The inclusion n
8
Ker(A.) c
i•l
1
is obvious.
If we write
A. • [ 1
. --
AdP.(>.)
we must show that 52
-
n t n Ker(A.) 1 i•l
1
n n
n
P!(O)H c 8 P.(O)H .• 1
i•l
i•l
-
i To prove this let ej'
1
1
j • 1, ••• , i • 1, ••• , n, be an orthonormal basis in
Hi' and let E:
ln
f •
La.l1l2• . • ·ln. P 1 (0)e~l1 8
t n P.(O)H • 1
n
n
..• 8 e.
••• 8 P (O)e~ n ln
n E:
8 P.(O)H. 1 i•l 1
and the proof is complete. As in the previous chapter we make the definition of eigenvalue and eigenvector as follows. Definition 4.2:
An eigenvalue and eigenvector for the multiparameter array
{A.,s •• } is defined to be ann-tuple 1 1J
A~
(A1 , ••• , A) of complex numbers
and a decomposable tensor f • f 1 8 ••• 8 fn A.f. + 1 1 Theorem 4. 5:
A. J
s 1J .. f.1
•
o,
n
E:
H such that (4.9)
i • 1, ••• , n.
Let A • (A 1 , ••• , An) be an eigenvalue for the system (4.9).
Then if S • det{S •• } is positive definite on H each A., 1J
1
i • 1, ••• , n is
real. If f • f 1 8 ••• 8 f is an eigenvector corresponding to the eigenn value A, we have
~:
A.(S •• f., f.). J
1J
1
1
1
and (f.,A.f.). •1 1 1 1
I'.(f., J
1
s 1J .. f.). 1 1 53
and since A. is self-adjoint and each s .. Hermitian 1
1J
n
r J. .. 1
.. J
x. ><s .. f .• f.>. J
1J 1
i = 1, ... , n •
0,
1 1
It now follows from the positive definiteness of S that>..= J
X., J
j
= 1,
••• ,n
thus proving the result. If A € a is such that E({).}) ; 0 then A is an eigenvalue.
Theorem 4.6:
Conversely if A is an eigenvalue then A Proof:
Suppose A
€
£
a and E({>.»; 0.
a is such that E({>.}) ; 0 and g
€
E({>.})H,
since the operators E.({>..}) commute we have E.({>..})g • g 1 1 1 1 i
= 1,
each i and r.g =>..g. 1 1
From theorem 4.1 we conclude that g
t t n t t + n A.g >.. S .. g = A.g + S .. r. g 1 J 1J 1 j=l 1J J j=l
r
r
= 0,
i
= 1,
Then
for each
From ordinary spectral theory we deduce that g
.•. , n.
g ; 0.
€
€
D(r.) for 1
D and
... , n.
Consequently, using lemma 4.5, we have 0 ; g
Ker{A. +
€
1
n t r >..S .. } j•l J
1J
=
n 8
i•l
Ker{A. + 1
n r A.S .. }, j=l J
1J
and so there must exist a non-zero decomposable element f .. £1 8 ••• 8 fn' f.1
€
D(A.), such that 1 n
A.£.+ 1 1
r
j•l
>..S .. f. J
1J
1
= 0,
i •1, ... , n.
This shows that A is an eigenvalue. Conversely, if A • (>.1 , ••• , An) is an eigenvalue with non-zero decomposable eigenvector f • £1 8 ••• 8 fn
54
€
D we have
or n·
n
n
i•l
j•l
i•l
t t ~ AOik Ai f + ~ >..J ~ AOik 8 ij f .. o.
i.e.
- Ak f + >.k Ao f • o.
or
rk f .. ).kf' ~
from which it follows that ). E 0 and E({>.}) If for ). ERn we define operators S.(>.)
D(A.) c H.+ H. by 1
1
n
s.CA> 1
... ~ ).,J s 1J
=A.1 +
i
j=l
= 1,
0. 1
1
•• •' n,
then as in theorem 3.7 of chapter 3 we have Theorem 4.7: (i) (ii)
A Eo
1
A is an eigenvalue if and only if 0 is an eigenvalue of each
4. 4
if and only if 0 is in the spectrum of each S.(A).
s·.1 (A).
THE COMPACT CASE
In this section we shall investigate the spectrum of the multiparameter system under the additional hypothesis that for each i • 1, ••• , n -1
A.1
: H.1
+
H.1 is compact.
Theorem 4.8:
-1 If A. : H. +H., 1
1
1
i • 1, ••• , n, then the spectrum o of the
multiparameter system ~ •• s .. } consists entirely of eigenvalues and 0 ~a. 1
1J
-1
Proof:
From theorem 4.7 and the fact that A.1
0
Also from theorem 4.7 we must show that if 0 is in the spectrum of
~a.
Si (A), .then it is an eigenvalue of Si (A). of S.(A) and consider a sequence 1
f'!1
E H1. ,
is compact it is clear that
Thus suppose 0 is in the spectrum
llf'!ll· • 1 1
1,
n • 1, 2, ... ,
so
that 55
A.~+ 1 1
n
l
A. S .. ~
j•l
J
1J
+
O,
as
m + oo.
1
Then f~ + 1
n l A. A-1• S. • ('! j•l J 1 1J 1
0,
+
as m
+ oo.
Since A: 1 is compact it follows that there is a sequence ~ 1
A.-1 S •. f.m converges. 1
1J
so that
1
1
..m Consequently E. converges to f. say as m + 1
1
oo.
Thus
llfilli • 1 and n
f. + 1
-1 A. A. S •• f. r j=l J
1
n
A.f. + 1 1
or
r
j•l
1J
A.
s ..
J
1J
1
f ••
• 0
o,
1
and hence 0 is an eigenvalue of each Si(A),
i • 1, ••. , n.
We now turn to the question of multiplicity of eigenvalues. Definition 4.3:
If
~
is an eigenvalue its multiplicity is defined to be
(from lemma 4.5) n t n t n n dim n Ker(A. + l A. S •• ) • dim 8 Ker(A. + l A. S .• ) 1 . j•l J 1J i•l 1 j•l J 1J i•l
•
Theorem 4.9:
n n R dimKer(A. + l A. S •• ). 1 i•l j•l J 1 l
Under the compactness hypothesis of this section each eigen-
value bas finite multiplicity. n ~:
For each i
Ker(A. + 1
l
jO.l
A. S .• ) is a closed subspace of J
may select an orthonormal bases e~, 1
1J
m • 1, ••• ,
for it.
H 1•
and so we
If this basis is
infinite in number the argument of the previous theorem shows that lim e~ ur+oo
56
1
exists which is Theorem 4.10: orthogonal.
~possible.
Eigenvectors corresponding to different eigenvalues are[·,·]Furthermore the eigenvalues have no finite point of accumulation.
The orthogonality of eigenvectors follows a standard argument and is omitted. Suppose Am • (Am Am) is a sequence of distinct eigenvalues 1,
~:
....
I
eigenVeCtOrS with (fm, en )
n
t: be corresponding
Let fm • ~ 8 ••• 8
converging to A= (A 1 , ••• , An). ~
1 o Again following the argument Of theorem mm 4.8 we may find a sequence en converging to f. Thus on the one hand
[f,f]
111
= lim[fm,en] = 1 while ~
on the other [f,f]
= 1~
I
l~[en,enJ
~~~
= 0.
Thus
the eigenvalues have no finite point of accumulation. Let us now define the eigensubspace corresponding to the eigenvalue A .. (A 1 ,
... '
A ) to be n n n ·8 Ker(A1 A.S •• ). j•l J 1J i•l
+I
Then we may prove Theorem 4.11:
Each eigensubspace is finite
d~ensional
and has a basis of
[·,·]-orthonormal decomposable tensors. That the eigensubspace is finite
~:
d~ensional
follows immediately from
theorem 4.9. Now let A be an eigenvalue and put n
G. • 1
Ker(A. + 1
Then G.1 is a finite
I A. S •• ). j=l J 1J d~nsional
subspace of B1••
Let P.1 be the projection of
B.1 onto G.1 and consider the operators P.T., P.S.j : G.1 +G 1•• 11 11
These operators
are Hermitian in G.1 and the array
57
p n sn 1
PA
n n
p
s
n nn
satisfies all the conditions of Ll, §§7.4-7.6].
Thus appealing to theorem
7.6.2 of that reference the result is proved. In summary, the content of this-section may be summarised in Theorem 4.12:
-1
Let A.1
: H. +H. exist as a compact operator, 1
i .. 1, ... , n.
1
Then the spectrum a of the system {A.,S •• } consists entirely of eigenvalues 1
having no finite point of accumulation.
1J
If
p
A ,P
= 1, 2, ••• ,
is an
enumeration of the eigenvalues repeated according to multiplicity then there is a set of[·,·] orthonormal eigenvectors
such that for any f f ...
I
H we have
£
[f,hp]hP,
p
where the series converges in the norm of H. 4. 5
AN APPLICATION TO ORDINARY DIFFERENTIAL EQUATIONS
In this section we apply the abstract theory of this chapter to the classic multiparameter eigenvalue problem for ordinary differential equations. Thus we consider the system n
+ q.(x.)y.(x.) + 2
dx. 1
where 58
1
1
1
1
I A.a .• (x.)y.(x.) j=l J 1J 1 1 1
""0,
i
= 1,
... , n,
(4.10)
q.(x.) 1
E
1
C[a.,b.], 1
i,j
1
= 1,
... , n.
Furthermore we assume a .. , q. to be real valued and 1J
1
(4.11)
n
for all x
X
[a.,b.J. 1
i=l
1
An eigenvalue problem is formulated for the system (4.10) by seeking non-
of (4.10) satisfying the end conditions y.(a.)cos a. - y!(a.)sin a. = 0, 1
1
1
e1. -
y.(b.)cos 1
1
1
1
y!(b.)sin 1
(4.12)
1
1
e.1
= 0,
o all!
(i)
is convergent.
p•l
Furthermore 00
I II (Wmt Tn Ep >n.... 11 n2
(ii)
p•l
t
~ (W t T W ). m n m
(Bessel's inequality)
Proof: M
I p•N
M
E • (W t T E >~t Tt E • (W t T E \... p m n p n n q•N q m n q1i
I
M
M
• p•N I q•N I
(B 8 (W t T E ) At Tn Eq 8 (W t T E ) .,..) P m n p n m n q n
( (W t T E ) At (W t T E ) A) • (E t T E ) A• m npn m nqnn p nqn But (E t T.. E ).,. • a 6 (Kronecker-delta) since E is an orthonormal .p .. q n n pq p system in ( • t al T • )A • n
a
M
I II (Wm np•N
t
n
n
T E
>... II
Hence the above sum reduces to
2
npnn
.
N
It follows that
only if
I
p•l
N
I II (Wmt p•l
E 8 (W t'r· E )A is a Cauchy sequence in (. t Tt •) if and p m n p n n 2
T E >... II n p n n
is a Cauchy;sequence.
If we evaluate the inner
product N
(W
m
-
I p-l
N
E
p
• (W
T E )A m n p n t
t
I
Tt (W E 8 (W t T E )..J n m p-l p m n p n
we find Bessel's inequality in the usual way.
'l'his completes the proof of
the leuma.
69
Since Tn is positive definite and bounded the ( ·, T tn • )-metric is equivalent to the (·,·)-metric in Hand hence it follows that
(...!.. I a
(Wm' un)n •
n
E
p
p
(W , T E ) ,. , un) • m npn n
8
Now form the (•,•),.-product of this with u 1 8 ..• 8 un-1 to obtain n
(W , u) •
m
. -
(...!.. I a
n p
E
8 (W , T E ) ,. , u) •
p
m
Since this holds for all u
wm = ....!. a
I
np
E
p
€
npn
D'
whi~
is dense in HA and H we deduce that
8 (W • T E ) ,..
m
npn
With this result it remains to show that the factor (Wm, Tn Ep ),., assumed n non zero is in fact a solution of (5.1) with i • n and A.
J
...'
j - 1,
n • Here we have defined An,p by
-1
A
n,p
= A.J ,p ,
an
n-1 (A
m
-
I
k=l
~A
k,p
)
.
Clearly not all the factors (W , T E ),. can be zero, for otherwise W would m n p n m be identically zero. Let (5.12)
f • (W , T E ),., m n p n.
then, as proved in lemma 5.3 below, A f = n
-
((A -
n-1 t t I A. T. )W • E ),. ) i•l 1 1 m p n
(AW • E ) A m p n
-
n-1 I i•l
t (T. 1
wm• A.1
E ) .L'. p n
However the definition of E implies that p
70
(5.13)
n
A.E 1
p
.. I
A.
j=l
i .. 1, .•. , n -1,
S •• E ,
J ,p 1J p
and we also have AW
= A SW • m m
A f
=A (SW ,E),.-
m
Hence n-1 n
m
m
n
... (T~ w. J, p 1 m
I I i=l j•l
p n
A.
s ..
E),..
1J p n
(5.14)
Consider the second term in the equation (5.14) which can be written as n-1 (
I
n
I
i•l j•l
A.
t t
T.S •• W ,E)
J,p 1 1J m
p 6
On using (5.3) we can write
n -r l.
n l.-r
j•l k=l
•
Am s
-
'
A.
J,p
.r: ak <s u.k J
-
s"tk st . > n
nJ
In A. Tt st .. j•l JoP n nJ
Thus (5.14) reduces to
71
An
=
f
A.
loP
t t
(T S • W
0
n nl m
A•
E ),. •
p n
loP
S • (W o T E ) ,..
nl
m
n p n
n
- j•l I A.loP snl. f. Thus we have shown that f is in-fact a solution of the remaining equation in (5.1) with the same eigenvalue as the first n -1 equations.
This means
that W can be written m
W
m
•
Ip
where each u
C u 1 8 ••• 8 un pm p p i
(5.15)
is a solution of (5.1) for i • 1 0
p
n.
••• 0
The sum in (5.15)
must also be a finite sum with the number of terms not exceeding the multiplicity of the eigenvalue Amandeacb u!8 ••• 8 u: isaneigenvector of (5.5) co with eigenvalue A • From the fact that {W } 1 is a complete set in m m m• · f o 11 ows t ba t (up1 8 8 un)co · a comp1 ete set as we 11 • HA 8 HA (co ) 1t p p-1 1s This proves the theorem. Lemma 5.3:
p
eigenvector of the n -!-parameter system (5.10).
(i) (ii)
Proof:
(W 0 T E ) ,.
n p n
£
n
n p n •
p n
m0
"and by
72
~bapter
n-1
I
i•l
t
(T.W 0 A.E ),. 1 1 p n
n
£
D• such that W m
T E ),. £ D(A )
n p n
p
Then
linear bull of u 1 8 ••• 8 u· 0
Choose a sequence W m
(W
8 un-l be an
p
D(A ) n
A (~ T E ) " • (AW 0 E ),. -
Recall that D1
(a)
• u1 8
Let W £ D(A) (A • closure of A) and E
n
2 §2.2) that
+ W
u
•
i
and A W m
£ D(A.) 0 is dense 1
+ AW.
It follows
(b)
(W , T E ) .... m npn
+
(W T E ) .... in H • ,npn n
But A (W , T E ) ""• (At Tt W
n m
n p n
n n m'
E ) ....
p n
n-1
•
I
( (A -
•
(A
w•E m
A: T:)
i•l
1
n-1 ) .... -
p n
w• E m
1
I
i•l
)
p
6
(T.t W, A. E ) ..... 1
m
p n
1
Let m + ~ and since Tt is bounded it follows that n
1 im A (W • T E ) .... •
n m
art-
n p n
(A w. E
n-1 ) .... -
p n
I
i=l
t
(T. W, A. E ) ..... 1
1
p n
But A is a closed operator and so (i) and (ii) follow. n
We apply lemma 5.3 to the eigenvector W to arrive at equation (5.14) and m may continue as before.
5. 4
AN APPLICATION TO ORDINARY DIFFERENTIAL EQUATIONS
As in chapter 4 §4.5, we illustrate the above theory in application to the multiparameter eigenvalue problem. 2
d y.(x.) 1
1
2
dx.
n
+ q.(x.)y.(x.) + 1
1
1
1
I
j•l
A. a •• (x.)y.(x.) • 0, J
1J
1
1
1
i • 1, ••• , n
1
"(5.16) where -
~ O, r•l r sr
s • 1, ••• , n,
(5.19)
for all x £,. I n , where a*sr denotes the cofactor of a sr in the determinant
74
• A·· det{a }n n rs r,s• 1 For the Hilbert spaces H.
i • 1, ••. , n, of the abstract theory we take
1
H. • 1
2
L (a., b.), and define S .• : H. +H. by 1
1
1J
1
1
(S •. f.) (x.) =a .. (x.)f.(x). 1J
1
1
1J
1
1
The self-adjoint operatorsA. are Sturm-Liouville operators generated from the 1
differential expressions d2
2 - q.(x.), dx.
1
i • 1, ••• , n
1
1
and the end conditions (5.17). n
Next, the operators
T.
1
+
8
k=l k"i
~·
i • 1, ••• , n
are defined as multiplication by the continuous function h.1 given by (5.19). Following the formulation in section 1 of this chapter we see that the eigenvalue problem (5.5) becomes the boundary value problem n
AY
- i•l l
(h. 1
2
a Y - h.q.Y) • - AA Y, 1 1 n ax.2
(5.20)
1
where n A=
r
j•l
]J. J
>.. J
•
(5.21)
and whe•Y is subject to the boundary conditions (5.17) on the hypercube I • n In order to cons true t the Hilbert spaces HA we consider the following sets of boundary conditions
75
(A)
Robin condition
(i)
ay - cot a. y • axi 1 ay -a x. - cot 1
a.1
o, x •• b.,
y - 0,
1
(5.22)
1
i • l , ... ,n. (B)
Neumann condition (ii)
J[! • 1
Dirichlet condition (iii)
1
1
i • 1, ••• , n.
1
a.1 = 'lf/2.
a •• (C)
x.•a.,b.,
0,
axi
(5.23)
Y•O, x. •a.,b., 1
a • • 0, 1
1
i • 1, .•• , n
1
a1.• 'If.
(5.24)
In the case of Dirichlet boundary conditions HA is the completion of C~(In) with respect to the inner product D(u,v) • JI n
n "1. (h • a au a av h -) dx 1 x.1 x.1 + 1.q.1 uv - '
(5.25)
"•1 1
while for the Neumann problem HA is the completion of C (I ) with respect to n
the inner product (5.25).
.
In the case of the Robin boundary conditions
(5.22) HA is the completion of C (In) with respect to the inner product
fI
D(u,v) •
where
Iin
n X
j•l #
~ 1.
n
i•l
(h. .au av + h.q. uv) - dx _ -a 1 GXi xi 1 1
[a., b.], and dxi • dx J
J
j"i
the boundary of In consists of two parts 76
••• , dxi-l dxi+l' ••• , dxn. Suppose
n1, n2 for which we have Robin or Neumann
conditions on g 1 and Dirichlet conditions on g 2 •
Then we take the inner
product D(u,v) as defined in (5.25) but with boundary integrals only over g 2 • The Hilbert space HA is then the completion of the set{u g 2 } with respect to this modified inner product.
£
c1 (In); u = 0 on
Because of the conditions
imposed on the coefficients a .. , q. we see that (5.25) defines a positive 1J
definite Dirichlet integral.
1
If we also suppose, in the case of Robin
boundary conditions, that
a.£ (0,'11"/2], 1
a.£ ['11"/2,'11"), 1
i .. 1, ... , n
then (5.26) also defines a positive definite Dirichlet integral. Since S : H + H defined as multiplication by the function det{a •• }~ . 1 1J 1,J• is continuous it follows that S is compact relative to A as defined by (5,20). Consequently all the conditions of the abstract theory are met and we have Theorem 5.2:
Under the stated hypotheses the spectrum of the system (5.16)
(5.17) consists of a countable set, having no finite point of accumulation, of real eigenvalues with finite multiplicities.
Furthermore the correspondiqg
eigenfunctions form an orthonormal set with respect to the D metric (defined by (5.25) or (5.26)) and are complete in the space HA 8 HA(=) where HA(=) is the set {u
£
H,
~
n u • 0 and u satisfies the boundary conditions on 3I n }.
5.5 A COMPARISON OF THE DEFINITENESS CONDITIONS To conclude this chapter we compare the main hypothesis of chapter 4 that S be positive definite and the present assumption (5.3) that each of the operators Ti' i
= 1,
••• , n be positive definite.
In
~he
case n • 2 it is
easily proved using a theorem of Atkinson [1, p. 151 Theorem 9.4.1] that if S is positive definite then there exists a pair a 1, a 2 such that T1 , T2 are 77
positive definite.
The converse however need not be true as may be seen from
the following simple example (5.27) 0 S
X
S
1,
together with Sturm-Liouville boundary conditions for both equations. The condition that S be positive definite is equivalent to requiring -1
1
This is obviously not true except for special choices of p and q.
The
assumption that T1 , T2 be positive definite in this case is equivalent to seeking two real numbers a and
a such
that -1
a
> 0
>
o.
a
1
Clearly if we choose a • 1, choice of p and q.
a•
0 then the assumption is satisfied for any
Notice that in this example assumption 2 is also
satisfied since -yl + y 1 has a positive definite Dirichlet integral. If n
~
3 then there is no connection between assumption 1 and the
condition S be positive definite.
s
-
1
cos xl
sin x 1
1
cos x2
sin x 2
1.
cos x3
sin x 3
on I • [0, '11'/3] Then
78
X
[2'11'/3,
'II']
X
Consider
[4'11'/3, 5'11'/3].
for all_x1 , x 2 , x 3 £I. The determinant of cofactor& is
Using assumption 1 suppose there exist real numbers a 1 , a 2 , a 3 such that cos x2) >
o,
cos x3) > 1 -
o,
T3 •al sin(x2 - x 1 ) +a 2 (sin x 1 - sin x 2 ) +a 3 (cos x2 - cos xl) >
o.
Tl •al sin(x 3 - x 2 ) +a2 (sin x 2 - sin x 3 ) +a 3 (cos
X
T2 =al sin(x1 - x 3 ) +a 2 (sin x 3 - sin x 1 ) +a 3 (cos
X
Then for x 2
and for x 1 •
= 2~/3,
~/3,
x3
x3 •
= 5~/3
3
-
we have
4~/3
From this it follows that a 3 > 0 but for x1
= O,
x2 •
~
T3 • -2a3 > 0 which gives a contradiction.
Hence there are no numbers a 1 , a 2 , a 3 such that
T1 , T2 , T3 are all positive. In the reverse direction consider the determinant
2
-1
-1
-1
2
-1
-1
-1
2
•
o.
and here
79
and each Ti' Notes:
i = 1, 2, 3 is positive if for example a 1 = a 2
= a3
• 1.
The abstract theory given in this chapter is based on the work of
Klllstr8m and Sleeman [3].
It seems likely that the theory here is still
valid without the assumed "compactness" requirement used in §5.2 but the proof seems difficult, certainly for n
~
3.
The illustrative application
to differential equations is largely taken from [4] wherein it is shown that the conditions on the coefficients q.{x.) may be relaxed considerably. 1
1
References
1
F. V. Atkinson,
2
A. KUllstr8m and B. D. Sleeman, A multiparameter Sturm-Liouville problem. Proc. Con£. Theory of Ordinary and Partial Differential Equations. Lecture Notes in Mathematics, Vol. 415 {1974) 394-401, Springer-Verlag, Berlin.
3
~KHllstr8m
4
A. KHllstr8m and B. D. Sleeman, A left-definite multiparameter eigenvalue problem in ordinary differential equations. Proc. Roy. Soc. Edin. {A) 74 {1976) 145-155.
and B. D. Sleeman, An abstract multiparameter eigenvlaue problem. Uppsala University Mathematics Report No 1975:2.
5 S. G. Mikhlin,
80
Multiparameter eigenvalue problems, Vol. 1. Matrices and compact operators. Academic Press, New York {1972).
The problem of the minimum value of a quadratic functional. Holden-Day, San Francisco, London, Amsterdam {1965).
6 An abstract relation
6.1
INTRODUCTION
In pursuing the study of multiparameter spectral theory perhaps the most important stimulus arises from the conjecture that any aspect of the one parameter case should have its multiparameter analogue.
For example in one
parameter spectral theory for differential equations it is often advantageous to replace the problem by its integral equation equivalent, thus making available the somewhat easier theory of bounded operators in Hilbert space. It is the purpose of this chapter to consider such a generalisation of this idea to the multiparameter system
A.u i +
A.S .. u
1
where u
i
£
J
1J
i
•0,
i • 1, ••• , n, S .. :H.+ H.,
Hi'
(6.1)
i • 1, •.• • n
1J
1
1
j • 1, ·•·• n is bounded and
symmetric and A. : D(A.) +H. is self-adjoint. 1
1
1
For ordinary differential equations such a generalisation has been outlined in [5], and, under additional hypotheses, an abstract approach for the case n • 2 has been given by Arscott [2].
6.2
THE PROBLEM
In addition to the system (6.1) consider the operator equation n
Bv +
I I. T. v
j•l
J
• 0,
(6.2)
J
where B is densely defined and closed in a separable Hilbert space h and
81
TJ·•
j = 1, ••• , n is a bounded operator in b.
taken tobe aneigenvalue9f thesystem (6.1).
In applications the operator
B may be identified with any of the operators A1• and similarly T. with S •• , 1J
J
for some fixed i, so that h and Hi are topologically equivalent. also happen that the null space of (6.2) be empty.
It may
We assume that this is
not the case. The problem to be discussed is to seek an expression for a solution v of
(6.2) in terms of the eigenvectors Qf the system (6.1).
To this end we
introduce the new Hilbert space H* • h 8 H, (H
=
n 8
Hi)'
i•l
in which the inner product (norm) is denoted by ( ·, • )* We also introduce the following notation.
H
++
€
Then
D(B)
= H -
""'tt t In H. i=l 1 1
This lemma is merely a restatement-of lemma 5.3 applied to the system (6.1)
(6.2). Proof of Theorem 6.1: and Bv • -
n I i•l
= <S ttK,
u>H
£
D(B)
~ t ~ t H where A. is the cofactor of A. in the determin1
antal array (6.3).
Bv •
From lemma 6.1 we have that v
n
I
n
I
i•j j=l
1
1
1
Since u is an eigenvector of (6.1) we obtain ....tt t H. 1
(6.6)
J 1J
Observe from (6.3) and (6.5) that
is equivalent to a determinantal array with the first and j-th row equal and so must be zero. Let '
€
h be arbitrary; then on taking the inner product of Bv with ' we
obtain from (6.6)
The summation over i can be written as
84
n
I i•l
( 8 , cjl)h 1 1J
n
=
I i•l
t (h' sij u)H 1
n
•
I i•l
--ttK, (S t.. h' u)H 1J 1
-
I i=l
n
--tt Stt (h' u)H 1 1J
.-
----
(<SttT1tK, cjl>h' u)H (
h' u)H
Hence we have
-and the result follows. We make the following remarks regarding theorem 6.1. (1)
Observe that in the determinantal arrays (6.3) (6.5) the operators
in different columns cODDute. are uniquely defined.
This implies that the operators
A
and Stt
Note also that from (6.6) and onwards all operators
used in the proof are induced by bounded operators and hence are bounded themselves.
Therefore K is always in the domain of these operators.
85
(2)
In the multiparameter eigenvalue problema treated in the previous
two chapters the eigenvalues of the system (6.1) are real. case we can deduce something more. K ..
This being the
Let v be a solution of (6.2) and define
· an e1genvector · u1 8 • • • 8 un 8 v, wh ere u 1 8 • • • 8 un 1s o f (6 • 1) •
Then
K is in the null space of (6.3) and we easily obtain
<S (3)
tt
K, u>H
= v(St u,
u)H.
If the operators Ai'
. i
= 1,
•..• n are not necessarily self-adjoint
j - 1, ••• ;-n are bounded not necessarily symmetric
and the operators S .• • 1J
operators then we may establish the following generalisation of theorem 6.1. (We use * to denote adjoint.) Theorem 6.2:
Let K
£
A*tt
A*tt
..........
s*tt
8 .tt 21
.......... S*tt nl
1
A=
H* be an element in the null space of the array.
11
2
A*tt n
(6.7) 'tt
T
n
and let u • u1 8 ••• 8 un be an eigenvector of the system (6.1). v • <s*tt K, u>H is a solution of (6.2) where s*tt is the determinantal array.
86
Then (6.8)
~
SOME APPLICATIONS TO ORDINARY DIFFERENTIAL EQUATIONS
(I)
The one-parameter case (n • 1)
Let H1
= L2 (0,1)
(Lebesque measure) and let A1
D(A1 ) c H1
+
H1 be the
Sturm-Liouville operator
A1 • with domain D(A1 ) • {u, u' absolutely continuous locally on [0,1] and A1u
€
= 0,
L2[0,1], u(O) cos a - u' (0) sin a
u(l) cos a - u' (1) sin a = 0}. Then A1 is self-adjoint.
For the operator
s 11
: H1
+
H1 we take
Then if a 11 Cx) is real valued, positive and continuous on [0,1], Hermitian operator on H1 •
s 11 is an
Now let h be a separable Hilbert space and define
B to be a self-adjoint ordinary differential operator with domain D(B).
For
T1 : h + h we take
where t(y) is real valued and continuous.
If K € D(A1 ) •a D(B) is a
solution of the partial differential equation
a~ (t(y)q(x) + a (x)B(y))K • O, t(y) --a11
ax
then
v(y) •
J:
K(x,y)a 11 (x)u(x)dx.
(6.9)
If h • H1 and B is. identified with A1 and t identified with a 11 then since the eigenvalues A1 are real and simple it is easily verified that v must ,n
87
be at most a constant multiple of u and that K must satisfy the partial differential equation
Consequently u(x) must satisfy the integral equation (6.10) Results of the form (6.9) (6.10) are well known and may be found for example in Ince [3]. (II)
The multiparameter case
Let H1 • H2 • ••• • Hn • L2 (0,l) (Lebesque measure) and let A. : D(A.) 1
c:
1
H. +H. 1
1
be the sturm-Liouville operator q.(x.), 1
1
i • 1, ••• , n,
with domain • dui D(A.) • {u1 , ---d 1 x.1 absolutely continuous locally on [0,1] and A.u i 1
€
L2[0,1], cos a. ui (0)
sina.1 • 0,
1
cos Then each Ai is self-adjoint.
a. ui (1) 1
-
dui(l) dx.
sin
a. -
1
For the operator Sij
1
H. + H. 1
1
0} •
we take
S •• ui • a •. (x.)ui(x.). 1J
1J
1
1
Then with a •• (x.) real valued and continuous on [0,1], ~
operator on Hi. 88
1
s~ ..
is a Hermitian
Let B be the self-adjoint ordinary differential operator
with domain D(B)
c
h and T.
1
h
+
h real valued and continuous function.
Then if n K £
8
D(A.) 8
a i=l
1
a
D(B)
and satisfies the partial differential equation
AK = o, where A is defined
by (6.3), then
•• (x.)}u1 Cx1 ) ••• un(xn)dx 1 , ••• , dx v(t) "" JI K(t, x 1 , • • •, xn )det{a1J n 1 n (6.11) where I
denotes the Cartesian product of the n intervals [0,1].
n
If equation
(6.2) is a member of the system (6.1) in this case, then we obtain the multiparameter generalisation of the integral equation (6.10).
This bas been
studied in [5]. Finally, for n • 2, there are a number of specific applications in ordinary differential equations of theorem 6.1; we give one such example, others may be found in (1] or [4]. Let E:(y) be a
Lame
polynomial, then we have the integral equation
2K JK+2iK 1 Em(y) • A J n -2K K-2iK 1 where y
£
(K + iK', y 0 ) and R(P, P0 ) denotes the distance, in ellipsoidal
coordinates, of
P~,
a,
a0 ,
Y0 ). In this example & 1 • L2[-2K, 2K], & 2 • L2[K- 2iK', K + 2iK 1 ] and R •
y) from P0 (a0 ,
1
a(p,po)
is a solution of
89
2 2 a2a + (sn2 y - sn2 a) -a2a2 + (sn2 a - sn2 B) -a2a • O, (sn B - sn y) ~2
~
~2
wbich is simply Laplace's equation expressed in terms of ellipsoidal coordinates. References 1
F. M. Arscott,
Periodic differential equations. Oxford (1964) •
2
F. M. Arscott,
Transform theorems for two-parameter eigenvalue-problems in Hilbert space. Proc. Conference on the Theory of Ordinary and Partial Differential Equations. Lecture Notes in Mathematics 415, ·302-307, Berlin, Springer-Verlag (1974).
3
E. L. Ince
Ordinary differential equations. New York (1956).
4
A. Killstr8m and B. D. Sleeman,
5
B. D. Sleeman,
90
Pergamon,
Dover,
An abstract relation for multi-parameter eigenvalue problems. Proc. Roy. Soc. Edin. 74A 135-143 (1976).
Multiparameter eigenvalue problems and K-linear operators. Proc. Conference on the Theory of Ordinary and Partial Equations Equat_ions. Lecture Notes in Mathematics 280, 347-353 Berlin, Springer-Verlag (1972).
7 Coupled operator systems
7.1
INTRODUCTION
The spectral theory studied so far in this book bas been related to systems of operator equations which are coupled through the spectral parameters
A.,
j = 1, ••• , n.
J
"weakly coupled".
From now on we shall refer to such systems as being When in a given system of operator equations coupling
is effected through the "unknowns" in the system we shall say that the system is "strongly coupled"
Finally a system of operator equations which is
both weakly and strongly coupled will be called a "completely coupled" system. In this chapter we shall examine some particular completely coupled systems regarding them as a generalisation of weakly coupled systems. Typically these systems will take the form of sets of operator matrix equations of the form (7.1)
k • 1, ••• , n,
where
~·
Bk are n x n matrices with operator entries, A is an n x n diagonal
matrix with complex scalar entries and
~
is an n x 1 column vector.
In ordeE
to be able to consider the structure of such equations we introduce the following notation for various components.
.... A • diag{A.}, A. 1
1
€
c,
i •
k • 1, ••• , nand
1, ••• , n.
Any interpretation of the system (7.1) as a system of operator equations depends intimately on the properties of the component operators Ak and B~ •• ij 1J 91
Briefly, suppose for some fixed k we are given a collection of n separable k k k Hilbert spaces Hi, i = 1, ••• , n. Then the operators Aij• Bij may be interpreted in one or other of the following ways k Aij• B~. 1J
H.k +H.k
t
(7.2a)
k k •• A.1J·• B1J
H.+ H.·
(7.2b)
J
1
k J
k J
In both of these cases the system (7.1) can be interpreted as an operator k
equation in some suitably defined direct sum of the Hilbert spaces Hi• i • 1, ••• , n.
Systems of the form (7.1) are not entirely new to the literature. example in the case when k
=1
1
and Hi • H,
i
contributions by Dunford [2] and Anselone [1].
= 1,
For
•••• n, there are
However the general system
(7.1) does not appear to have been considered before. 7.2
DIRECT SUMS OF HILBERT SPACES
Suppose Hi' i
c
1, •••• n are separable Hilbert spaces.
The direct sum Hd
of these not necessarily distinct spaces is denoted by n
Hd • H1 I ••• t Hn •
t
i•l
(7.3)
Hi.
Any element of Hd is an ordered n-tuple { g 1 , ••• , gn} gi
€
Hi,
i "" 1, ••• , n.
Addition and multiplication by a scalar are defined by
(7.4) and c{g 1 , ••• • gn } • {c s 1 , .
••• , c gn } •
(7.5)
If e.1 denotes the zero element of H., then the zero element of Hd is 1 specified as {e 1 , ••• , en}. ( ·, ·) i ( II 92
.ll)
If the scalar product (norm) in Hi is denoted by
then the induced scalar product ( •, ·) and norm
II· II
in Hd is
defined as n
<s.h> • I
(g.,h.). 1
i•l
(7.6)
1 1
and
for all g
= {g1 ,
••• , gn} and h • { hl' ••• , hn} in Hd.
Clearly Hd is a
Hilbert space. I~
as we shall assume from now on the spaces H.1 are mutually orthogonal
then Hd admits a direct sum decomposition in the following way.
If g
Hd
€
then g has the unique representation g • &1 + • • • + g n • g.1
€
H., 1
i • 1, ••• , n.
For the remainder of this chapter we shall assume that the operator entrie1 in the matrix system (7.l)satisfy (7.2a).
In this case we can give the k
system (7.1) a suitable formulation in the direct . sum of the spaces Hi' i
= 1,
••• , n.
For simplicity we consider the case of
two
such spaces.
That .is consider
(7.8) Suppose we have a collection of operators A.. : H.+ H., 1J
J.
1
i,j • 1, 2,
then we may associate with these operators the matrix A : Hd + Hd with operator entries A •• defined as follows. 1J
Let x
€
Hd' then x has the unique decomposition (7.9)
where xi Ax
€
Hi'
= Ax1
i • 1,2.
Let A be a linear operator on Hd, then
+ Ax 2 •
93
However Axj'
= 1,
j
2, being,
an element of Hd has the unique decomposition (7 .10) .
Now they •. depend on x. and the dependence is 1J
J
linear and continuous, that is (7 .11)
Y1·J· =A •• x. 1J J
and
A.• : H. + H.• J
1J
1
Thus corresponding to each A on Hd there is a matrix {A •• } whose entry in 1J
row i and column j is the projection onto the i·th component of the restriction of A to H..
7.3 REDUCTION OF STRONGLY COUPLED SYSTEMS As before consider matrix operators A • {Aij}' is A•• : H. +H., B•• : H. +H., 1J
J
1
operator equation 94
1J
J
1
B • {Bij} : Hd + Hd; that
i,j • 1, ••• , n.
Then
we
have the
.Ax • ABx,
where x A.
1
£
£
a:,
(7 .15)
Hd and A • {Ai} is a
diagonal matrix with scalar entries
i • 1, ••• , n.
Again, for ease of presentation, consider the case n • 2,
i.e.
Thus in the notation of the previous section we may write (7.15) as
(7.17)
If we set
A.1 • A a.1 + J.1 B1. , where a., 1
B.1
i • 1, 2,
(7 .18)
are arbitrary real constants then (7.17) may be expressed as (7.19)
C.x • a. B.x, 1
1
1
D.x • B. B.x, 1
1
1
i - 1,2.
Finally, combining the two members of the system (7.19) we have Ax • A Cx + J.1 Dx
(7.20)
where
A•
The procedure for the reduction of the general system (7.1) is now clear. 95
Here A is a diagonal matrix with entries
~ii'
i • 1, ••• ,nand for each k
we have the direct sum Hilbert space k k k Hd • H1 t ... t Hn• with
elements~
If we let
~ ••
11
k
e Hd,
(7.21) k • 1, ••• , n.
a ..
•
(7 .22)
~.
1J J
then we arrive at the system
k
= 1,
••• , n
k
= 1,
••• , n,
(7.23)
where A... • -K
k {A •• } 1J
Hk d
+
Hk
d'
k
aljBln k
a2jB2n
........ a nJ.Bknn k,j • 1, ••• , n. 7.4
SPECTBAL THEORY FOR WEAKLY COUPLED SYSTEMS
Having described the method by which systems of the form (7.1) are reducible to weakly coupled systems of the form (7.23), the spectral theory for such systems is now clear.
Indeed the theory developed in chapters 2, 3 and 4
can now be carried over "en bloc" to the system (7.23).
We shall, therefore,
indicate bow the theory pxoceeds in the case of bounded operators and state the corresponding Parseval equality. 96
The extension to unbounded operators
is straightforward. Let H:.
Jt.
let
k = 1, •••• n, defined by (7.21) be separable Hilbert spaces and n
=
k
operators~·
the
k
8 Hd be their tensor product. k=l k
Ckj : Hd
+
k
Hd,
In each space Hd we assume that
j = 1, ••• n are Hermitian and continuous.
As in chapter 3 (3.1) we can define operators fl.1 :
Je +It via
the determin-
antal expansion of the equation n
flf
r y.!:.. f = i=O 1 1
.. det
...........
Yo
yl
-Alfl
Cllfl
-Ann f
c n lfn •••••••••• c nn f n
yn
.......... clnfl
(7.24)
for any n-tuple of real numbers y 0 , y 1 , •••• yn and where f = f 1 8 ••• 8 fn• fk
£
H:.
k • 1, •••• n is a decomposed element of
It.
For simplicity
assume that fl 0 defined in this way is positive definite in the sense of chapter 3 (3.2), i.e. (7.25)
for some constant C > 0 and all f in
i{ and HI ·Ill -1
=fl0 fl., 1
1l where denotes
the associated norm.
self-adjoint operators
r.1
£
i
r.1 : ;It = 1,
+ '/{
the inner product
As • is now familiar we construct
defined as
.•. , n.
(7.26)
and let E.(·) denote the resolution of the identity for these operators. 1
Next if we define the spectrum a for the system (7.23) in the same way as was done in chapter 3 section (3.2) then we arrive
at the fundamental result, 97
Theorem 7.1: f
Let
J.J
€
(i)
(ii)
/'t. •
n
k
8 Hd. k-1
• f •
Then
t
<E(dA)f, A0 f>.
Ja E(dA)f,
where this integral converges in
the uorm of ;/{ , and E( •) denotes the
Cartesian product of the spectral
mea~ures
Notes:
E.(·), 1
i
= 1,
••• , n.
The material of this chapter is based on the results of Roach and
Sleeman [3,4].
It remains however to apply the theory to strongly coupled
systems of differential equations which arise in a number of physical applications. lleferences 1
P. M. Anselone,
2
N. Dunford,
3
4
98
Matrices of linear operators. Math. 9 191-197 (1964).
Enseignement
A spectral theory for certain operators on a direct sum of Hilbert spaces. Math. Ann. 162 294-330 (1965/1966). G. F. Roach and B. D. Sleeman, Generalized multiparameter spectral theory. Proc. Conference on Function Theoretic Methods in Partial Differential Equations. Lecture Notes in Mathematics. Springer-Verlag, Berlin (1976), 394-411 G. F. Roach and B. D. Sleeman, Coupled operator systems and multiparameter spectral theory. Proc. Roy. Soc. Edin. (A) (to appear).
8 Spectral theory of operator bundles
8.1
INtRODUCTION
Let H be a complex separable Hilbert space on which are defined linear Hermitian and continuous operators A. : H + H, 1
i • 0. 1, •••• n.
We shall
be interested in the spectral properties of operator bundles of order n, denoted by LD (A), and having a form LD (A)u • A0 u - AA 1 u - A2 A2 u - ••• - A0 AD u. Such operators arise frequently in the literature. operators Ai•
i • 0, 1 1
••••
Indeed, when the
n are assumed to be completely continuous,
equations of the form
(I - L (A))u D
=0
have been studied by M. V. Keldys [3].
In the particular case when n • 2
the associated operator L2 (A) is referred to as a quadratic bundle.
Equatioms
of the form (8.1) play an important rSle in the linear theory of small damped oscillations of systems having an infinite number of degrees of freedom.
For an account of
this particular problem in an abstract setting we cite the book of I. C. Gohberg and M. G. Krein [2].
In the following discussion we shall
confine our attention to operators of the form
~(A).
Some extensions of
the results obtained to higher order bundles are indicated [4].
To be
specific we are interested in the spectral properties of the operator L2 (A) which for couvenience we redefine in the form 99
L(A)u • (I - L2 (>.))u • Au - ). B u - >. 2 C u
(8.2)
where A, B, C : H + H are Hermitian and continuous linear operators. The usual procedure adopted for an investigation of the spectral properties of L(A) is to reformulate the equation
(8.3)
L(A)u • 0 as an array of equations which is linear in the spectral parameter >..
This
can be done in several ways; for instance it is easy to see that the equation
(8.3) is equivalent to the pair of equations Au - >.v • 0,
= 0.
v - Bu - >.cu
(8.4)
Consequently, the spectral properties of the operator L(A) are related to the spectral properties of the operators f,7lz:HIH+HtH defined by
(-: :) (:) (: :)(:) - >.
where U • [u,v] T
€
H t H ~ u, v
€
H.
(8.5)
Further progress along this line
requires the introduction of the concept of "multiple completeness" of a system of eigenvectors, a notion which eigenvectors [u,v]T
€
H t H.
is related to the completeness of
Although this particular approach has been
adopted by certain writers, (c.£. I. C. Gohberg and M. G. Krein [2]), the problem of establishing the completeness of eigenvectors of L(A) in H together with an associated expansion theorem seems to be outstanding. the main purpose of this chapter to examine this problem.
100
It is
The essential idea is to reformulate (8.3) as a two-parameter spectral problem and then to appeal to the theory of chapter 3 in order to arrive at a a spectral theory for the operator L(A).
In the case when the operator A
appearing in (8.2) is unbounded we shall make use of the theory described in chapter 5. 8. 2
THE FUNDAMENTAL REFORMULATION
To begin with we establish some notation.
Let H denote a complex separable
Hilbert space with inner product (.,.)Hand induced norm II·IIH· denote an arbitrary Hilbert space with structure ( ·, • >tt by h .. '/( t
K
the direct sum of two copies of
defined in a natural way as follows.
where xi' yi
€
1r,
i • 1, 2.
/l.
, 11·11:/l
Let '/{, and denote
The structure on h is
Let
Then we define the inner product
and norm
With the operators A, B, C
H + H defined as in (8.2) we now consider
the system of equations
(8.6)
1.1 Au - Bu - A Cu • 0
• o. (8.7) where u
€
H, yi
€
j{ ,
i • 1, 2, A, 1.1
£
C and I denotes the identity 101
operator onjl.
It is clear that equation (8.7) has a non-trivial solution
if and only if A p
= 1.
When this is the case the equivalence of (8.6) and
(8.2) is obvious and we shall refer to (8.6) (8.7) as a non-degenerate system.
With this in mind we notice that the system (8.6) (8.7) is a
typical two-parameter system of the form studied in chapters 3-5.
In sub-
sequent sections we shall exploit.the theory for multiparameter systems in order to yield a spectral theory for the operator bundle L(A) defined by (8.2). 8.3
TWO-PARAMETER SPECTRAL THEORY
Following the pattern set in earlier chapters, we pose the problem defined by (8.6) (8.7) in the tensor product space T • H 8 h. construct operators A. : T 1
+
i
T,
c
Furthermore we can
O, 1, 2 via the determinantal expansion
of 2
r a. A. i•l 1
1
- det
ao
al
a2
-Bt
-ct
At
yt
t -Yl
-Yt
0
where y! : T y
0
=
+
(8.8)
2
T are matrix operators induced by the operators
t: :) .
y1 -
defined on h. In addition to the assumption that the operators A, B and C be Hermitian we shall need a definiteness condition, which for simplicity we take to be Al.
A0 : T
+
T is positive definite in the sense that (8.9)
for all f
102
€
T and where q is a positive constant.
This condition may be realised if we assume A2.
A, C : H + H are positive definite.
(8.10) -1
is assumed positive definite it follows that A0 : T
+ T
exists as
a bounded operator. The spectral theory now proceeds in precisely the same way as in chapter 3, and gives the main result
Theorem 8.1:
Let E(·) be the spectral measure defined as in chapter 3 §3.2
then for any f (i)
(ii)
£
T
(A0 f, f)T •
f •
t (E(~)f,
A0 f),
~ •
()., }.1)
J 0E(d~)f,
where the integral converges in the norm of T. The spectrum a of this theorem is defined as in chapter 3 §3.2. We close this section by establishing a result which will be required subsequently. Lemma 8.1:
(xi)~•l'
Given Hilbert spaces X, Y and Z with orthonormal bases
(yi)~-l and {zi}~. 1 then
(X I Y) 8 Z a!. (X 8 Z) I (Y 8 Z).
Proof:
The space X I Y has a basis {[xi' 0], [O,yj]
I
i, j • 1, 2, ••• }
and so the space (X I Y) 8 Z has a basis
On the other hand the space X 8 Z has a basis
i, k - 1, 2,
••• J
and the space Y 8 Z has a basis
j, k - 1, 2,
....
consequently (X 8 Z) I (Y 8 Z) has a basis
103
i, j, k, n
= 1,
2, ••• }.
Consider the map
[0, y.] 8 z
n
J
+
[0, y. 8 z ]. n
J
This map is 1 - 1 in the sense that· there is a 1 - 1 correspondence between orthonormal bases of (X I Y) 8 Z and (X 8 Z) t spaces are unitarily equivalent by continuity.
~~ns
(Y 8 Z).
Thus these two
of f extended by linearity and
In particular for a decomposed element [x, y] 8 z we have
f([x, y] 8 z) • f (( ~ a.[x., OJ+~ a.[O, y.]) 8 ~ Yk zk) i11 jl J k • f(~ a. yk[x., 0] 8 zk + ~ a.yk[O, yl.J 8 zk) 1 ik 1 jk J •
~a. yk[x. 8 zk, 0] + ~ a. yk[O, y. 8 zk]
ik
:1.
1
jk J
J
• [x 8 z, y 8 z] •
8.4
CONCERNING EIGENVALUES
By an eigenvalue of the two-parameter system (8.6) (8.7) we mean an ordered pair (A, p)
c2
£
f • X 8 x, X
£
for which there exists a non-trivial decomposed element
H, x
- BX - A C X
£
h such that
+ p AX • 0
(8.11)
(8.12) If (A, p) is an eigenvalue of this system then the self-adjointness of the operators A, B, C together with the condition A2. forces A and p
to be real.
Furthermore when all the operators are self-adjoint there exists a form of orthogonalit~
104
among the associated eigenvectors which is characterised by
(8.13) where f.,
i • 1, 2 are distinct eigenvectors for the system (8.11), (8.12),
1
corresponding to the eigenvalues (A., p.), 1 1
and
x. .
1J
E:
'It
i • 1, 2.
Furthermore for
i, j • 1, 2
we obtain on writing (8.13) out in full, (8.14) On using (8.12) we see that this may be rewritten in the form (8.15) In the particular case when the spectrum a consists entirely of eigenvalues it is instructive to write out in full the expansion theorem (8.1). To this end, let f • X 8 x, X
E:
H, x
E:
h be an arbitrary decomposed element
of T aDd f.1 • X.1 8 x.1 an eigeuvector of the ponding to the eigeuvalue (A., p.). 1
1
syst~
.
(8.11), (8.12) corres-
In this case, the definition of the
spectral measure E(·) yields E({A})f • -
I
(A0 f, f.)f. 1
(8.16)
1
where the summation is taken over all i for which (A., p.) •A. < A • (A, p) 1
i.e.
for which A. < A, p. < p. 1
1
1
-1
-
As a consequence of theorem 8.1 we have
the expansion (8.17) which when written in full yields (8.18) On using lemma 8.1 equation (.8.18) implies that
105
x 8 x2 • 8. 5
Ii
(A 0 f,
f.)x. 8 x. 2 • 1 1 1
THE CASE OF UNBOUNDED OPERATORS
In order to apply the theory outl.in_ed above to polynomial eigenvalue problems in ordinary differential equations we need to consider the case where A : D(A)
~
H + H is self-adjoint and possibly unbounded.
In this
situation we proceed somewhat differently by basing the theory on that of chapter 5.
To this· end we assume
A3.
A : D(A)
(i)
~
H + H is positive definite, i.e. for all u
£
D(A), (8.19)
for some positive constant c. (ii) C : H + H is positive definite. (iii) h is finite dimensional. Following the construction in chapter 5 §5.1 it is clear that 110
=P• A\i + c\~
(8.20)
is positive definite on D(At) 8 h c T. Finally we assume -1
A
A4:
is compact.
We need to consider the eigeuvalue problem A2u
i.e.
(B
• A-A0 U
\i + Cty~)U
• A PU.
(C.F. chapter 5 §5.2).
106
(8.21)
If we introduce the inner product [u,v] hull of all formal products x 1 8 x 2 , x 1 to a Hilbert space H . p
£
p
= (Pu, v) on T' (the linear)
H, x 2
£
h) we can complete T'
Furthermore P is bounded below which implies that P
has an extension (the Friedrichs extension) to a self-adjoint operator in T. P will, in the sequel,
always denote this extended operator.
remark that since P is positive definite !lull c and hence H c p
-
p
:!:
We further
cllull for some constant
T topologically and algebraically.
Also the operator
P- 1~ 2 will be a bounded symmetric operator in Hp . As in chapter 5 §5.2 we note that Lemma 8.2:
A=~
cannot be an eigenvalue and
The eigenvalues of (8.21), if they exist, must be real.
If A1 ,
A2 are two distinct eigenvalues and u 1 , u 2 the corresponding eigenvectors then
Furthermore Theorem 8.2:
we
have the fundamental result
The problem (8.21) has a set of eigenvalues A ,
n
n
= 1,
2, ••.
and corresponding eigenvectors W which are complete in H • p
n
Proof:
To show that the spectrum of the problem (8.21) consists entirely
' 1ues we argue as f o 11ows. o f e1genva
S'1nce A-l 1s ' compact and B and C are
Hermitian it follows that B, C are compact relative to A, (c.f. [1] p 184). Also because of the special forms of the matrix operators
Yo·
the fact that h is finite dimensional we see that Btyt +
cty~
relative to P.
-1
Consequently the spectrum of the operator P
entirely of eigenvalues An •
Furthermore lA I -+ 0 as n-+ n
~.
yl and y 2 and is compact
~2
consists
If
we
denote by
Wn the corresponding eigenvectors then the closed linear hull of {W }~ 1 is n n= the orthogonal complement of the eigenspace corresponding to A = 0.
This
107
eigenspace consists of all u
£
-1
Hp for which P
~ 2u
• 0 or
However since H is dense in T p
this implies
~ 2u
= 0.
~2
Again the particular form of
shows that u • 0.
Hence {W }~ 1 spans exactly the space H • This completes the proof. n n• P With the aid of Theorem 8.2 and the analysis of chapter 5 §5.3 we obtain the expansion theorem Theorem 8. 3:
The system (8.6), (8.7) has a set of eigenvalues
and a corresponding set of eigenvectors
+r
£
H,
~
r
(~
r
,
~)
~
r r=l
h such that
£
8 ~ }~ 1 is a complete orthonormal system in H • r r• p
{f
r
As a corollary we have Corollary 8.1: X
Let X 8 x
£
Hp be a separable element with llxllh •1, then
D(A) can be expanded as
£
(8.22) where a nr • (X 8 x, P(fr 8 ~
AN APPLICATION
to
~ ))(~
r
(8.23)
r , x)h.
ORDINARY DIFFERENTIAL EQUATIONS
Consider the two point boundary value problem defined by 2
Ly • ~ + q(x)y - A&(x)y - ~ 2b(x)y • 0, dx 2 x
£
(0,1),
y(O) • y(l) • 0.
For this problem we take H • L2 (0,l) and define as follows A : D(A) c H + H is defined as A=
-d2
2
dx
108
+
q(x),
(8.24)
linear operators A, B, C
D(A)
{u, u1 E ACLoc[O,l],
m
The operators B, C continuous functions
Au
E
L2 (0,1),
H + H denote respectively multiplication by the real a(x), b(x).
If we assume q(x) > 0 for all x
E
[0,1] then A is positive definite with
compact resolvent, and if b(x) > 0 for all x positive definite.
u(O) = u(l) • 0}.
E
[0,1] then C : H + H is
For h we take the Euclidean 2-space E2 with the usual
norm and inner product. With these conditions (8.24) is reformulated as 2
d u1 l.l (- - 2 + q(x)u1 ) - a(x)u 1 dx
~b(x)u 1
(8.25a)
= 0,
and (8.25b)
The eigenfunction expansion theorem is then given in terms of the tensor product of the eigenvectors of (8.25a,b) respectively via (8.23). Notes:
The theory developed in this section is based on the theory contained
in [4, 5] wherein the possibility of extending the ideas to polynomial bundles of order higher than the second is considered.
The proof of lemma
8.1 is due toP. J. Browne. References 1
F. V. Atkinson,
Multiparameter eigenvalue problems Vol. 1. Matrices and compact operators. Academic Press, New York (1972).
2
I.C.Gohberg and M.G. Krein, Introduction to the theory of linear nonself-adjoint operators. A.M.S. Translations of Mathe Monographs Vol. 15. Providence, Rhode Island (1969). 109
3
M. V. Keldys,
4
G. F. Roach and B. D. Sleeman, On the spectral theory of operator bundles. Applicable Analysis 7 1-14 (1977).
5.
G. F. Roach and B. D. Sleeman, On the spectral theory of operator bundles II. Applicable Analysis (to appear).
110
On the characteristic values and characteristic functions of certain classes .of non-selfadjoint equations. Dokl. Akad. Nauk. SSR 77, 11-14 (1951)
9 Open problems
9.1
SOLVABILITY OF LINEAR OPERATOR SYSTEMS
The development of multiparameter spectral theory covered in this book, particularly chapters 3, 4 and 5, relies heavily on the solvability of a system of linear operator equations.
This question was considered in
section 2.5 of chapter 2 wherein it was shown that if S
= det{s!.}, 1J
(see
2.20), considered as an operator in the tensor product space H was positive
definite then the system
s!.u ... f., 1J J
i • 1, ••. , n,
1
(9.1)
has a unique Cramer's rule solution for any set of given vectors •••• f
E
n
H.
Thus in order to extend and generalise multiparameter
spectral theory it is of fundamental importance to investigate the possibility of solving (9.1) under somewhat weaker hypotheses.
In pursuing this question
P. J. Browne and the author [6] have proved the following:
Theorem 9.1: (i) (ii)
Let
S be densely invertible and injective.
The given vectors f 1 , f 2 , ••• , fn satisfy the condition n
At ~ S.1n f.1
E
R(S)
(the range of S).
i=l
....
If S is positive definite in H then the system (9.1) has a unique solution. nn In this and the following results we have used the nomenclature adopted in chapter 2.
111
Let G, assumed nonempty, be the set defined by G - {(fl .... t f n )
I f.1
E
H
i = 1, .... n. skjfk
E
R(S),
k,j=l ..... n}
c H x ••• x H (n factors).
Theorem 9.2: ••• t
f
n
(9.2)
Let S be densely invertible and injective and let E
G.
(9.3)
k = 1, ••• • n,
then the system (9.1) has a unique solution. Both these results, like Theorem 2.9, give rise to some unexpected commutativity relations enjoyed by the elements of the determinant S. To summarize, we know that Theorem 2.9 holds for all f 1 , ...• fn the commutativity relations (9.3) may be deduced as a corollary.
E
Hand
(See [7].)
In Theorem 9.1 the assumption that
I ~:
f.
i•l 1n 1
E
R(S)
is a necessary condition for solvability.
,.,
That S
nn
is required to be positive
definite comes about in view of the method of solution and could possibly be relaxed.
For example if S is positive and so having zero as a point of
its continuous spectrum we can argue as in lemma 2.1 to arrange that
stnn
is also positive and this may be sufficient to establish solvability.
. we d1spense . . the requ1rement . Theorem 9.2 shows that 1f W1th that ,.,t S be nn
positive definite .then we must assume the commutativity relations (9.3). 9. 2
MULTIPARAMETER SPECTRAL THEORY FOR BOUNDED OPERATORS
In chapter 3 a fairly comprehensive account of multiparameter spectral theory for 112
~ermitian
and continuous operators was given under the main
hypothesis that the operator A defined by (3.1) on decomposed elements induces a positive definite operator on H.
(See 3.2.)
It is known that if
an operator, defined on the linear hull of H, is positive definite then its extension to the whole of H need not preserve this property. being the case when H is finite dimensional.
The exception
It is therefore of interest to
study conditions under which the positive definiteness property of an operator is retained when the operator, defined on the linear hull of H, is extended.
Some results in this direction have recently been given by
Binding and Browne [3]. If the condition that A be positive definite is relaxed to the requirement that A be positive and so having zero as a point of its continuous spectrum then a spectral theory may also be developed. the theory is less complete than that of chapter 3. the case a 0 = 1, ai • 0,
i
1, ••• , n is studied.
However, as expected
See Browne [4] wherein As a further extension
of the theory we mention that Browne [5] has also developedatheory under the assumption that A. is Hermitian and the operators S •• : H., i,j 1
i, j
= 1,
1J
L
=
••• , n are self adjoint and pairwise commutative in the sense of
having commuting spectral resolutions.
9.3 MULTIPARAMETER SPECTRAL THEORY FOR UNBOUNDED OPERATORS If the operators S •• A., i, j LJ 1
§4.1 of chapter 4 then with S
= l, ••• ,n
enjoy the properties set out in
det{S •• } assumed positive definite the
spectral theory is well developed.
LJ
However if S is not positive definite
then alternative methods seem called for.
For example if the
s .. satisfy LJ
the assumption 1 of chapter 5 then with certain extra conditions one can obtain a Parseval equality and expansion theorem.
One of these extra
conditions is a certain "compactness" requirement (see §5.2) which ought to be relaxed.
In attempting to do this the author in [8] has developed an 113
an alternative theory which certainly obviates the need of the compactness condition but at the same time retains some of the weaknesses of the theory outlined in Browne [4].
Nevertheless the theory in [8J is sufficiently
general as to claUn some unification of the theories developed here. general one cannot
In
expect to relax all structural conditions in multi-
parameter spectral theory; for even in the one parameter case of the operator (T + -1
V
~V)
compatibility conditions on the ranges and domains of T, V,
etc are necessary. It should be clear by now that much remains to be explored in multi-
parameter spectral theory.
We have spectral theorems and associated Parseval
equalities; however the nature of the spectrum is far from understood. aspect is particularly important
in
This
relation to multiparameter spectral
problems for differential equations.
9.4
THE ABSTRACT RELATION
The problem treated in chapter 6 may appear to be an isolated result in multiparameter spectral theory, however in view of its application to integral equations satisfied by certain special functions of mathematical physics, which in themselves arise as solutions to multiparameter eigenvalue problems for ordinary differential, it is believed that Theorem a completely new approach to multiparameter spectral theory. be seen how far this approach can be taken.
6~1
suggests
It remains to
For example Theorem 6.1 may
form the basis of an extended Fredholm theory in the multiparameter case analogous to that well known in the one parameter case. 9.5
APPLICATIONS
It was pointed out in chapter 2 that multiparameter spectral theory receives its main motivation from the study of systems of ordinary differential 114
equations arising from the separation of variables technique applied to boundary value problems for partial differential equations.
The topics
treated in chapters 7 and 8 provide examples which do not come from this source.
Indeed treating coupled systems of operator equations and poly-
nomial bundles as special cases of the multiparameter structure may give new insights into these problems and at the same time suggest new areas of investigation. To conclude we mention that a variational approach to multiparameter spectral theory has been initiated by Binding and
Bro~e
[1, 2].
Here the
theory has been outlined in the case of multiparameter eigenvalue problems for matrices, (i.e. the H. are finite dimensional) and to abstract multi1
parameter eigenvalue problems in infinite dimensional Hilbert spaces. References 1
P. Binding and P. J. Browne, A variational approach to multiparameter eigenvalue problems for matrices. S.I.A.M. J. Math. Anal 8 (1977) 763-777.
2
P.Binding and P. J. Browne, A variational approach to multiparameter eigenvalue problems in Hilbert space. (submitted)
3
P. Binding and P. J. Browne, Positivity results for determinatal operators. Proc. Roy. Soc. Edin. (a) (to appear)
4
P. J. Browne,
Abstract multiparameter theory II. Anal. Applies. 60 (1977) 274-279.
5
P. J. Browne,
Abstract multiparameter theory III. (submitted)
6
P. J. Browne and B. D. Sleeman, Solvability of a linear operator system II. (submitted)
7
A.KHllstrBm and B. D. Sleeman, Multiparameter spectral theory. fBr Matematik 15 (1977) 93-99.
8
B. D. Sleeman,
J. Math.
Arkiv.
Multiparameter spectral theory in Hilbert space. J. Math. Anal. Applies. (to appear).
115
Index
Adjoint 31, 43 Anselone, P. M. 92, 98 Arscott, F. M. 2, 3, 7, 81, 90 Atkinson, F. v. 6, 7, 31, 41, 46, 59, 60, 77, 80 Basis - Orthonormal 12 Binding, P. 113, 115 B3cher, M. 7 Borel Measure 33 Borel Measurable Function 16, 17 Borel Set 16, 17, 33, 43, 44, 50, 51 Boundary Conditions 3 Boundary Value Problems 1 Browne, P. J. 41, 60, 61, 111, 113, 114, 115 Camp, C. C. 6, 7 Carmichael, R. D. 6 7, 8 Cartesian Product 9, 32, 43, 52, 89 Cauchy Sequence 69 Circular Membr~ne, Vibrations of 1 Cofactor 20, 23 Commutativity 32 Compact - Relative 65 Compact - Resolvent 65 116
Faierman, M. 3, 8, 60, 61 Fourier Coefficient 68 Fredholm Theory 115 Friedrich's Extension 64, 107
2
Mathieu 2 Paraboloidal Wave Special 115 Spheroidal Wave Gohberg, I. C.
88, 90
Inequality - Bessel 69 Integrals - Dirichlet 76, 77 Integral Equations 81, 115 Klllstr8m, A.
27, 41, 80, 115.
Keldy• M~ V. 99, 110 Kernel 31, 35, 52, 56, 57 Klein, F. 6, 8 Krein, M. G. 99, 100, 109 Kronecker Delta 20 Left Definite Problems 45 Linear Operator System 20 Solvability of 18, 30, 111
2, 8
Functions: Bessel 1 Ellipsoidal Wave :r.ame 2, 89
Ince, E. L.
92, 100
2 2
99, 109
Halmos, P. 26, 27 Hilbert, D. 3, 4, 6, 9
Manifold - Linear 14 Matrices 6 Matrix Equation 91 Mikhlin, S. G. 64, 80 Multiple Completeness 100 Murray, F. J. 26, 27 Operator: Abstract Linear 6 Bounded Linear 14, 112 Bundles 99 Compact 37, 55, 65, 107 Commuting Self-adjoint 16, 32, 43, 46, 47, 51 Hermitian 28, 32, 47, 87, 99, 112 Induced 15, 30, 63 Linear 1, 6 117
Rectangular Membrane Vibrations of 1 Rellich, F. 51 Resolution of the identity
Von Neumann, .J. 15, 33,
42, 50, 51, 52, 98 Richardson, R. D. 6, 8
Riesz, F. 61 Right Definite Problema 4, Roach, G. F. 98, 110 Schatten, R. 26, 27 Self-adjoint: Differential Expression Extension 38 Separation Constants 1 Separation of Variables 1, Spectral: Measure 16, 33, .Parameters 1, 6 Theory 1, 4
118
Algebraic 9, 15, 83 Hilbert 11 of Linear Operators 14 Space 11
