AN ABSTRACT OF THE THESIS OF
CHRISTINA JOSEPHINE MIRKOVICH for the MASTER OF SCIENCE
(Name)
(Degree)
in
MATHEMATICS
(Major)
presented on
May 31, 1973
(Date)
Title: APPLICATIONS OF COLLECTIVELY COMPACT OPERATOR
THEORY TO THE EXISTENCE OF EIGENVALUES OF
INTEGRAL OPERATORS
Abstract approved:
Redacted for privacy
John W. Lee
The existence of eigenvalues is shown for certain types of
integral equations with continuous kernels, the proofs utilizing some
basic results of collectively compact operator approximation theory.
Applications of Collectively Compact Operator Theory
to the Existence of Eigenvalues of Integral
Equations
by
Christina Josephine Mirkovich
A THESIS
submitted to
Oregon State University
in partial fulfillment of
the requirements for the
degree of
Master of Science
Completed May 1973
Commencement June 1974
APPROVED:
Redacted for privacy
Asst. Professoi-Of Mathematics
in charge of major
Redacted for privacy
Cha,i-trnan of Department of Mathematics
Redacted for privacy
Dean of Graduate School
Date thesis is presented
May 31. 1973
Typed by Clover Redfern for
Christina Josephine Mirkovich
TABLE OF CONTENTS
Chapter
Page
I. INTRODUCTION
II.
COLLECTIVELY COMPACT OPERATOR THEORY
Collectively Compact Sets of Operators
Reduction of Integral Operators to Finite Algebraic
1
3
4
Systems
Iterated Kernels and Matrix Theory
III.
THE EXISTENCE OF EIGENVALUES OF INTEGRAL
OPERATORS WITH CONTINUOUS KERNELS
12
BIBLIOGRAPHY
24
APPLICATIONS OF COLLECTIVELY COMPACT OPERATOR
THEORY TO THE EXISTENCE OF EIGENVALUES OF
INTEGRAL OPERATORS
I.
INTRODUCTION
In a recent paper by Russell and Shampine [7], the existence of
eigenvalues for certain types of integral operators with continuous
kernels is demonstrated. The proofs in [7] employ several basic
concepts of numerical integration, finite matrix theory and calculus
to derive the existence results for eigenvalues. One of the types of
kernels treated in [7] is continuous and positive on the unit square.
Russell and Shampine use several uniform continuity and compactness
arguments to extend the eigenvalue properties of positive matrices to
integral equations with positive, continuous kernels. Anse lone and
Lee [2] treat the same problem, but instead of continuity arguments,
they employ collectively compact operator theory. Two additional
classes of kernels discussed in [7] are:
(a)
k(s, t) is complex-valued and continuous on the unit
square; the trace of kP,
the
p -th
iterated kernel,
is nonzero for some p > 3;
and
(b)
k(s,t) is continuous and nonnegative on the unit square;
the trace of
kP
is nonzero for some p > 1.
The objective of this paper is to illustrate how collectively
compact operator theory can be used to obtain simple derivations of
existence and related results for eigenvalues of integral operators.
Specifically, as in [7], it is shown here that integral operators with
kernels (a) and (b) have nonzero eigenvalues. However, the collec-
tively compact operator theory allows us to sharpen the results
obtained in [7]. Namely, we prove that the spectral radius of an
integral operator with kernel (b) is a nonzero eigenvalue with an
associated nonnegative eigenfunction.
In Chapter II, some fundamental definitions and results of col,
lectively compact operator theory and matrix theory, of use in the
subsequent analysis, are reviewed. Then, in Chapter III, the theory
is applied to the classes of integral equations with kernels described
by (a) and (b). The kernels in this paper are defined on the closed
unit interval for convenience, any interval [a, b] would suffice.
3
II.
COLLECTIVELY COMPACT OPERATOR THEORY
The following results of collectively compact operator theory,
along with the accompanying notation, will be employed throughout
this thesis. For proof of these results, the reader is referred to
[1, Ch. 1, 2 and 4].
Let
X
be a complex Banach space, Q.
the closed unit ball and
operators on
C',
and
X.
{x E XI II xII
1
the Banach space of bounded linear
[X]
Of particular importance in this paper are
the complex Banach spaces of continuous functions defined
on the unit interval and the unit square, respectively. Both
C'
C
and
have the uniform norm
max
0<t<1
114 =
If
Tn, T E [X]
00
I x(t) I
max
0 < S,t < 1
for each x E X. For
,
XEC
Ik(s,t)I,
for n = 1, 2, ... ,
pointwise convergence on X,
n
C
then
k E C'.
T
n
denotes
--" T
that is,
II T x
n
T E [X],
the spectrum of T
Tx II
0
as
is
denoted by ci(T),
c(T) = {Xlthere does not exist (X-T)-1 E [X]}
The spectrum is compact and contains the eigenvalues of
T.
The
4
number
r(T) = max.(' XI
I
the case where
T
IX
E
is the spectral radius of T.
o-(T)}
is a compact operator,
cr(T)
In
has the following
des cription.
Recall that an operator K E [X] is compact if
relatively compact, i.e. ,
Kfit
is
KE3
is compact. The Arzea-Ascoli
theorem provides a means for identifying relatively compact sets in
C
and
C'.
The Fredholm Alternative [1, p. 119] states that for
K E [X]
a compact operator, there exists
(-K)X = X,
in which case
the nonzero elements of
infinite dimensional, then
(X-K)
o-(K)
-1
(X-K)
-1
X
if and only if
is bounded. This implies that
are all eigenvalues of K. If
0 E cr(K),
0 and
X
is
and the eigenvalues of K are
either finite in number or form an infinite sequence converging to
zero [1, p.
120].
Collectively Compact Sets of Operators
A set of operators X C [X] is collectively compact if
R.ta={1(xIK EX, xE
is relatively compact. Every element of a
collectively compact set is compact, and
(1)
if
A is any bounded set of scalars, then AX is
collectively compact.
Of interest to this thesis is the situation in which K,Kn E [x]l
5
n = 1, 2, ...,
such that
{Kn}
(2)
is collectively compact and
(3)
as
n
Together, (2) and (3) imply that
K
n
co.
is compact; and the following
results concerning the spectral properties of the operators
Kn
K hold [1, Ch. 4]:
any neighborhood of
(4)
cr- (K)
contains
o-(Kn)
for all
sufficiently large;
if
(5)
X
n
then
if
(6)
(7)
E o-(Kn)
for
n = 1, 2, ... ,
and
X
n
-- X ,
X E cr(K);
X E cr(K),
then there exist
X.n E 6(K ),
n
n = 1, 2, ... ,
such that Xn -- X .
r(Kn)
as
r(K)
00
Reduction of Integral Operators to Finite Algebraic Systems
Let
k(s, t) E C'. The eigenvalue problem (EVP)
1
k(s, t)4)(t)dt = X4 (s),
(8)
0
0<s<1
and
6
where
(I)
can be approximated by means of numerical integra-
E C,
tion using the rectangular quadrature formula:
n
1k(s , j /n)(1)n(j /n) = AncOn(s),
(9)
0 < s < 1.
j=1
For
(9) suggests an interpolation formula for
0,
An
of the values
An
.4)n
in terms
j = 1, ...,n. In fact (see (17) below), when
st.ri(j
EVP (9) is equivalent to the matrix EVP
0,
(10)
n
j in).0n(j in) = Xnitori(i
11c(i
i = 1,
j=1
This discretization procedure allows us to consider the EVP (8) by
means of a passage to the limit as
Define operators
n
co
in EVPs (9) and (10).
L,Ln E [C], n = 1, 2, ...,
by
1
(L4)(s) = J
k(s, t)(1)(t)dt
0
n
(12)
where
(Lnsb)(s)
-1
k(s, j in)(13.(j in)
,
k(s,t) e C'. The convergence of the rectangular quadrature
formula on
operators
C
L
and the Arzela-Ascoli theorem imply that the
and Ln, n = 1, 2,
,
satisfy (2) and (3), so that
7
results (4)-(7) apply to the EVPs
(13)
L4 = X4
and
(14)
Ln4n = Xn4)n
Define
Ln
Ln
(15)
Let
as the
2
)n = (O1 4)n,
4-
,
n,
=
n x n matrix
[n-lk(i/n, j/n)],
where
4)n),
n
1 < i, j < n.
for
(1)n(i in)
4ni
i = 1,
,n.
Then (10) can be rewritten as
(16)
Consider
n-tuples
L
nn
4)n
=X
to be an element of
v = (v1,
, vn)
nn
oo
the Banach space of
in,
with complex components, equipped with
the uniform norm
II 711 =
oo
Then Ln E [in ],
max
1<j<n
I vil
and the following easily verified relationship
between EVPs (14) and (16) holds [1, p. 19]:
(17)
for
An
0
and
v=
,v
oo
E in,
n
)
8
n
Ln v = X
nv
and
prOn(s) =
if and only if
Ln.tin = Xn (1)n
and v = (v1,
vn)
where
v. = n (j/n),
j=
1,
It follows that the nonzero eigenvalues of Ln and Ln are the
same, hence
(18)
r(Ln) = r(Z, n)
.
Along with results of certain compactness arguments, (18) will be
used to extend several properties of specific types of matrices to the
operators
Ln,
to the operator
and ultimately in passing to the limit as
00,
L.
Iterated Kernels and Matrix Theory
The following definitions and results of integral equations and
matrix theory will be used in the subsequent analysis.
Let
k(s, t) E C'. For a positive integer
P,
iterated kernel is defined recursively [5, p. 22] by
the
p-th
k1 (s,t) = k(s,t)
1
kP(s , t)
k13-
(s, w)k(w, t)dw,
p > 1.
0
The trace of
kP
is
1
kP(s, s)ds,
-r(kP)
p > 1.
0
The
p-th iterated kernel is an element of
bounded linear functional on
and
T
is a
C'.
The trace of a complex-valued n x n matrix A = {a..]
iJ
is
defined to be
a..
T(A) =
(19)
i=1
The Euclidean and uniform norms of A are defined respectively by
n
II All
E
i,j=1
and
II A 11 oo =
If
max
1 < <n
j=1
X1, ...,Xn are the eigenvalues of A,
then Schur's inequality
10
[8] states that
n
(20)
i=1
Let
46p,
n
= {(t 1) .
3t
)
P
I t.= 1 ,
for i = 1, 2, .
,n
.
,p}.
Then, the
p-th power of A is
1<i, j<n,
AP =[c..],
13
where
c..
=
13
(21)
a
A
at
itl a
.
a
(p- l)j
p- 1, n
It is an elementary result of matrix theory [3, p. 95] that
(22)
p>1.
XP;
T(AP) =
1=1
The
n x n matrix P = [p..]
is said to bepositive if
Lj
p., > 0
lj
for all
(i, j). Therefore,
0<a= min p,. < max. p..13
.
i,3
1.3
.
1,3
The properties of positive matrices that will be useful in the subsequent analysis are the following:
11
(23)
The positive n x n
value
X.
matrix P has a unique eigen-
of maximum modulus; X > 0
= r(P) )
and simple; and
(hence
has a corresponding
X.
eigenvector with positive components, (positive
eigenvector) [6];
(24)
If
is an n x n
A = [a..]
for all (i, j),
then
I a..I < p..
matrix with
1.3
r(A) < r(P). (In fact, the
inequality holds for any nonnegative matrix P)
[9, p. 47].
(25)
If
v = (v 1,
,
is any positive vector, then [4]
vn)
(P v).
1
min
v.
i.
For
1
v = (1,
,
1),
--..
(P v).
< r(P) < maxv.--1.
1
the above inequalities become
n
n
0 < na. < min
i.
p.. < r(P) < max
p.. < nP
i=1
12
III. THE EXISTENCE OF EIGENVALUES OF INTEGRAL
OPERATORS WITH CONTINUOUS KERNELS
The results developed in the previous chapter will now be
applied to prove two theorems concerning the existence of eigenvalues
L, Ln and
of specific types of integral operators. The operators
Lin
'ID'
referred to below are defined by (11), (12) and (15). The letter
always denotes a positive integer.
Theorem 1. Let
k(s, t) E C'. If T(kP)
is not zero for some
then there exists a nonzero eigenvalue of the operator L.
p > 3,
Let M = max
Proof.
I k(s, t) I
0 < s, t < 1.
,
(s,t)
Consider the matrix EVP (16). Let
mum modulus for
n
1, 2,
sequence
.
{X
n1
Ln,
X
be the eigenvalue of maxi-
n1
with corresponding eigenvector
14)n'
The basic idea of the proof is to establish that the
is bounded away from zero and
co
for all n
sufficiently large. Then, by means of a simple collectively compact
operator approximation theory argument, it will, follow that
a nonzero eigenvalue.
From the definitions of
Xn1
e.'1,
n,
Xnl
=L
nn
and
we have
On,
.
Thus,
<M,
n>1
.
L
has
13
To show that the sequence
{x
but a finite number of
we make use of the following lemma.
Lemma 1.
If
n,
is bounded away from zero for all
n1}
k(s,t) E C1
then -r[(tn)P]
-r(kP)
as
for p > 1.
Proof. Define
T, Tn
C'
,
n = 1,2, .
,
by
(Tf)(s,t)=.5- k(s,w),(w,t)dw
0
lk(s, j /n)f(j in, t),
t) =
(T
f E C'
j=1
Letting f = k,
we obtain for any p > 1
(TP-1k)(s, t) = kP(s,t)
(26)
and
(TP-ik.)(s,t)
(27)
n-r(p-1)k(s, t 1 /n)k(t 1 In, t2 /n).
=
A
p-1,n
, . ,
(T
0
0
= Tn = I,
k(t
P-1
/21,0
the identity operator on C').
From (21) and (27), we see that the p-th power of
(28)
(r, )P
n
=
[n-1(TP-1k)(i
j /n)J,
rn
is
1 < i, j < n .
n
00,
14
The operators
are bounded linear operators, hence continu-
T, Tn
ous, on C'. Also, the convergence of the rectangular quadrature
formula on
implies that
C
Hence, for any p > 1 ,
Tn
TP
n
TP
as
.,
n = 1,2,
TP, TP E [C1,
(29)
as
T
n
and
00
By (26) and (29) we obtain
(30)
II TP- lk-kP II
as
0
Therefore, by the continuity of the trace function,
(31)
IT(TP-1k)--r(kP)1
as
0
n
oo
Convergence of the rectangular quadrature formula on
(32)
-1
C
1
Tn
1
j in) -
k)(j
(TP- 'k)(s, s)ds I -4.0
0
as
n ~ co.
However, by (19)
n-1
(33)
and
P-1k)(j/n, j/n) =
[(11,n)P]
j=1
1
(34)
(Tn
0
1
k)(s, s)ds =
T 13-1k)
implies
15
Making the substitutions of (33) and (34) into (32),
(35)
I
n
as
(TP- lk) I -- 0
)131
n
00
Together, (31) and (35) give the desired result
n)P1
-r(kP)
as n
00
Now, suppose there exists a subsequence of the eigenvalues
n1
denoted by {x,1 },
such that
Denote the eigenvalues of tn by
I
0
n'l
Let
X
By (22)
(36)
n
,)P]I
n'
n
i=1
i=1
=
Schur's inequality implies
n'
Xn,i I
< II L
n' E
2
i =1
so that (36) becomes
IT[(1-in,)13]1 < Ix
11D-27,A.2
n'll
Hence,
(37)
ndp]
0
as
as
n
00
n'
p > 3.
co.
16
Consequently, Lemma 1 and (37) imply that r(kP) = 0
p > 3,
for all
which contradicts the hypothesis of Theorem 1.
Therefore, there exists m > 0 and a positive integer
N
such that
m < IX
(38)
< M for all n > N.
n.1
(Without loss of generality, let N = 1. ) Thus, there exists a scalar
X.
and a subsequence
For each n',
let
corresponding to
{X.
n'l } such that
4n ,(t)
X
n '1.
X.
n'l
X
and I X.I >m > 0.
be the eigenfunction of the operator
As sume
II 4:8n, II
=1
Ln,
for all n'.
The following argument employs the results of collectively com-
pact operator theory to show that there exists a subsequence of
{4)
that converges in norm to an eigenfunction 4 of the operator
L,
with associated eigenvalue
By (38) the sequence
,}
X.
1}
{(X.
n'l
is bounded; hence (1) implies
)
that the set
= {(xn'l)
is collectively compact. Thus,
4)n'
of
(39)
(Xn 1) 1Ln14)n?
{4)n,},
ta.
is relatively compact. Since
for all n',
there exists a subsequence
E
denoted by
4110,
114)n-4)II
eC
and a function
as
n"
00.
such that
17
Also, successive application of the triangle inequality and the Banach-
Steinhaus theorem reveals that
(40)
114)n-x-11411
=
11(Xn1)-1Les.n-c1L4
as
n"
11.
0
CO
Together, (39) and (40) imply
1_,01) = X(1) .
is nontrivial because for all n"
The function
11(on11
= 1
and
11
$111
(On
Sin -4)11
0
as
n"
oo.
Hence
110 = 1.
Therefore,
).
is a nonzero eigenvalue of L,
which is the desired
result.
Notation.
In the next theorem, the following notation will be
used:
S
is the unit square;
)P
The
is defined by (28).
p-th power of the integral operator L
is denoted by L.
18
It is easily verified that LP is an integral operator on
kernel kP,
iterated kernel. Thus,
p-th
the
with
C
1
(LPCI)(S)
=
E C.
kP(S,t)01)(t)dt,
1
0
Let
(LP)
n
be the
n x n matrix
"-or,
1
(Vin =
Theorem Z.
T(kP) > 0
L,
Let
p
,
k (i in, /n)
,
j
kE C' such that k > 0 on
for some p > 1. Then r(L)
Because
k
S,
and
is a positive eigenvalue of
with corresponding nontrivial eigenfunction
Proof.
, n.
1, .
(I) > 0
[0, 1].
on
is continuous and nonnegative on S, and
7(kp) 5- kP(s, s)ds > 0 for some p > 1,
0
it follows that there exists a point
0
of
(s0' sO)
in
S
(s
0
,s 0 )ES and a neighborhood
such that kP(s, t) > 0
0 < a < kP(so, so). Without loss of generality,
on
S2
be a closed square, symmetric about the diagonal
parallel to the sides of
S,
such that kP(s, t) > a
Cl.
Let
can be chosen to
s = t,
with sides
for all (s, t) E O.
19
Figure 1
be the set of points in
Let An
defined by
An = {(i in, j in) i = 1, .
Then, the image of An under
. .
, n and j =
there exists a positive integer
intersects
C2.
Therefore, for
mn x mn submatrix of
n- 1(T-n lk)(i. in, j in)
element of
O.
ger
N
2
N
1
of
(1%)13
(tn)P
n>
N2.
2
n}
.
It is easy to see that
define
Pn
N
,
1
A
n
to be the
obtained by deleting all entries
such that
n -0-
mr
B
(Ln)
00.
and
(LP)n
On, j in)
is not an
goes to zero, uniform-
Thus, there exists a positive inte-
such that
a
p,. > n1 (-2
) > 0 for all entries
when
13
(Ln)
such that for all n
n ..> N1,
,
By (28) and (30), the difference between the
i-jth entries of
ly in i-j, as
N1
no
.
arranged in the
(TP- 1k),
n
appropriate square array, is the matrix
1,
p.,
Pn
20
By simultaneously permuting the rows and columns of
we see that (L n )'
is similar to
A
n
Qn
(7C
B
A, B and C are nonnegative block matrices, and both Pn
where
and
Ln
B
are square. Thus,
r(6-#,n)P) = r(Q n)
Let N
Qn be the
n x n matrix obtained from Qn by replacing all
the entries of matrices A, B and C by zeros. Then,
QO
n
Note that triangularization of
n
0
n
0
Qn
0
is essentially triangularization of
so that
ev0
es,
r(Qn) = r(P n )
.
Application of (24) and (25) yields
r((Ln)P)
for n > N2.
2
,y
=
-
r(rin) > r(Un6 ) = r(Pn) >mn(iln-)
21
Also, it is not difficult to see from Figure 1 that
lim mn(-1)
n =b - a.
co
Hence, the above limit and the fact that
r((irn)P)
(r(lin))P
=
imply
lim (r(Ln )) p
n
>
a( b-a)
2
> O.
Therefore,
r(L) = lim r(Ln) = lim r(L n) > a(b-a
n co
n-00
(41)
It remains to be shown that
11p > o.
is an eigenvalue of L. The proof
r(L)
will be simplified by the following lemma.
Lemma 2.
Let
entries. Then r(B)
B
be an n x n matrix with nonnegative
having a correspond-
is an eigenvalue of B,
ing nontrivial eigenvector
x,
such that the coordinates of x are
nonnegative.
Proof. Let
adding
(42)
B
m
denote the matrix obtained from B by
lim to each entry of
11B -Bm
m = 1, 2, .
B,
0
as m
.
.
oo
Then
22
Bm
is a positive matrix for all m > 1. Therefore, by (23) Bm
has a positive eigenvalue
Il x
m = r(B m ),
and
X
m has an associated
m = (xml, ...,xmn). Assume that
positive eigenvector
(43)
X
x
II
=
max
1x
1<i<n
mi
1
=
m
1,
=
1, 2.
By (24)
> r(B) > 0.
Xi = r(Bi) > X2 = r(B2) >
Therefore, there exists a nonnegative number
(44)
X
m --- X > r(B) >0
as m
X
such that
°0.
Also, (43) implies that there exists a subsequence Ix 111 r} and a
vector
x such that
Hence,
m'
m,-x 11 - 0 as
(45)
11x
11;11 =
and the coordinates of
(42) and (45) and the fact that
Ilx
m m 1-33;11
11 X
m Ixm 1-Xx11 -- 0
17c
11B
B
m,xm
,
1-3c
x
=X
m m 1-13;11
00.
are nonnegative. By
m,xm
-1-
0
as
we have
m'
00 ,
and
Therefore Bx =Xx
eigenvalue of B.
as
m'
00.
and recalling (44), we see that X = r(B)
is an
23
Thus, by Lemma 2,
r(Ln)
is an eigenvalue of Ln with
=
corresponding nonnegative eigenvector
such that
(I)n
II c
II
=
1,
for all n > 1. Let r(L) > 13 > 0. Then, from (17) and (41) it
follows that there exists a positive integer
N,
such that for all
the following results are obtained:
n > N,
r(Ln) = r(Ln) = Xnl > P ;
ii) r(Ln) is an eigenvalue of Ln,
i)
function
iii) Ln
iv) r(L
L
n
)
defined by (17), such that
(I)n >
as
with corresponding eigen-
n
r(L),
II Sin II
= 1;
00;
hence
r(L) = X.
is an eigenvalue of L.
In addition, application of the compactness argument used in the
proof of Theorem 1 yields a subsequence
EC
{(1)
n,}
such that
II 43'n
t
-4)11 1- 0
(1) > 0,
as
11 (H1
and
L(I) = 4.
co
and a function
24
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2.
Anse lone, P. M. and J.W. Lee. Applications of Collectively
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publication.
3.
Bellman, R. Introduction to Matrix Analysis. New York,
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4.
Collatz, L. EinschliePungenssatz fir die characteristischen
Zahlen von Matrizen. Math. Zeitschr. 48 :221 -226. 1942.
5.
Mikhlin, S.G. Linear Integral Equations. Vol. 2. New York,
Gordon and Breach, 1960. 223 p.
6.
Perron, 0. Zur Theorie der Matrices. Math. Ann. 64:248-263.
1907.
7.
Russell, R. D. and L.F. Shampine. Existence of Eigenvalues of
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8.
.
SIAM Review.
13:209 -219.
Schur, I. Uber die characteristischen Wurzeln einer linearen
Substitution mit einer Anwendung auf die Theorie der Integralgleichungen. Math. Ann. 66:488-5 10.
9.
1971.
1909.
Varga, R.S. Matrix Iterative Analysis. Englewood Cliffs,
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