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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
<
1T,
(1. 2)
dy (1) r cos e y (0) - sine r r r dx r
0,
o<
e
r
s
1r,
r = 1, .•. , n.
If
~
can be so chosen, then it is called an eigenvalue of the system (1.1)
(1.2); if {y (x ,A)}n_ 1 is a corresponding set of simultaneous solutions of r r rn IT y (x ,A) is called an eigenfunction of this r=l r r system corresponding to the eigenvalue ~·
(1.1) (1.2) then the product
Much of the early work regarding the system (1.1) (1.2) was concerned with certain extensions of the sturmian oscillation theory. [5, 16, 17, 19].
See for example,
More recent contributions in this direction are due to
"' Neuman and Arscott and Sleeman [13, 14, 20-22]. Faierman, Gregus,
The multi-
parameter eigenvalue problem did not escape the attention of Hilbert [15] who made the first contribution to the question of completeness of eigenfunctions. As regards spectral theory and in particular questions related to completeness of eigenfunctions, the Parseval equality and the like, some further structure must be added to the system (1.1) (1.2).
It is clear from the
formulation that since the eigenfunctions are considered as products of solutions of each separate equation which in turn may be
thoug~of
as being
generated by self-adjoint differential expressions in the Hilbert space L 2 (0,l), an appropriate setting for the spectral theory is some tensor product of n- copies of L2 (0,1). more is needed.
However, in order to progress, something
To see how this comes about we first recall some fundamental
notions related to the one-parameter case.
Here we have the classical
Sturm-Liouville problem defined by 3
d2
- ~ + q(x)y
= Ap(x)y,
(1.3)
dx 0 s x s 1, with p(x), q(x) continuous and real valued functions defined on the interval 0 s x s 1, and we seek solutions satisfying the homogeneous conditions cos ay(O) - sin a
dy~~) • 0,
0 s a < 1r, (1.4)
. dy(l) cos ay(l) - 81n a dx =
o.
o
In order to treat the problem (1.3) (1.4), particularly as regards questions of completeness of eigenfunctions and the development of a spectral theory, it is desirable to interpret it in terms of linear operators in Hilbert space. ways.
Such a Hilbert space structure may be realised in one or two
Firstly, if we assume p(x) is positive on [0,1] then we take our
Hilbert space as L2[0,lJ. p
With this condition on the coefficient p(x) we
are led to the study of what may be conveniently termed "right definite" problems for (1.3) (1.4).
On the other hand if p(x) is not identically
zero and is allowed to change sign in [0,1] then clearly a different Hilbert space setting is called for. simplicity, suppose a
€
Suppose q(x) is positive on [0,1] and for
(0,1f/2], a
€
[1f/2,1f) then a positive definite
Dirichlet integral may be associated with (1.3) (1.4) and a spectral theory may be developed in the Hilbert space which is the completion of
c1[0,l]
with respect to the inner product (u,v) •
JO (dxdu dxdv + q(x)uv)dx + cot auV(O) l ·
cot auv(l)
(1.5)
This leads to the study of what may be appropriately called "left definite" problems.
4
For the multiparameter eigenvalue problem (1,1) (1,2) the appropriate generalisations of the above assumptions turn out to be ~
(A)
n
• det{a
}n > 0 rs r,s• 1
for all x•(x 1 , ••• xn)
€
(1,6)
In (the cartesian product of then intervals
r • 1, .•• ,n),
~
(B)
n
Ill
~
0 and
......
a21
IJn a2n
8
>
o.
11 ••••••••••
8
......
> 0, a
a nl
a
1n
nn
lll
a
r-l,n
. • .. • • . • • • lin
r+l,l ••••••• ar+l,n
......... a nn
(1. 7)
......... > 0
J.ll • • • • • • • • • • • 11n
for some non-trivial n-tuple of real numbers p 1 , ••• ,1Jn• the inequalities holding for all x • (x 1 , ••• ,xn)
€
In.
Problems defined by (1.1) (1.2) and
condition (A) will be called "right definite" multiparameter eigenvalue problems whereas problems defined by (1.1) (1.2) and condition (B) will be termed "left definite" problems.
s
The conditions (A) and (B) have an extention to analagous conditions for abstract linear operators in Hilbert space and these extended conditions are crucial to the theories developed in chapters 3, 4 and 5 of this work. Away from the immediate area of Sturm-Liouville problems containing several spectral parameters we call attention to the work of R. D. Carmichael [8-lOJ who, among other things, suggested a method of attack on multiparameter spectral theory for matrices an area to be investigated much later by F.V. Atkinson.
We shall return to this in .a moment.
In 1922 A. J. Pell
[18] studied pairs of Fredholm integral equations, coupled by a pair of parameters.
This work was followed a few years later by studies of multi-
parameter problems for first order partial differential equations. example the work of
c.
See for
C. Camp [6, 7] and H. P. Doole [11].
Since the 1930's, apart from the continued interest in the special functions mentioned earlier, multiparameter theory remained somewhat neglected until the early 1960's when F. V. Atkinson took up the multiparameter matrix case suggested by R. D. Carmichael.
Atkinson's work began in 1964 with the
report [2] and culminated in the book [4].
The work in [41 is a comprehen-
sive treatment of multiparameter spectral theory in finite dimensional spaces and includes suggestions for proceeding to the infinite dimensional case. The approach developed in this book is somewhat different to that of Atkinson in that it provides an introduction to the infinite dimensional case via the theory of several commuting operators in Hilbert space.
In this sense
it is hoped that this work will form a companion to Atkinson's book.
It is
appropriate to call attention to the excellent survey article of Atkinson (3] which appeared in 1968.
In this paper we are led through the many
ramifications of multiparameter spectral theory, beginning with the early work of Klein, Richardson etc. on differential equations through multi6
parameter spectral problems for arrays of linear operators to multiparameter problems embedded in a modern algebraic setting.
Further more this article
contains a comprehensive bibliography which includes most of the important references to the literature prior to 1968. This book therefore concentrates on a small but significant portion of multiparameter spectral theory and hopefully illustrates the variety of problems that arise and the richness of this yet to be fully explored field of study. References 1
F. M. Arscott,
Periodic differential equations. Oxford 1964.
Pergamon Press:
2
F. V. AtKinson,
Multivariate spectral theory: the linked eigenvalue problem for matrices. Technical Summary Report No 431, U.S. Army Mathematics Research .Center, Madison, Wisconsin 1964.
Multiparameter eigenvalue problems. Matrices and compact operators. Academic Press: New York and London 1972.
5
M. B8cher,
The theorems of oscillation of Sturm and Klein I. Bull. Amer. Math. Soc 4 (1897-1898), 295-313; II ibid 365-376; III ibid 5 (1898-1899) 22-43.
6
C. C. Camp,
An expansion involving P inseparable parameters associated with a partial differential equation, Amer. J. Math. 50 (1928) 259-268,
7
C. C. Camp,
On multiparameter expansions associated with a differential system and auxiliary conditions at several points in each variable, Amer. J. Math 60 (1930) 447-452.
8
R. D. Carmichael,
Boundary value and expansion problems; Algebraic basis of the theory. Amer. J, Math, 43 (1921) 69-101.
7
R. D. Carmichael,
Boundary value and expansion problems. Formulation of various transcendental problems. Amer. J. Math. 43 (1921) 232-270.
10 R. D. Carmichael,
Boundary value and expansion problems. Oscillatory, comparison and expansion problems. Amer. J. Math. 44 (1922) 129-152.
11 H. P. Doole,
A certain multiparameter expansion. Math. Soc. 37 (1931) 439-446.
12
Higher transcendental functions Vola I, II, III. McGraw-Hill, New York 1953.
9
A. Erdelyi et al,
13 M. Faierman,
Bull. Amer.
Boundary value problems in differential equations. Ph.D. Thesis. Toronto 1966.
14 M. Greg~s. F. Neumann and F. M. Arscott, Three point boundary value problems in differential equations. J. Lond. Math. Soc. 3 (1971) 429-436. 15
D. Hilbert,
GrundzUge einer allgemeinen theorie der linearen integralgleichungen. Berlin 1912.
16
E. L. Ince,
Ordinary differential.
Dover, New York 1944,
17 F. Klein,
Bemerkungen zur theorie der linearen dif·ferentialgleichungen zweiter ordnung. Math. Ann. 64 (1907) 175-196.
18 Anna. J. Pell,
Linear equations with two parameters. Amer. Math. Soc. 23 (1922) 198-211.
19
R. G. D. Richardson,
Theorems of oscillation for two linear differential equations of the second order with two parameters. Trans. Amer. Math. Soc. 13 (1912) 22-34.
20
B. D. Sleeman,
Multiparameter eigenvalue problems in ordinary differential equations. Bul. Inst. Politechn. lasi 17 (21 (1971) 51-60.
21
J. D. Sleeman,
The two parameter Sturm-Liouville problem for ordinary differential equations, Proc. Roy. Soc. Edin. A 69 (1971) 139-148.
22
B. D. Sleeman,
The two parameter Sturm-Liouville problem for ordinary differential equations II. Proc. Amer. Math. Soc, 34 (1972) 165-170.
8
Trans.
2 Tensor products of Hilbert spaces
2.1
THE ALGEBRAIC TENSOR
~RODUCT
Denote by H1 x ••• x Hn the Cartesian product of n Hilbert spaces H1 , ... ,Hn and introduce in H1 x ••• x Hn equivalence classes by the indentification (ah(l)) x h( 2 ) x .•. x h(n) = h(l) x (ah( 2 ))x ••. x h(n) = h(l)
for all complex numbers a and all h
(1)
X
h( 2 )
x ..• x h
X ••• X
(n)
(ah(n)),
£ H1 x •.• x Hn.
(2.1) Let
(H 1 x ... x Hn)~ denote the family of all such equivalence classes and · 1 ence c 1ass conta1n1ng · · h(l) x .•. x h(n) . h (l) 8 ... 8 h(n) the equ1va
a
Next define in (H 1 x ••• x Hn) c
£
t
~
£
a:, Ia I >
Th us
0, k = 1, •.• , n}.
the operation of multiplication by a scalar
as (2.2)
If 8k is the zero element of Hk it follows that ch(l) 8 •.• 8 ek 8 •.• 8 h(n)- h(l) 8 •.• 8 ek 8 ••• 8 h(n) for all c £ C, that is (2.3) for all h(i) £H., i • l, ••. ,n. 1
element of (Hl
X ••• X
The element (2.3) will be called the zero
Hn)~ and be denoted by 8.
9
Consider now the family of all m-tuples (f 1 , ••• , fm) of elements fj
E
(H 1 x ••• x Hn)~, j • 1, ••• , m and introduce in this family equival-
ence classes by the following identifications. (i)
(2.4)
(fl' .... f ) - (fk ..... fk ) m 1 m
for any permutation fk , 1
is arbitrary. (ii)
for any f 1 ,
••• J
••• J
f m,
e> =
f
(H 1
x ••• - }(
m
E
(iii) (h(l) 8 ... 8
h~k)
... , f
m>
(2.5)
Hn)~, and
8 ... 8 h(n), h(l) 8 ... 8
... , fm) • (h(l) 8 ... 8 (h~k) + h~k)) 8 any f 1 , ... , fm' h Denote by H1 8
(1)
8 .. , 8 h
(n) E
(Hl
h~k)
8 .. ·• 8 h(n),
8 h(n), f 1 , ... , fm)' ~
X
.. ,
X
Hn) ,
H the family of all such equivalence classes. n introduce in (H 1 x ••• x Hn)~ the operations a
••• 8
a
(2.6) If we
(2. )
for all p-tuples fi' gj'
j • 1, ••• , q
E
i - 1,
••• J
cf)
••• J
p, and q-tuples
q
C
E
C
(2.8)
(H 1 x ••• x Hn)~ then it can be shown that (2.7) (2.8)
leave the equivalence classes of
a1
... 8 H invariant and that, they a a n satisfy all the usual axioms on vector operations. 8
All the above constructions may be summarised in the following theorem; Theorem 2.1: · The set a 1 8 a ... 8 a Hn of equivalence classes constructed from elements of the Cartesian product a 1 x ••• x Hn of Hilbert spaces
a1 , •••,Hn according to the equivalence relations (2.1) (2.2),(2.4)-(2.6) is
10
a vector space (called the algebraic tensor Eroduct of H1 , ..• ,Hn) under the operations (2.7) (2.8) and having as a zero element the equivalence class containing (9), where ~
e
is the zero element of (H 1
X
...
X
"'
H ) • n
HILBERT TENSOR PRODUCT OF HILBERT SPACES
Let H1 ... ,Hn be Hilbert spaces with inner products (.,.) 1 ,
(
respectively.
8
In the algebraic tensor product space H1 8a
... ) n a
H
n
of
H1 , ... ,Hn denote by(·,·) the inner product, having the value n
(f,g) •
II
k•l
( f(k) ,g (k)) k
for all f,g e H1 8a
8
a
(2.9)
H of the form n g = g
(1) ,.. v
•••
,.. v
g
(n)
.
The relation (2.9) defines a unique inner product and we define the Hilbert tensor product space (or simply the tensor product space) H1 8 •.• 8 Hn of the Hilbert spaces H1 , ... ,Hn to be the completion of H1 8a ..• 8a Hn with respect to the inner product (2.9). Since, in this book, we shall be dealing exclusively with Hilbert spaces we shall drop the terminology Hilbert tensor product and simply refer to the space constructed above as the tensor product space. Factorising elements in H1 8 ••• 8 Hn Let f = f(l) 8 ••• 8 f(n) e H 8 a··· 8 a Hn and g(n) e Hn. 1
Define a mapping
by
and extend this definition to H1 8a ••• 8a Hn by linearity.
If we denote
11
this mapping by f
(f,g(n))n an
+
for separable elements f. is the norm in H1
e8
•••
easy calculation shows that
The norm on the left hand side of this inequality
e8
Hn·l·
By introducing complete orthonormal sets
in Hl' ••• ,Hn we find that for--the mapping extended to H1 ea ... ea Hn (2.9) is still valid. It follows that the mapping f + (f,g(n)) is bounded in n
H1
e8
•••
e8
Hn and so can be extended to H1
e ... e
Hn by continuity so
that (2.10) still holds. In a similar way we may define
and correspondingly for more factors.
The analogue of Fubini's theorem is
true, i.e. (2.11)
On occasion we shall make use of the mapping
f
+ (( ••• (f,g
denoted by (f,g
(1)
(1)
)lg
(2)
e ... e
g
)2 ••• )n-2' 1 (n-1)
(n-1)
)A or (f,g n
>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
<
(2.20)
co,
:Rn
We usually denote A by F(A 1 , ••• ,An). Theorem 2.8:
If A • F(A 1 , ••• , An) in D(A) is dense in H, then
A* • F*(A1, ••• ,An) where
_F~(~)
is the complex congugate
ofF(~).
We note that in general D(A) defined by (2.20) is not dense in H. if D(A) is dense and
F(~)
However
is real then it follows from theorem (2.8) that
F(A1 , ••• , An) is self-adjoint. 2.5
SOLVABILITY OF A LINEAR OPERATOR SYSTEM.
Throughout this book a key role is played by the solution of certain abstract systems of linear operator equations defined on tensor product spaces.
In this section we investigate the existence of solutions to such
systems and derive a form for the solution.
The system to be treated is
formulated in the following way; Let (i)
S •• : H. +H., j 1J
1
1
= 1,
••• J
n be bounded symmetric operators
defined on the separable Hilbert space H., i 1
(ii)
= 1,
••• , n;
H • H1 8 ••• 8 Hn be the tensor product of the spaces H., i • 1, ••• , 1
Every operatorS .. in H. induces a corresponding operators!. in H defined ~
~
1
as in (2.13) first on separable elements and then extended by linearity and continuity.
It is readily verified that s!. is bounded and symmetric on 1J
H and
II s!.ull 1J where II
·II
II s 1J .. ll.llull 1
s
denotes the tensor product norm in H and II S .. 11 • is the opera tor 1J
norm of S .. in H.• 1J
18
(2.21)
1
1
~
Since s .. and skl operate in different spaces when i ~ k the corresponding 1J t t operator s •. and skl in H will commute for all choices of j and 1, 1J 1 s j, 1 s n.
det
s Let f
Thus it is possible to define in a unique way the determinant
.... 1.... "'1oJ"'n
= f1
(S t.. ), as follows.
(2.22)
1J
8 ••• 8
fn be a decomposed element of H then Sf is defined by
the equation Sf •
snn f n where the determinant is to be expanded formally using the tensor product, i.e.
where a runs through all permutations of { 1,2, ••• , n} and €a is +1 or -1 according as a is even or odd.
This defines Sf for decomposable f
we extend the definition to arbitrary f
£
£
H and
H by linearity and continuity.
Clearly (2.22) defines a bounded symmetric operator on H. Throughout this section we make the basic hypothesis Hypothesis 2.1:
S is a positive definite operator in H, i.e. there is a
constant C > 0 such that ( Su, u)
i!:
C II u 11 2 •
(2.23)
19
This hypothesis implies in particular that S has a bounded inverse defined on H. For the remainder of this section all operators, unless otherwise stated, will be considered as acting in H and the t-notation will be omitted. By expanding det(S .• ) we see that S can be expressed in the form 1J
where Sik is the ~ofactor 1 of Sik defined in the usual way.
Similarly we
obtain n
Is1..
i•l
J
j,k • 1, ••. , n, where
s1.k .. s6.k,
is the Kronecker-delta. for
j,k
(2.24)
J
a
Furthermore we note that S •• commutes with 1J
1, ••• , n since ~ik contains noaements from the i-th rowS.
Consider now the linear system n
I
S .. u. • f., . 1 1J J 1 J•
(2.25)
i • 1, ••• , n
where f 1 , ••• , fn are given vectoEBin H. Theorem 2.9:
We prove the following result.
Under the hypothesis (2.1) the linear operator system (2.25)
has a unique solution given by Cramer's rule. That is
k
~:
••• , n.
(2.26)
First of all we observe that if the systems (2.25) has a solution at
all it must be unique.
20
= 1,
For if we apply Sik to the i-th equation in (2.25)
and then sum over i, we find that
wbich by (2.24) reduces to
or k • 1, ••• , n.
This proves the uniqueness of the solution and also the form of solution. In order to establish existence of the solution it is natural to insert the ~·
k • 1, ••• , n defined by (2.26) into the system (2.25) and verify that
they are solutions.
This however leads to sums of the form
n -1~ r sij s l)kj j•l
which cannot be reduced to simpler terms unless some commutativity conditions are imposed on the operators S •• , 1J
ikJ..
To avoid this we establish
existence of a solution by an inductive argument.
To this end we require
the following lemma Lemma 2.1:
If S • det(S •• ) is positive definite on H there exists a linear 1J
combination of cofactors n'{'
t. k•l
,..
!1..
k
SJ.k
for some j • 1, ••• , n
which is positive definite on H.
21
Proof of the lemma. There is a '
n
n n 0 in Hn such that at least one of (S n k' · ,, ) n , k = 1, ••• , n
~
is non-zero for otherwise we get a contradiction to the assumption that S is positive definit~ Define ak • (Snk'n,,n)n• without loss of generality that a
n
~
k = 1, ••• , n
and assume
Now consider the operator
0.
determinant T acting on
H-
n-1 8
H
i-=1
and defined by 8 11
•••••••••••• 8 ln
sn-1,1
s n-l,n an
Let u
£
H then
(Tu,u),.. n • (Su
(2.23) and Where u
8'
n
8' , 8' ) n
u
n
0!:
ell'nn 11 2 11ull~n
by assumption
is regarded as an element of
n H •
8
H ••
i=l
1
" and so its induced operator on H is also Hence T is positive definite on H positive definite. Returning to the existence part of the proof of theorem 2.9 we will first show that it can always be arranged that g ·
a 1 , ••• , an be chosen as in lemma 2.1.
22
nn
is positive definite.
Make the substitution
Let
ul
-
0
................. 1 .................
0
0
1
- a2 a n
- ....!!:.! a
1
0
al a n
u n
0
vl
0 0
a
1
v
n
n
Equation (2.25) is then transformed into a2
al
su -a- s ln n
8 12
-a- s ln n
8 ln
vl
fl
snn This new system has the same determinant as (2.25).
v
n
f
n
Furthermore the cofactor
of Snn is
a. 1 n det (S •• - ~ S. ) • Q: ak S k lSi,jsn-1 1 J an 1 n n k•l n
r
which by lemma 2.1 is positive definite (if a a n < 0).
n
> 0, negative definite if
Thus there is no restriction in assuming that
Snn
is positive
definite in (2.25). Assume now, for the purposes of induction; that every (n - 1) x (n - 1) system with a positive definite determinant is solvable and write (2.25) as n-1
r
j•l
s .• u.- f. - sinun' 1J
l
i . 1, •••• n- 1
(2.27)
1
and (2.28)
23
From the preceding discussion we know that the system (2.27) which has the determinant § , has the solution nn
s-nn1 rnil i "k(f.1 . 1 nn,1
uk -
s.1nun
L1•
>]
(2.29)
k•l, ••• ,n-1
,.. where §nn.ik is the cofactor ..of Sik in §nn"
Now if we can determine un in
(2.29) so that (2.28) is satisfied then we have a solution to the system (2. 25). Substituting
from (2.29) in (2.28) we get
for~
,._ 1 [ n-1 ~
n-1
r k•l
r
s s
nk nn
s
.
8 nk
g-1 nn
i•l
nn,1k
s.1nun ) ]
(f. 1
+
snnun
= f
n
which gives n-1 n-1 (S
r r
nn
k•l i•l
i
nn,ik
s.1n )un
-
(2.30)
F n
where n-1 n-1
F
n
• f
But Snk and
~
1 r r s k s- s .kf .• k•l i•l n nn nn,1 1
n ~-1
~nn
(2.31)
,..
commute, since Snk and Snn commute, and hence the double sum
in (2.30) can be rewritten as n-1 n-1
r r k•l i•l
1
S-nn
~
Snk S "k S. • nn,1 1n
n~l ,._ 1 ( n~l L
i~l
S nn
S k=l nk L
,..
Snn,1"k
)
s 1n ..
(2.32)
,.. But Snk commuteswith every element in Snn and so the summation over k in (2.32) results in the determinant
24
..
su
51 n-1
snl
sn,n-1
sn-1,1
sn-l,n
.
s l,n-1 ( row number 1.) • ( -1) n-l+i
sn,n--1 (with row sil = -
S. deleted) 1,n- 1
... s1n ..
Consequently equation (2.30) reduces to n-1
+
(S
nn
S-l l S. nn i=l 1n
S. )u 1n n
•
F n
that is .....-1
S
(
nn
n
l S.
i=l 1n
S. )u
1n n
= F ,
n
which on using (2.24) further reduces to .....-1
S S u • F nn n n
that is u
n
=
s- 1
snn Fn •
(2.33)
In a similar way the expression (2.31) reduces to
F
n
..
r
n"" s. f. nn i•l 1n 1
s-1
and so from (2.31) we obtain the result
un -
s -1
r s. n ""
i•l 1n
f .•
1
25
=1
In the case n
the solvability of (2.25) is obvious.
In the above
argument we have used second minors which means that the induction step is only valid when n
~
3.
To complete the induction proof it remains to
establish solvability when n
= 2.
This problem is considered under somewhat
weaker.hypotheses in Halmos [l pp 55-57J.
A proof can however be given
along the same lines as above. Consider the system
~2lul + 8 22u2 - f2
with
s 11 (= ~22 >
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
E
II fmll
H,
E
a. Then
• 1 such that
Such a sequence may be constructed, for example, by
forming intervals I~ • [A. - 1/n, A. + 1/n] and selecting fn from 1 1 1 E (Im1 ) ••• E (Im). 'l'hen since n
t
-Ai r O +
n
rn St•• r . j•l 1J J
n
•
0
and
r a. r. j•O 1
1
•
I,
we see that i • 1, ••• , n
and
37
n
r ai Ai j•O
"' 1 •
Hence 0 is in the spectrum of each s!(A) and since the spectra of S.(A) and 1 1 si(A) coincide, we see that 0 is in the spectrum of each Si(A). Now assume that 0 is in the spectrum of each Si(A). sequences fm1•
H1., 11~11. ""lauch that S.(A)f~+ 0.
t:
1
1
1
1
Then we can find Set fm
= fm
8 ... 8 fm. n
Consider A Afm 1
=8
- - Alal
ao -Al£7
ao
- 8
-Al£7
-A
£"1
nn
Alsn£7
............ ............
an
sln~
n a r A.a. ............ n j•O J J m s 1 (A)f~ ............ slnfl
sn (A)fan
from which we see that
and so (r1 - A1 )fm + 0.
Similar reasoning shows that the sequence ~has
the property that II~~~- 1 and Consequently A Theorem 3.8:
t:
A.)~+ 0 for each i • 0, 1, ... ,n. 1
a.
If each of the operators Ai
following situations prevail 38
(r.1 -
Hi+ Hi is compact and any of the
o.
a 0 • 1, a.1 •
(i)
(ii)
a0 •
(iii)
).0 •
o. o.
i - 1.
n and ).
••• J
and not all a. = o. i • 1, 1 and at least one >. •• 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
w!(Bn).
Now consider the array of
operators
(4.2)
43
This array may be used to define operators A.(B), J
precisely the same manner as the operators A., J
above.
j • 0,1, ••• , n in
j • 0,1, ••• , n are defined
The operators A.(B) are Hermitian and defined on the whole of H. J
= A0
Furthermore A0 (B)
and for all fED,
Aj(B)f + Ajf as B +Rn.
also satisfies all the conditions using in chapter 3 if we set a 0
The array
= 1,
ai •
i • 1, ••• , nand so from Theorem 3.1 we see that the operators r.(B) • A0- 1A.(B) are[·,·] self-adjoint. 1 1
Lemma 4.2:
r!" 8 1
Proof:
If g E His such that lim ri(B)g exists, then g E D(ri> and - lim B-+Jin
B-+Jin
r. (B)g. 1
Suppose lim r 1.(B)g • g0 , then for any f E D, B-+Jin [f,g 0 J • lim [f,r.(B)gJ • lim [r.(B)f,gJ • [r.f,gJ. B-tlin 1 B-tlin 1 1
*
Thus g E D(r. ) 1
Lemma 4.3:
Each of the operators r., j • 1, ••• , n has deficiency. indices J .,
(0,0); that is r.g J
= ig,
11<
r. g ... -ih J
( i . -1-1) implies that g. hIf;
Proof:
To begin with we obtain an explicit expression for r .• J
*J
Let
gED(r.), then
*J
[r .f,g] • [f,r. g] J
and
if
(A.f,g) • (f,A0 r. g) J
J
for all f E D for all f E D.
Next, if f E H then W(B)f E D for any Borel subset B of Rn and so A.W(B)f J
= A.(B)W(B)f. J
Consequently
*'
(A.(B)W(B)f,g) • (W(B)f,A0 r. g) J J
.,
for all f E H,
(f,Wb)Aj(B)g) • (f,W(B)AO r j g) for all f E H that is 44
o.
~
(4.3) Now define operators P and Q by (A1 - il)t •••••• (An - il)t: D+H . -lt •••• (A - il) -lt : H + D. Q- (Al - 11) n p-
Clearly Q is bounded and for all f QP
E
H we have PQ • I while for all g
= I.
Furthermore P and Q commute with W(B).
Let g
E
E
D
1F
D(r. ), then J
,. Q6j(B)g • Q(-l)j
Aiwi(B 1 )
.•
s'!1n "
•••••••• j
g
........
.
~ st l · · · · · · · · · nn
( t•• ).n 1. where j" indicates the omission of the columns 1J 1•
Thus
~
J
g
.
(A -i~)-ltst n nl
.
,.
(A
j
n
-i~>-ltst
nn
and as B +Rn we have lim Qll.(B)g •
Jttlin
J
(-l)j
l+i(A -il)-lt 1
g
.
"'
J
(A :u>-ltst n nn 45
1 g
. I
- iQ
A
(A -il)-ltst n nl
j
.,.
I
• • • • • • • • • • J ••••••••••
g.
.......... r
I
By expanding the right band side of this expression by the operator analoguesofdeterminantal operations, seft for example Atkinson [1, Theorem 6.4.1 p 106], we obtain lb. ~.(B)I •
B-tlln
J
-
-lt r (A -ii) k•l 1 n
n
r
(4.4)
- iQ AOk'g, k•l J where AOkj is the cofactor of s!j in the expansion of A0 • Since IIW(B) II ~ 1 for all B and W(B)
+
I
strongly as B
follows tba t lim
~.(B)g
B+RD
J
• lb.
W(B)~.(B)g
• lim
~(B)A.(B)g
B-llln
B-llln
since Q and W(B) commute. lim.. ~.(B)g • lim
JttllD
46
J
:&-tilD
J
J
Now if we use (4.3) we have
~(B)A.(B)g J
•
~ 0r""!s. J
+
ltn, it
and so (4.4) results in
--
••• k •••
We now observe that the right hand side of this equation is in D and so we may apply P throughout to obtain
-t
n
= - k=l I AkAOk'g. J (4.5)
That is iF
and we have an explicit expression for r. g. J
Suppose next that g i[g,g]
€
.-
However the operators the quantities
*
D(r.) and that r.1F g • ig. J
J
Then from (4.5) we have
~Wk(Bk) and AOkj commute and are Hermitian so that
(~Wk(Bk)AOkjg,g) are real. Consequently i[g,g] is real and D(r~) and r~h • -ih then h • 0. This
hence g • 0.
Similarly if h
completes
proof of lemma 4.3.
~he
Furthermore it is clear that
€
J
J
47
As a Corollary to lemma 4.3 we have
r ., j • 1, ••• n has a unique [•,•] self-adjoint extension,
Corollary 4.1:
J
namely the closure of
r., J
j • 1, ••• , n.
continue to use the notation
r.J
Unless otherwise stated we shall
to denote this closure so that we have
D c D(r.) and r! • r., -
j • 1, ••• , n with r. given by (4.5). .... n t J We know already that D(r.) c n D(Ak !1(k.) ; here we prove something more J - k•l J J
J
J
namely that
For f e: D
Also '.rhus
-1
- !10
In t10k. Akt
~1
r.J •
-48
J
has domain
n t n D(.Ak) • D.
~1
Accordingly n
4.4:
LeDIII4
D ..
n D(r.> 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<
e.
1
:s; 'II'.
i .. 1, ... , n.
In order to apply the abstract theory we set Hi S..
1J
= L2 (a 1. , b.1 )
and define
H. +H. by 1
1
(S .. f.)(x.) "'a .. (x.)f.(x.). 1J 1
1
1J
1
1
1
For the self-adjoint operators A. we take the Sturm-Liouville operators 1
generated from d2
A. "' --2 1
dx.1
- qi (xi)
subject to the end conditions
(4.12).
We note that there is no loss of
generality in assuming that 0 is in the resolvent set of each A., for if 1
this is not the case an affine transformation of the parameters A1 , ••• , An together with use of condition (4.11)
allows one to appeal to a theorem of
Atkinson [1, §9] in order to transform the given system (4.10) into one for which the associated Sturm-Liouville operators have zero in the resolvent set. 59
It is now a simple matter to check that all the abstract hypothesis of this chapter are fulfilled. Theorem 4.13:
Thus we may state
The eigenfunctions of the system (4.10-4.12) form a complete
orthonormal set in the tensor product space H•
n
8
~1
2
L (a.,b.) 1
1
with respect to the weight function det{a •. }~ • 1 • 1J 1,J= Notes: The abstract theory of this chapter is due to Browne [4], and apart from a few notational changes follows closely the account set out in [4]. Browne [5] has extended this theory to the case where as a point of its continuous spectrum.
~O
(see §4.1) has zero
This theory is not as nearly complete
as that in [4J but is of interest in application to systemsof ordinary differential equations defined on infinite or semi-infinite intervals.
See
Sleeman [10], Browne [3]. The application treated in §4.5 is the classic problem which is the motivation for most of the abstract multiparameter spectral theory developed to date.
Alternative approaches to this particular application have been
discussed by Browne [2j, Faierman [7] and Sleeman [9]. References 1
F. V. Atkinson,
Multiparameter eigenvalue problems, Vol I Matrices and compact operators. Academic Press, N~ York.
2
P. J. Browne,
A multiparameter eigenvalue problem. Anal. Appl. 38 (1972) 552-568.
3
P. J. Browne,
A singular multiparameter eigenvalue problem in second order ordinary differential equations. J. Differential Equations 12 (1972) 81-94.
4.
P. J. Browne,
Abstract multiparameter theory I. Anal. Appl. 60 (1977) 259-273.
60
J. Math.
J. Math.
5
P. J. Browne,
6
N. Dunford and J. T. Schwartz, Linear operators Part II. New York a963).
7
M. Faierman,
The completeness and expansion theorems associated with the multiparameter eigenvalue problem in ordinary differential equations. J. Differential Equations 5 (1969) 197-213.
8
F. Riesz and B. Sz-Nagy,
Functional Analysis • (L.F. Baron, Trans.), Frederick Ungar, New York (1971).
9
B. D. Sleeman,
Completeness and expansion theorems for a two parameter eigenvalue problem in ordinary differential equations using variational principles. J. London Math. Soc. 6 No 2 (1973) 705-712.
10
B. D. Sleeman,
Singular linear differential operators with many parameters. Proc. Roy. Soc. Edinburgh Ser. A 71 (1973) 199-232.
Abstract multiparameter theory II. Anal. Appl. 60 (1977) 274-279.
J. Math.
Interscience,
61
5 Multipammeter spectral theory for unbounded opemtors (The left defmite case) 5.1
INTRODUCTION
Recall that in chapter 4 we developed a spectral theory for the system n
r >-. s .. u j=l J 1J
i
A.u • 1
• 1.
i
= 1,
(5.1)
... ' n,
under the hypothesis S = det{S .• }
(5.2)
1J
be positive definite on the tensor product space H.
However, as we remarked
in chapter 1, the system (5.1) often arises as a result of separation of variables in a partial differential equation of elliptic type.
In such
problems (5.2) may not be true and the metric of H must be given in some other way.
In this chapter we study the eigenvalue problem for (5.1) under
conditions which are natural to the afore-mentioned elliptic boundary value problems.
We continue to adopt the same nomenclature concerning the operators
A., S •• appearing in (5.1), that is, A., 1
·1J
1
of chapter 4 §4.1.
Assumption 1
s 1J .. possess the properties (i)
In addition we make the following assumptions.
(Elliptigity condition)
Lets •• • cofactor of S .• in the determinantS. 1J
... '
1J
a
n
(ii)
Then there exists ann-tuple
of real numbers, not all zero, such that
(5.3)
62
n
is positive definite on 8 k•l k;'i Assumption 2
for i • 1, •.. , n.
(Definiteness condition)
At least one of the operators A. is positive definite on D(A.), 1
Remark:
1
i • 1, ... , n.
S is the operator induced in H by (5.2) when extended by linearity
,. and continuity.
In the same way Sik is the operator induced by (5.2) (with n
the i-th row and k-th column deleted) in 8 H.. j•l j;'i ~
In particular we note that
J
n
i
if Sik is applied to u' 8 ••• 8 u, it has no effect on u • This means that t ,.t the operators Ai and Sik' k = 1, ••• , n, commute. 1 i-1 i+l n If we formally "multiply" (5.1) byu 8 ••• 8u 8u 8 ••• 8u we obtain (since AT, 1
only operate on u i )
t A.S •. u,
n
t A.u "" 1
s!. 1J
I
j•l J 1J
(5.4)
where u "' u1 8 • • • 8 un.
.
,.t
Apply1ng Sik to (5.4) and summing over i gives n I i•l
t "t
A. S "k u • AkSu • 1
1
Using assumption 1 we obtain from this the following equation
Au
= In
i•l
t t
A. T. u • A Su • 1
(5.5)
1
where (5.6)
By assumptions 1 and 2 A is a positive definite operator on D' • linear hull
63
of u 1 8 ••• 8 u0 where ui
D{A.),
€
1
i = 1, . . .
and Ti commute and Ti has a square root Ti1/2 • 2u, = {A.T~/ 1 1
{A.T.u, u) 1
1
This follows since A.
n.
1
Hence we see that
2 u) ~ 0. T~/ 1
Also, if A. is positive definite it follows that 1
where M is the lower bound of A. and d ofT .. 1
We also remark that A defined
1
by {5.5) is essentially (for bounded_Ai) the operator A defined by (chapter 3. (3.1)) with a 0 a0
=0
= 0.
Indeed assumptions 1 and 2 above together with
are sufficient to ensure that A defined by {3.1) be positive definite.
Furthermore, by writing the operator A in terms of the operators ai'
i = 0, 1, •.• , n (see 3.1) we can argue as in Lemma 4.3 of chapter 4 to
prove that A in D' is essentially self-adjoint. If we introduce the inner product [u,v]A D' to a Hilbert space HA.
= {Au,v)
on D' we can complete
Furthermore A is bounded below which implies that
A has an extension (the Friedrichs extension) to a self-adjoint operator in H.
A will, in the sequel, always denote this extended operator.
When A is
positive definite, that is lluiiA ~ cllull for some constant C > 0, and all u
€
D{A) then HA
~
H topologically and algebraicly.
will be a bounded symmetric operator in HA. in the sense that {Au,u) > for all u which are not in H. operator on HA.
~
0, u
-1
Also the operator A S
If, however, A is only positive €
D{A), HA may contain elements
In this case A- 1s would be an unbounded symmetric
See for example Mikhlin [5].
5.2 AN EIGENVALUE PROBLEM We now study the eigenvalue problem (5.5) in HA.
As a preliminary we note
that A = 0 cannot be an eigenvalue; for Au = 0 implies {Au,u) turn implies u 64
= 0.
s
0 which in
Lemma 5.1:
The eigenvalues of (5.5), if they exist, must be real.
If A1
and A2 are two different eigenvalues and u 1 , u2 the corresponding eigenvectors then
~:
= A(Su,u)
Au • ASu implies (Au,u)
Hence (A - A)(Su,u)
=0
~ 0 and (u,Au) =A(u,Su) =A(Su,u).
and since (Su,u) ~ 0 we conclude that A • A.
Also,
if Au 1 • A1S u 1 and Au2 • A2su2 then
from which it follows that
(~ 1
- A2 )(Su1 ,u2 ) = 0.
But A1
~
A2 implies
(Su1 ,u2 ) • 0 and finally it follows that (Au1 ,u2 ) • 0. We assume from now on that we have the "compact" case in the sense that all occurring eigenvalue problems have only discrete spectra and the eigenvalues have finite multiplicity.
This is true for'· instance if we consider
the regular Sturm-Liouville problem (see [2], [4] and section 5.4 below). Another situation in which the above compactness criteria seem to hold is the case where all the operators Ai' or where all the S •• , 1J
i
= 1,
••• , n have compact resolvent&
j • 1, ••• ,n, are compact relative to A.• 1
Then the
operator A would be expected to have compact resolvent or S would be compact relative to A.
In the case n • 2, (i.e. the two parameter eigen-
value problem) this can be shown to be true. remains open.
However for n > 2 the problem
To avoid this difficulty we make the further assumption that
all the A.1 have compact resolvents in H.1 and that S is compact relative to A in the HA topology. We are now in a position to state our main result.
65
Theorem 5.1:
The system (5.1), under the assumptions 1 and 2 above together
with the compactness assumptions on the operator A, has a set of eigenvalues CD . 1 n f ().l,p• • • • • ).n,p)p=l and a corresponding set o e1genvectors up • .•• • up nCD such that (up1 8 8 up)p=l is a complete orthonormal system in HA 8 HA (CD) where Remark:
e denotes orthogonal complement and HA (CD) = { u e HA : Su =0}.
HA(CD) can be thought of as the eigenspace belonging to the eigen-
values A = CD and A = -CD. The proof of theorem 5.1 falls into two parts.
In the first part we
prove the completeness of the eigenfunctions Wm of problem (5.5) and in the second part we prove that each Wm can be expressed as a finite linear combination
of the eigenfunctionsof the original problem (5.1).
In this
section we prove the first part and defer discussion of the second part to the next section. Equation (5.5) can be written as A- 1su
= 1.1 u.
(5.7)
1.1 • 1/A.
-1
The operator A S is compact and hence we have a system of eigenvectors Wm and eigenvalues 1.1m where l1.1m I + 0 as m +CD.
The closed linear hull of
{ Wm}CD is the orthogonal complement of the eigenspace corresponding to m• 1
l.l = 0.
-1
This eigenspace consists of u e HA such that A Su = 0 or -1
[A Su,vJA • (Su,v) = 0 for all v e HA which, since HA is dense in H, implies Su = 0. 5.3
Hence {Wm}:. 1 spans exactly the space HAS HA(CD).
THE FACtoRISATION OF W
Fix one eigenvector Wm and the
i
cor~esponding
AW • A!i!w • A SW • m i=l 1 1 m m m
66
eigevnalue Am so that (5.8)
Since not all the numbers ai' suppose a
n
i = 1, ... , n in (5.6) are zero we may
f 0 and write 1 ( A
A
a
n
(5.9)
m
n
If we replace An by this value in the first n- 1 equations of (5.1) we obtain the system
A
.
( A. - .....!!! S. ) u 1 1 a 1n
i
n
= 1,
... , n-1. (5.10)
This system has a determinant corresponding to the right hand side, namely d
et
{s
a ~
ik - a
n
s }n-l
in i,k=l
which is evaluated to be
,.. snn and by assumption 1 is a positive definite operator (or negative definite By the positivity of the determinant it can
depending on the sign of an).
always be arranged, by a shifting of the spectrum, that 0 belongs to the A resolvent set of all the operators A1• - .....!!! S i • 1, ••• , n- 1, (see a in' n
the remarks preceding theorem 4.13 of chapter 4).
It follows then from
theorem 4.12 that the system (5.10) has a set of vectors E of the form p
1 n-1 E =u 8 ••• 8u p p p
= 1,
p
corresponding to eigenvalues A1
'P
n-1
equality holds in
2,
, ... , A 1 . Furthermore, the Parseval n- 'P n-1
8 H. for any vector u, and where the metric in 8 Hi i=l i•l 1
67
1 (See chapter 2, §2.2). Thus is given by the inner product (--a Tn u,v)~. n n if we let un € H be arbitrary then for the eigenfunction W of problem (5.5) m n n we can expand (W ,u ) as m n (W ,un)
m
n
=Ip
C (un)E ,
p
(5.11)
p
where the coefficient C (un) is· determined as the Fourier coefficient with p
respect to E , i.e. p
C (un) p
1
= ((Wm,un) n ,
a T E ) ......
n
n p n
From chapter 2. §2.2 this can be written as 1 C (un) .. ((W , -T E ) "' , un) n p m an n p n
and inserting this in (5.11) it follows that (W , un)
m
n
(W , T E )~, un) E . . r (~ an m n p n n p
p
=
1 E 8 rp (-an p
• (..!. rE a.p np
(W ,
m
T E )~, un ) n p n n
8 (W , T E
)~, un) •
mnpn
n
This last equality follows if we can prove that N
r E 8 N-- p=l P lim
m
(W, T E )..,•
m
n p n
r
p=l
E 8 (W, T E)~ p m n p n
exists in the B-metric
To this end we prove the following m
Le1111118 5.2:
r
p•l
Ep 8 (W , T E )~ m n p n
is convergent with respect to the metric
given by ( ·, T! • ) in H if and only i f
68
00
I II (Wmt Tn Ep>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 -
~<
a.1 :s; x.1 s b.1 <
~.
and
y.(a.)cos a.1 - y!(a.)sin a.1 • 0, 1 1 1 1
o
:s;
a.<
1r
o<
B. :s;
1r,.
1
1
(5.17)
i • 1, ••• , n.
73
As before we ass\Dile a .. (x.) 1J
£
1
C[a., b.J, q.(x.) 1
1
1
1
£
C[a., bJ and that a •. (x.) 1
i
1J
1
and q.(x.) are real valued. 1
1
In addition we ass\Dile q.(x.) 1 1
:?!
0
x.1
[a., b.], 1 1
£
i • 1, ••• , n.
Now
instead of the definiteness condition (4.11) we suppose the following "ellipticity" condition holds; namely
........ \.In a22 ....... a2n ·.• an2 ....... a nn
\.11
\.12
a21
a
nl
> 0,--- ••
all
au
a r-1,1
a r-1,2
\.11
\.12
a r+l,l
a r+l,2
....... ....... ....... .......
aln a r-l,n \.In a r+l,n
• ••• • • • • a
>0
> 0,
nn
(5.18)
for some real n-tuple of numbers \.1 1 , ••• , \.In not all zero and the inequalities holding for all
These conditions may be conveniently expressed in the £orm n
h
s
•
I \.1 a* > 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.
The superscript t will be
reserved, as in the previous chapters, to denote operators induced in H by operators from the factor spaces Hi'
i
= 1,
••• , n.
Correspondingly the
superscript tt is used to denote those operators induced in H* by .operators from the spaces Hi. mappings from H* out in
§
+
Finally the symbols < •, • >H (< ·, ·>h) are used to denote h
2.1 chapter 2.
(H*
+
H) in the sense of factorising elements as set
Finally inner products in H and h will be denoted by
(·,·) 8 and (.,.)h respectively. Define the operator~ by the determinantal array Att n
Btt
tt sll
......... .........
stt nl
Ttt
8 tt
.........
.
.
stt nn
Ttt n
Att 1
A
-
ln
The domain D(A) of A 82
1
is taken to be the algebraic tensor product
(6.3)
n
( 6a D(Ai)) i•l
8a D(B) c H*
and A maps this set into H* again.
A
is not necessarily a closed operator
in H* but it will always have a closed extension. we assume that wn £ D(.A) are such that wn If we choose f.£ D(A.), 1 1
i
= 1,
+ 0
In order to prove this
and.A. wn
••• ,nand g £ D(B*)
+
w in H* as n
+ ....
(B* is the adjoint
of Bin h), then it is easily seen that f 1 8 ••• 8 fn 8 g is in the domain of the adjoint
