<< стр. 9(всего 11)СОДЕРЖАНИЕ >>
Aв€’1 Ajk = О±jk .
E
jj

Moreover, due to Lemma 15.1.4, with the notation
jв€’1
n
vA := max О±jk and wA := max О±jk
j j
k=j+1 k=1

we have
Dв€’1 V в‰¤ vA and Dв€’1 W в‰¤ wA .
Now Corollary 15.1.3 implies
Lemma 15.3.1 Let the conditions (3.1)-(3.3) and
nв€’1
kj
vA wA < 1 (3.4)
j,k=1

hold. Then operator (1.1) is invertible. In particular, let vA , wA = 1. Then
(3.4) can be written in the form
nв€’1 nв€’1
(1 в€’ vA )(1 в€’ wA )
vA wA < 1. (3.5)
(1 в€’ vA )(1 в€’ wA )
15.4. Matrices with Bounded oп¬Ђ Diagonals 205

Furthermore, under (3.1), (3.2) assume that for all regular О» of D, the rela-
tions
n
sup |П†k (t)(О» в€’ t)в€’1 | < в€ћ and
v (О») := max
Лњ
j
k=j+1 tв€€Пѓ(Sj )

jв€’1
sup |П†k (t)(О» в€’ t)в€’1 | < в€ћ.
w(О») := max
Лњ (3.6)
j
k=1 tв€€Пѓ(Sj )

are fulп¬Ѓlled. Due to Lemma 15.1.4 with the notation from (2.1), we have
Лњ Лњ
W (О») в‰¤ w(О») and V (О») в‰¤ v (О»).
Лњ Лњ

Now Theorem 15.2.1 gives.
Lemma 15.3.2 Under conditions (3.1), (3.2) and (3.6), the spectrum of
operator A deп¬Ѓned by (1.1) lies in the union of the sets Пѓ(D) and
nв€’1
v k (О»)wj (О») в‰Ґ 1}.
{О» в€€ C : Лњ Лњ
j,k=1

15.4 Operator Matrices with Bounded
oп¬Ђ Diagonal Entries
Again assume that H is an orthogonal sum of the same Hilbert spaces Ek в‰Ў E
(k = 1, ..., n). In addition, Ajk (j = k) are arbitrary bounded operators:

v0 в‰Ў V < в€ћ, w0 в‰Ў W < в€ћ. (4.1)

Lemma 15.4.1 Let the conditions (3.1),(4.1) and
nв€’1
kj
ПЃв€’kв€’j (D, 0)v0 w0 < 1
j,k=1

hold. Then operator (1.1) is invertible.
Proof: Since D is normal,

(D в€’ О»IH )в€’1 = ПЃв€’1 (D, О»). (4.2)

Therefore Dв€’1 = ПЃв€’1 (D, 0). Due to (4.1)

Dв€’1 V в‰¤ Dв€’1 в‰¤ ПЃв€’1 (D, 0)v0 .
V

Similarly, Dв€’1 W в‰¤ ПЃв€’1 (D, 0)w0 . Now Corollary 15.1.3 implies the required
result. 2
206 15. Operator Matrices

Lemma 15.4.2 Under conditions (3.1) and (4.1), let z1 be the unique non-
negative root of the algebraic equation
nв€’2
nв€’jв€’1 nв€’kв€’1
z k+j v0 = z 2(nв€’1) .
w0 (4.3)
j,k=0

Then the spectral variation of A with respect to D satisп¬Ѓes the inequality

svD (A) в‰¤ z1 .

Proof: With the substitution y = 1/z equation (4.3) takes the form
nв€’1
jk
y k+j v0 w0 = 1. (4.4)
j,k=1

So under consideration equation (2.5) can be rewritten as (4.4). Due to
Lemma 15.2.3 we arrive at the result. 2

Due to Lemma 1.6.1, if
nв€’1
jk
p(A) := v0 w0 < 1, (4.5)
j,k=1

2(nв€’1)
в‰¤ p(A). So under (4.5), the previous lemma gives the inequality
then z1

svD (A) в‰¤ 2(nв€’1)
p(A). (4.6)

Let us improve Lemma 15.4.1 in the case, when Ajk (j = k) are Hilbert-
Schmidt operators. Recall that N2 (.) denotes the Hilbert-Schmidt norm.
Clearly,
2 2
N2 (V ) = N2 (Ajk ),
1в‰¤j<kв‰¤n

and
2 2
N2 (W ) = N2 (Ajk ).
1в‰¤k<jв‰¤n

Lemma 15.4.3 Under condition (3.1), let V and W be Hilbert-Schmidt op-
erators and the inequality
nв€’1
j
(k!j!)в€’1/2 N2 (V )N2 (W )ПЃв€’kв€’j (D, 0) < 1
k
(4.7)
j,k=1

hold. Then operator A deп¬Ѓned by (1.1) is invertible.
15.5. Hilbert-Schmidt Diagonals 207

Proof: Due to (4.7) Dв€’1 is bounded, Dв€’1 V and Dв€’1 W are Hilbert- Schmidt
operators. In addition, according to (1.5), they are nilpotent. Due to Corol-
lary 6.9.2,

(Dв€’1 V )k в‰¤ N2 (Dв€’1 V )(k!)в€’1/2 , (Dв€’1 W )k в‰¤ N2 (Dв€’1 W )(k!)в€’1/2
k k

(4.8)
for any natural k < n. But

N2 (Dв€’1 V ) в‰¤ Dв€’1 N2 (V ) = ПЃв€’1 (D, 0)N2 (V ).

Similarly,
N2 (Dв€’1 W ) в‰¤ ПЃв€’1 (D, 0)N2 (W ).
Now Theorem 15.1.1 implies the required result. 2

15.5 Operator Matrices with Hilbert-Schmidt
Diagonal Operators
Again, let H be an orthogonal sum of Hilbert spaces Ek , k = 1, ..., n. In
addition, Ek are separable, condition (4.1) holds and the diagonal operators
have the form

Ajj = IE + Kj where Kj (j = 1, ..., n) are Hilbert-Schmidt operators .
(5.1)
Due to Theorem 6.4.1, for any Hilbert-Schmidt operator K in H, we can
write
в€ћ
g k (K)
в€’1
в€љ
(K в€’О»I) в‰¤ G(K, ПЃ(О», K)) в‰Ў for all regular О», (5.2)
k!ПЃk+1 (K, О»)
k=0

where
в€ћ
g k (K)
в€љ
G(K, y) в‰Ў (y > 0)
k!y k+1
k=0

and
в€ћ
2
|О»k (K)|2 )1/2 (K в€€ C2 ).
в€’
g(K) = (N2 (K)
k=1

If K is a normal operator: KK в€— = K в€— K, then g(K) = 0. The following
relations are true:
g 2 (K) в‰¤ N2 (K) в€’ |T race(K 2 )|
2

and
12 в€—
g 2 (K) в‰¤ N (K в€’ K) (K в€€ C2 )
22
208 15. Operator Matrices

(see Section 6.3). So due to (5.1) and (5.2)

(Kj в€’ О»I)в€’1 в‰¤ G(Kj , ПЃ(Kj , О»)) (5.3)
Ej

and therefore with О» = в€’1 we get

Dв€’1 = max (IE + Kj )в€’1 в‰¤ b0 (D), (5.4)
Ej
j

where
в€ћ
g k (Kj )
в€љ
b0 (D) = max G(Kj , ПЃ(Kj , в€’1)) = max .
k!ПЃk+1 (Kj , в€’1)
j j
k=0

Lemma 15.5.1 Let the conditions (4.1), (5.1) and
nв€’1
bk+j (D)v0 w0 < 1
kj
(5.5)
0
j,k=1

be fulп¬Ѓlled. Then operator A deп¬Ѓned by (1.1) is invertible.
Proof: Due to (4.1) and (5.4)

Dв€’1 V в‰¤ Dв€’1 v0 в‰¤ b0 (D)v0 .

Similarly,
Dв€’1 W в‰¤ b0 (D)w0 .
Now Theorem 15.1.1 implies the required result. 2

Thanks to Theorem 15.1.1, we also get the following result.
Lemma 15.5.2 Under conditions (4.1) and (5.1) the spectrum of operator
A deп¬Ѓned by (1.1), lies in the set

в€Єn в„¦j (О»),
j=1

where
nв€’1
Gs+k (Kj , ПЃ(Kj , О» в€’ 1))v0 w0 в‰Ґ 1}.
ks
в„¦j (О») = {О» в€€ C :
s,k=1

In other words, for any Вµ в€€ Пѓ(A), there are an integer j and a

О»(Kj ) в€€ Пѓ(Kj ),

such that
nв€’1
Gs+k (Kj , |О»(Kj ) + 1 в€’ Вµ|)v0 w0 в‰Ґ 1.
ks

s,k=1
15.6. Example 209

Put
в€ћ
g0 y k+1
k
в€љ
h(y, D) = ,
k!
k=0

where g0 = maxj g(Kj ). Let us consider the scalar equation
nв€’1
hs+l (y, D)v0 w0 = 1.
ls
(5.6)
s,l=1

Thanks to (5.3), one can write

(D в€’ О»IH )в€’1 = max (IE + Kj в€’ О»IE )в€’1 в‰¤ h(ПЃв€’1 (D, О»), D).
j

Now Lemma 15.2.3 yields
Theorem 15.5.3 Let x0 be the extreme right (unique positive) root of equa-
tion (5.6). Then the spectral variation of operator A deп¬Ѓned by (1.1) with
respect to D satisп¬Ѓes the inequality

svD (A) в‰¤ xв€’1 .
0

15.6 Example
Let H = L2 ([0, ПЂ], Cn ) be the Hilbert space of functions deп¬Ѓned on [0, ПЂ] with
values in a Euclidean space Cn and the scalar product
ПЂ
(h, w)H = (h(x), w(x))C n dx,
0

where (., .)C n is the scalar product in Cn . Consider the operator A deп¬Ѓned
by the expression

d du(x)
Au(x) = в€’ + B0 (x)u(x) (u в€€ Dom (A), 0 < x < ПЂ) (6.1)
d0 (x)
dx dx
on the domain

Dom (A) = {u в€€ H, u в€€ H, u(0) = u(ПЂ) = 0 }, (6.2)

with continuous real n Г— n-matrices

d0 (x) = diag [a1 (x), ..., an (x)], B0 (x) = (bjk (x))n
j,k=1 ,

where functions aj (x) are diп¬Ђerentiable and positive:

Лњ
dj в‰Ў min aj (x) > 0. (6.3)
x
210 15. Operator Matrices

Take E = L2 ([0, ПЂ], C1 ) and deп¬Ѓne operators Ajk by

d dv(x)
Ajj v(x) = в€’ aj (x) + bjj (x)v(x)
dx dx

(v в€€ Dom (Ajj ), 0 < x < ПЂ),
where
Dom (Ajj ) = {v в€€ E : v в€€ E, v(0) = v(ПЂ) = 0 }
and
Ajk v(x) = bjk (x)v(x) (v в€€ E; 0 < x < ПЂ, j = k).
Assume that
Лњ
ОІj в‰Ў inf bjj (x) + dj > 0 (j = 1, ..., n). (6.4)
x

Omitting simple calculations, we have

Лњ
(Ajj v(x), v)E = (aj v , v )E + (bjj v, v)E в‰Ґ dj (v , v )E + (bjj v, v)E в‰Ґ

Лњ
dj (v, v)E + (bjj v, v)E = ОІj (v, v)E .
Consequently,
Aв€’1 в€’1
в‰¤ ОІj (j = 1, ..., n).
E
jj

Clearly,
в‰¤ max |bjk (x)|.
Ajk E
x

So
Aв€’1 Ajk в€’1
в‰¤ ОІj max |bjk (x)|.
E
jj x

With the notation
nв€’1 n
в€’2
ОІj max |bjk (x)|2 )1/2 .
vA в‰Ў (
Лњ
x
j=1 k=j+1

and
n jв€’1
в€’2
ОІj max |bjk (x)|2 )1/2
wA в‰Ў (
Лњ
x
j=2 k=1

we have
Dв€’1 V в‰¤ vA ,
Лњ
and
Dв€’1 W в‰¤ wA .
Лњ
Now Corollary 15.1.3 yields
15.6. Example 211

Proposition 15.6.1 Let the conditions (6.3), (6.4) and
nв€’1
Лњk Лњ j
vA wA < 1 (6.5)
j,k=1

hold. Then operator A deп¬Ѓned by (6.1), (6.2) is invertible. In particular, let
vA , wA = 1.
ЛњЛњ
Then (6.5) can be written in the form
Лњnв€’1 Лњ nв€’1
(1 в€’ vA )(1 в€’ wA )
vA wA
ЛњЛњ < 1.
(1 в€’ vA )(1 в€’ wA )
Лњ Лњ
In the case n = 2 one can write
в€’1 в€’1
vA = ОІ1 max |b12 (x)|, wA = ОІ2 max |b21 (x)|.
Лњ Лњ
x x

Inequality (6.5) takes the form
max |b12 (x)| max |b21 (x)| < ОІ1 ОІ2 .
x x

To investigate the spectrum of operator (6.1) assume for simplicity that
aj в‰Ў const > 0, bjj в‰Ў const (j = 1, 2, ...). (6.6)
Then it is simple to check that the eigenvalues of Ajj are
О»k (Ajj ) = aj k 2 + bjj (k = 1, 2, ...).
Denote
nв€’1 n
max |bjk (x)|2 )1/2
vb в‰Ў (
x
j=1 k=j+1

and
n jв€’1
max |bjk (x)|2 )1/2 .
wb в‰Ў (
x
j=1 k=2

Clearly,
в‰¤ vb , W в‰¤ wb .
V
Now Lemma 15.4.2 yields
Proposition 15.6.2 Let z2 be the unique non-negative root of the algebraic
equation
nв€’2
nв€’jв€’1 nв€’kв€’1
z k+j vb = z 2(nв€’1) .
wb (6.7)
j,k=0

Then the spectral variation of A with respect to D satisп¬Ѓes the inequality
svD (A) в‰¤ z2 .
212 15. Operator Matrices

In other words for any Вµ в€€ Пѓ(A), there are natural k = 1, 2, ... and j в‰¤ n,
such that
|Вµ в€’ aj k 2 в€’ bjj | в‰¤ z2 .
In particular, if n = 2, then
в€љ
z2 = vb wb = [max |b12 (x)| max |b21 (x)|]1/2 .
x x

Certainly, instead of the ordinary diп¬Ђerential operator, in (6.1) we can con-
sider an elliptic one.

15.7 Notes
The spectrum of operator matrices and related problems were investigated
in many works cf. (Kovarik, 1975, 1977 and 1980), (Kovarik and Sherif,
1985), (Gaur and Kovarik, 1991), (Stampli, 1964), (Davis, 1958) and refer-
ences given therein. In particular, in the paper (Kovarik, 1975), the Gersh-
gorin type bounds for spectra of operator matrices with bounded operator
entries are derived. They generalize the well-known results for block-matrices
(Varga, 1965), (Levinger and Varga, 1966). But the Gershgorin-type bounds
give good results in the cases when the diagonal operators are dominant.
Theorem 15.1.1 improves the Gershgorin type bounds for operator matri-
ces, which are close to triangular ones. Moreover, we also consider unbounded
operators.
Proposition 15.6.2 on the bounds for the spectrum of a matrix diп¬Ђerential
operator supplements the well known results on diп¬Ђerential operators, cf.
(Edmunds and Evans, 1990), (Egorov and Kondratiev, 1996) and references
therein.
The material in this chapter is taken from the paper (GilвЂ™, 2001).

References
 Davis, C. C. (1958). Separation of two subspaces. Acta Sci. Math.
(Szeged) 19, 172-187.
 Edmunds, D.E. and Evans W.D. (1990). Spectral Theory and Diп¬Ђerential
Operators. Clarendon Press, Oxford.
 Egorov, Y and Kondratiev, V. (1996). Spectral Theory of Elliptic Oper-
ators. BirkhВЁuser Verlag, Basel.
a
 Gaur A.K. and Kovarik, Z. V. (1991). Norms, states and numerical
ranges on direct sums, Analysis 11, 155-164.
 GilвЂ™, M. I. (2001). Invertibility conditions and bounds for spectra of
operator matrices. Acta Sci. Math, 67/1, 353-368
15.7. Notes 213

 Kato, T. (1966). Perturbation Theory for Linear Operators, Springer-
Verlag. New York.
 Kovarik, Z. V. (1975). Spectrum localization in Banach spaces II, Linear
Algebra and Appl. 12, 223-229 .
 Kovarik, Z. V. (1977). Similarity and interpolation between projectors,
Acta Sci. Math. (Szeged), 39, 341-351

 Kovarik, Z. V. (1980). Manifolds of frames of projectors, Linear Algebra
and Appl. 31, 151-158.
 Kovarik, Z. V. and Sherif, N. (1985). Perturbation of invariant subspaces,
Linear Algebra and Appl. 64, 93-113.
 Levinger, B.W. and Varga, R.S. (1966). Minimal Gershgorin sets II,
Paciп¬Ѓc. J. Math., 17, 199-210.
 Stampli, J. (1964), Sums of projectors, Duke Math. J., 31, 455-461.

 Varga, R.S. (1965). Minimal Gershgorin sets, Paciп¬Ѓc. J. Math., 15, 719-
729.
16. Hille - Tamarkin
Integral Operators

In the present chapter, the Hille-Tamarkin integral operators on space Lp [0, 1]
are considered. Invertibility conditions, estimates for the norm of the inverse
operators and positive invertibility conditions are established. In addition,
diп¬Ђerential operators and integro-diп¬Ђerential ones are also discussed.

16.1 Invertibility Conditions
Recall that Lp в‰Ў Lp [0, 1] (1 < p < в€ћ) is the space of scalar-valued functions
deп¬Ѓned on [0, 1] and equipped with the norm
1
|h(s)|p ds]1/p .
|h|Lp = [
0

Everywhere below K is a linear operator in Lp deп¬Ѓned by
Лњ
1
K(x, s)h(s)ds (h в€€ Lp , x в€€ [0, 1]),
Лњ
(Kh)(x) = (1.1)
0

where K(x, s) is a scalar kernel deп¬Ѓned on [0, 1]2 and having the property
1 1
|K(x, s)|q ds]p/q dx]1/p < в€ћ (pв€’1 + q в€’1 = 1).
Mp (K) в‰Ў [ [ (1.2)
0 0

Лњ
That is, K is a Hille-Tamarkin operator (Pietsch, 1987, p. 245). Deп¬Ѓne the
Volterra operators
x
(Vв€’ h)(x) = K(x, s)h(s)ds (1.3)
0

M.I. GilвЂ™: LNM 1830, pp. 215вЂ“226, 2003.
c Springer-Verlag Berlin Heidelberg 2003
216 16. Hille - Tamarkin Integral Operators

and
1
(V+ h)(x) = K(x, s)h(s)ds. (1.4)
x
Set
1 t
|K(t, s)|q ds)p/q dt]1/p ,
Mp (Vв€’ ) в‰Ў [ (
0 0
1 1
|K(t, s)|q ds)p/q dt]1/p
Mp (V+ ) в‰Ў [ (
0 t
and
в€ћ
k
Mp (VВ± )
В±
в€љ
в‰Ў
Jp .
p
k!
k=0

Now we are in a position to formulate the main result of the chapter.
Theorem 16.1.1 Let the conditions (1.2) and
+в€’ + в€’
Jp Jp < Jp + Jp (1.5)

hold. Then operator I в€’ K is boundedly invertible in Lp and the inverse
Лњ
operator satisп¬Ѓes the inequality
в€’+
Jp Jp
|(I в€’ K)в€’1 |Lp в‰¤
Лњ . (1.6)
+ в€’ +в€’
Jp + Jp в€’ Jp Jp
The proof of this theorem is presented in the next two sections.
Note that condition (1.5) is equivalent to the following one:
+ в€’
Оё(K) в‰Ў (Jp в€’ 1)(Jp в€’ 1) < 1. (1.7)

Besides (1.6) takes the form
в€’+
Jp Jp
|(I в€’ K)в€’1 |Lp в‰¤
Лњ . (1.8)
1 в€’ Оё(K)
Due to HВЁlderвЂ™s inequality, for arbitrary a > 1
o
в€ћ k
Mp (VВ± )
в€љ в‰¤
p
k!
k=0

в€ћ в€ћ
apk Mp (VВ± ) 1/p
kp
p p
в€’qk 1/q
= (1 в€’ aв€’q )в€’1/q ea Mp (VВ± )/p .
[ a ] [ ]
k!
k=0 k=0

Take a = 21/p . Then p
Jp в‰¤ mp e2Mp (VВ± )/p
В±
(1.9)
where
mp = (1 в€’ 2в€’q/p )в€’1/q .
16.2. Preliminaries 217

Since,
p p p
Mp (K) = Mp (Vв€’ ) + Mp (V+ ), (1.10)
we have
p
Jp Jp в‰¤ m2 e2Mp (K)/p .
в€’+
p

Now relation (1.9) and Theorem 16.1.1 imply

Corollary 16.1.2 Let the conditions (1.2) and
p p p
mp e2Mp (K)/p < e2Mp (Vв€’ )/p + e2Mp (V+ )/p

hold. Then operator I в€’ K is boundedly invertible in Lp and the inverse
Лњ
operator satisп¬Ѓes the inequality
p
mp e2Mp (K)/p
в€’1
Лњ
|(I в€’ K) |Lp в‰¤ 2M p (V )/p .
p p
e p в€’ + e2Mp (V+ )/p в€’ mp e2Mp (K)/p

16.2 Preliminaries
Лњ
Let X be a Banach space with a norm . . Recall that a linear operator V
in X is called a quasinilpotent one if

n
Лњ
V n = 0.
lim
nв†’в€ћ

Лњ
For a quasinilpotent operator V in X, put
в€ћ
Vk .
Лњ Лњ
j(V ) в‰Ў
k=0

Lemma 16.2.1 Let A be a bounded linear operator in X of the form

A = I + V + W, (2.1)

where operators V and W are quasinilpotent. If, in addition, the condition
в€ћ
(в€’1)k+j V k W j < 1
ОёA в‰Ў (2.2)
j,k=1

is fulп¬Ѓlled, then operator A is boundedly invertible and the inverse operator
satisп¬Ѓes the inequality
j(V )j(W )
Aв€’1 в‰¤ .
1 в€’ ОёA
218 16. Hille - Tamarkin Integral Operators

Proof: We have

A = I + V + W = (I + V )(I + W ) в€’ V W. (2.3)

Since W and V are quasinilpotent, the operators, I + V and I + W are
invertible:
в€ћ в€ћ
в€’1 k k в€’1
(в€’1)k W k .
(I + V ) = (в€’1) V , (I + W ) = (2.4)
k=0 k=0

Thus,

A = I + V + W = (I + V )[I в€’ (I + V )в€’1 V W (I + W )в€’1 ](I + W ) =

(I + V )(I в€’ BA )(I + W ) (2.5)
where
BA = (I + V )в€’1 V W (I + W )в€’1 . (2.6)
But according to (2.4)
в€ћ в€ћ
в€’1 kв€’1 k в€’1
(в€’1)kв€’1 W k .
V (I + V ) = (в€’1) V , (I + W ) = (2.7)
k=1 k=1

So
в€ћ
(в€’1)k+j V k W j .
BA = (2.8)
j,k=1

If (2.2) holds, then BA < 1 and

(I в€’ BA )в€’1 в‰¤ (1 в€’ ОёA )в€’1 .

So due to (2.5) I + V + W is invertible. Moreover,

Aв€’1 = (I + W )в€’1 (I в€’ BA )в€’1 (I + V )в€’1 . (2.9)

But (2.4) implies

(I + W )в€’1 в‰¤ j(W ), (I + V )в€’1 в‰¤ j(V ).

Now the required inequality for Aв€’1 follows from (2.9). 2
Furthermore, take into account that by (2.7)
в€ћ
в€’1
V k в‰¤ j(V ) в€’ 1.
в‰¤
V (I + V ) (2.10)
k=1

Similarly,
W (I + W )в€’1 в‰¤ j(W ) в€’ 1. (2.11)
16.3. Powers of Volterra Operators 219

Thus
ОёA в‰¤ (j(W ) в€’ 1)(j(V ) в€’ 1).
So condition (2.2) is provided by the inequality

(j(W ) в€’ 1)(j(V ) в€’ 1) < 1.

The latter inequality is equivalent to the following one:

j(W )j(V ) < j(W ) + j(V ) (2.12)

Lemma 16.2.1 yields
Corollary 16.2.2 Let V, W be quasinilpotent and condition (2.12) be ful-
п¬Ѓlled. Then operator A deп¬Ѓned by (2.1) is boundedly invertible and the in-
verse operator satisп¬Ѓes the inequality
j(V )j(W )
Aв€’1 в‰¤ .
j(W ) + j(V ) в€’ j(W )j(V )
Лњ
Let us turn now to integral operator K. Under condition (1.2), operators
VВ± are quasinilpotent due to the well-known Theorem V.6.2 (Zabreiko, et al.,
1968, p. 153). Now Corollary 16.2.2 yields.
Corollary 16.2.3 Let the conditions (1.2) and

j(V+ )j(Vв€’ ) < j(V+ ) + j(Vв€’ )
Лњ
be fulп¬Ѓlled. Then I в€’ K is boundedly invertible and the inverse operator
satisп¬Ѓes the inequality
j(Vв€’ )j(V+ )
|(I в€’ K)в€’1 |Lp в‰¤
Лњ .
j(Vв€’ ) + j(V+ ) в€’ j(Vв€’ )j(V+ )

16.3 Powers of Volterra Operators
Lemma 16.3.1 Under condition (1.2), operator Vв€’ deп¬Ѓned by (1.3) satisп¬Ѓes
the inequality
k
Mp (Vв€’ )
k
|Vв€’ |Lp в‰¤ в€љ (k = 1, 2, ...). (3.1)
p
k!
Proof: Employing HВЁlderвЂ™s inequality, we have
o
1 t
|Vв€’ h|p p K(t, s)h(s)ds|p dt в‰¤
|
=
L
0 0

1 t t
q p/q
|h(s1 )|p ds1 dt.
|K(t, s)| ds]
[
0 0 0
220 16. Hille - Tamarkin Integral Operators

Setting
t
|K(t, s)|q ds]p/q ,
w(t) = [ (3.2)
0
one can rewrite the latter relation in the form
1 s1
|Vв€’ h|p p |h(s2 )|p ds2 ds1 .
в‰¤ w(s1 )
L
0 0

Using this inequality, we obtain
1 s1
|Vв€’ h|p p kв€’1
k
|Vв€’ h(s2 )|p ds2 ds1 .
в‰¤ w(s1 )
L
0 0

Once more apply HВЁlderвЂ™s inequality :
o
1 s1 s2
|Vв€’ h|p p kв€’2
k
|Vв€’ h(s3 )|p ds3 ds2 ds1 .
в‰¤ w(s1 ) w(s2 )
L
0 0 0

Repeating these arguments, we arrive at the relation
1 s1 sk
|Vв€’ h|p p
k
|h(sk+1 )|p dsk+1 . . . ds2 ds1 .
в‰¤ w(s1 ) w(s2 ) . . .
L
0 0 0

Taking
1
|h|p p = |h(s)|p ds = 1,
L
0
we get
1 s1 skв€’1
|Vв€’ |p p в‰¤
k
w(s1 ) w(s2 ) . . . dsk . . . ds2 ds1 . (3.3)
L
0 0 0
It is simple to see that
1 skв€’1
w(s1 ) . . . w(sk )dsk . . . ds1 =
0 0

Вµ
Лњ z1 zkв€’1
Вµk
Лњ
... dzk dzkв€’1 . . . dz1 = ,
k!
0 0 0
where sk
zk = zk (sk ) в‰Ў w(s)ds
0
and
1
Вµ=
Лњ w(s)ds.
0

Thus (3.3) gives
1
w(s)ds)k
(
|Vв€’ |p p
k 0
в‰¤ .
L
k!

But according to (3.2)
1
p
Вµ=
Лњ w(s)ds = Mp (Vв€’ ).
0

Therefore,
M pk (Vв€’ )
|Vв€’ |p p
k
в‰¤ .
L
k!
As claimed. 2

Similarly, the inequality
k
Mp (V+ )
k
в€љ
|V+ |Lp в‰¤ (3.4)
p
k!
can be proved.
The assertion of Theorem 16.1.1 follows from Corollary 16.2.3 and rela-
tions (3.1), (3.4).

a Hille - Tamarkin Operator
Set
в€ћ
z k Mp (VВ± )
k
В±
в€љ
в‰Ў (z в‰Ґ 0).
Jp (z) p
k!
k=0
В± В±
So Jp = Jp (1). Clearly,

О»I в€’ K = О»(I в€’ О»в€’1 K) (О» = 0).
Лњ Лњ

Consequently, if
Jp (|О»|в€’1 )Jp (|О»|в€’1 ) < Jp (|О»|в€’1 ) + Jp (|О»|в€’1 ),
+ в€’ + в€’

Лњ
then due to Theorem 16.1.1, О»I в€’ K is boundedly invertible. We thus get
Lemma 16.4.1 Under condition (1.2), any point О» = 0 of the spectrum
Лњ Лњ
Пѓ(K) of operator K satisп¬Ѓes the inequality
Jp (|О»|в€’1 )Jp (|О»|в€’1 ) в‰Ґ Jp (|О»|в€’1 ) + Jp (|О»|в€’1 ).
+ в€’ + в€’
(4.1)
Лњ Лњ Лњ
Let rs (K) = sup |Пѓ(K)| be the spectral radius of K. Then (4.1) yields
+ в€’1 Лњ в€’ в€’1 Лњ + в€’1 Лњ в€’ в€’1 Лњ
Jp (rs (K))Jp (rs (K)) в‰Ґ Jp (rs (K)) + Jp (rs (K)). (4.2)
Note that according to (1.9) and (4.2) we have
p p p p
2(Mp (Vв€’ ) + Mp (V+ )) 2Mp (Vв€’ ) 2Mp (V+ )
] в‰Ґ exp [ p
mp exp [ ] + exp [ p ].
pЛњ Лњ Лњ
rs (K)p rs (K)p rs (K)p
222 16. Hille - Tamarkin Integral Operators

Theorem 16.4.2 Under condition (1.2), let Vв€’ = 0 and V+ = 0. Then the
equation
+ в€’ + в€’
Jp (z)Jp (z) = Jp (z) + Jp (z) (4.3)
has a unique positive zero z(K). Moreover, the inequality rs (K) в‰¤ z в€’1 (K)
Лњ
is valid.
Proof: Equation (4.3) is equivalent to the following one:
+ в€’
(Jp (z) в€’ 1)(Jp (z) в€’ 1) = 1. (4.4)

Clearly, this equation has a unique positive root. In addition, (4.2) is equiv-
alent to the relation
+ в€’1 Лњ в€’ в€’1 Лњ
(Jp (rs (K)) в€’ 1)(Jp (rs (K)) в€’ 1) в‰Ґ 1.

Hence the result follows, since the left part of equation (4.4) monotonically
increases. 2
Rewrite (4.4) as
в€ћ
z k Mp (Vв€’ ) в€ћ z j Mp (V+ )
k j
в€љ в€љ =1
p p
j!
k! j=1
k=1

Or
в€ћ
bk z k = 1
k=2

with
kв€’1 kв€’j j
Mp (Vв€’ )Mp (V+ )
bk = (k = 2, 3, ...).
j!(k в€’ j)!
p
j=1

Due to Lemma 8.3.1, with the notation
j
Оґ(K) = 2 max bj ,
j=2,3,...

we get z(K) в‰Ґ Оґ в€’1 (K). Now Theorem 16.4.2 yields
Лњ
Corollary 16.4.3 Under condition (1.2), the inequality rs (K) в‰¤ Оґ(K) is
true.
Theorem 16.4.2 and Corollary 16.4.3 are exact: if either Vв€’ в†’ 0, or V+ в†’ 0,
then z(K) в†’ в€ћ, Оґ(K) в†’ 0.

16.5 Nonnegative Invertibility
We will say that h в€€ Lp is nonnegative if h(t) is nonnegative for almost all
t в€€ [0, 1]; a linear operator A in Lp is nonnegative if Ah is nonnegative for
each nonnegative h в€€ Lp .
16.6. Applications 223

Theorem 16.5.1 Let the conditions (1.2), (1.5) and

K(t, s) в‰Ґ 0 (0 в‰¤ t, s в‰¤ 1) (5.1)

Лњ
hold. Then operator I в€’ K is boundedly invertible and the inverse operator
is nonnegative. Moreover,

(I в€’ K)в€’1 в‰Ґ I.
Лњ (5.2)

Лњ
Relation (2.9) with A = I в€’ K, W = Vв€’ and V = V+ implies
Proof:

(I в€’ K)в€’1 = (I в€’ V+ )в€’1 (I в€’ BK )в€’1 (I в€’ Vв€’ )в€’1
Лњ (5.3)

where
BK = (I в€’ V+ )в€’1 V+ Vв€’ (I в€’ Vв€’ )в€’1 .
Moreover, by (5.1) we have VВ± в‰Ґ 0. So due to (2.4), (I в€’ VВ± )в€’1 в‰Ґ 0 and
BK в‰Ґ 0. Relations (2.7) and (2.8) according to (2.10) and (2.11) imply

|BK |Lp в‰¤ (Jp (Vв€’ ) в€’ 1)(Jp (V+ ) в€’ 1),

since j(VВ± ) в‰¤ Jp (VВ± ). But (1.5) is equivalent to (1.7). We thus get |BK |Lp <
1. Consequently,
в€ћ
в€’1 k
(I в€’ BK ) BK в‰Ґ 0.
=
k=0

Now (5.3) implies the inequality (I в€’ K)в€’1 в‰Ґ 0. Since I в€’ K в‰¤ I, we have
Лњ Лњ
inequality (5.2). 2

16.6 Applications
Consider a diп¬Ђerential operator A deп¬Ѓned by

d2 h(x) dh(x)
(Ah)(x) = в€’ + w(x)h(x) (0 < x < 1, h в€€ Dom (A))
+ g(x)
dx2 dx
(6.1)
on the domain

Dom (A) = {h в€€ Lp : h в€€ Lp + boundary conditions }. (6.2)

the coeп¬ѓcients g, w в€€ Lp and are complex, in general. (6.3)
224 16. Hille - Tamarkin Integral Operators

Let an operator S be deп¬Ѓned on Dom (A) by
(Sh)(x) = в€’h (x), h в€€ Dom (A).
It is ssumed that S has the Green function G(t, s). So that,
1
в€’1
h)(x) в‰Ў G(x, s)h(s)ds в€€ Dom (A)
(S
0

for any h в€€ Lp . Besides, the derivative of the Green function in x satisп¬Ѓes
the inequality
1
|Gx (x, s)|q ds < в€ћ.
vrai supx (6.4)
0
Лњ
Thus, A = (I в€’ K)S, where
1 1
d
Лњ
(Kh)(x) = в€’(g(x) + w(x)) G(x, s)h(s)ds = K(x, s)h(s)ds
dx 0 0

with
K(x, s) = в€’g(x)Gx (x, s) в€’ w(x)G(x, s). (6.5)
We have
1 1
|g(x)Gx (x, s)|q ds]p/q dx =
[
0 0
1 1
p
|Gx (x, s)|q ds]p/q dx < в€ћ.
|g(x)| [
0 0
Similarly,
1 1
|w(x)G(x, s)|q ds]p/q dx =
[
0 0
1 1
p
|G(x, s)|q ds]p/q dx < в€ћ.
|w(x)| [
0 0
Thus, condition (1.2) holds. Take into account that by HВЁlderвЂ™s inequality
o
1 1
в€’1
G(x, s)h(s)ds|p dx]1/p в‰¤ bp (S)|h|Lp
|S |
h|Lp = [
0 0

where
1 1
|G(x, s)|q ds)p/q dx]1/p .
bp (S) = [ (
0 0
Since
Aв€’1 = S в€’1 (I в€’ K)в€’1 ,
Лњ
Theorem 16.1.1 immediately implies the following result:
Proposition 16.6.1 Under (6.3)-(6.5), let condition (1.5) hold. Then op-
erator A deп¬Ѓned by (6.1), (6.2) is boundedly invertible in Lp . In addition,
в€’+
bp (S)Jp Jp
в€’1
|A |Lp в‰¤ .
+ в€’ +в€’
Jp + Jp в€’ Jp Jp
16.7. Notes 225

16.6.2 An integro-diп¬Ђerential operator
On domain (6.2), let us consider the operator
1
d2 u(x)
(Eu)(x) = в€’ K0 (x, s)u(s)ds (u в€€ Dom (A), 0 < x < 1), (6.6)
+
dx2 0

where K0 is a kernel with the property
1 1
|K0 (x, s)|p ds dx < в€ћ. (6.7)
0 0

Let S be the same as in the previous subsection. Then we can write E =
Лњ Лњ
(I в€’ K)S where K is deп¬Ѓned by (1.1) with
1
K(x, s) = в€’ K0 (x, x1 )G(x1 , s)dx1 (6.8)
0

Лњ
So if I в€’ K is invertible, then E is invertible as well. By HВЁlderвЂ™s inequality
o
1 1 1
K0 (x, x1 )G(x1 , s)dx1 |q ds]p/q dx в‰¤
|
[
0 0 0

1 1 1
p
|G(x1 , s)|q dx1 ds]p/q .
|K0 (x, x1 )| dx1 dx [
0 0 0

That is, condition (1.2) holds. Since

E в€’1 = S в€’1 (I в€’ K)в€’1 ,
Лњ

Theorems 16.1.1 and 16.5.1 yield

Proposition 16.6.2 Under (6.4), (6.7) and (6.8), let condition (1.5) hold.
Then operator E deп¬Ѓned by (6.6), (6.2) is boundedly invertible in Lp and
в€’+
bp (S)Jp Jp
в€’1
|E |Lp в‰¤ .
+ в€’ +в€’
Jp + Jp в€’ Jp Jp

If, in addition, G в‰Ґ 0 and K0 в‰¤ 0, then E в€’1 is positive. Moreover,
1
в€’1 в€’1
h)(x) в‰Ґ (S
(E h)(x) = G(x, s)h(s)ds
0

for any nonnegative h в€€ Lp .
226 16. Hille - Tamarkin Integral Operators

16.7 Notes
A lot of papers and books are devoted to the spectrum of Hille-Tamarkin
integral operators. Mainly, the distributions of the eigenvalues are consid-
ered, cf. (Diestel et al., 1995), (KВЁnig, 1986), (Pietsch, 1987) and references
o
therein.
Theorem 16.4.2 and Corollary 16.4.3 improve the well-known estimate
1
Лњ
rs (K) в‰¤ sup |K(x, s)|ds
x 0

(KrasnoselвЂ™skii et al, 1989, Theorem 16.2) for operators, which are close to
Volterra ones.
The results of Section 16.6 supplement the well-known results on the
spectra of diп¬Ђerential operators, cf. (Edmunds and Evans, 1990), (Egorov
and Kondratiev, 1996), (Locker, 1999) and references therein.
The material in this chapter is taken from the paper (GilвЂ™, 2002).

References
 Diestel, D., Jarchow, H, Tonge, A. (1995), Absolutely Summing Opera-
tors, Cambridge University Press, Cambridge.
 Edmunds, D.E. and Evans W.D. (1990). Spectral Theory and Diп¬Ђerential
Operators. Clarendon Press, Oxford.
 Egorov, Y. and Kondratiev, V. (1996). Spectral Theory of Elliptic Oper-
ators. BirkhВЁuser Verlag, Basel.
a
 GilвЂ™, M.I. (2002). Invertibility and positive invertibility of Hille-Tamarkin
integral operators, Acta Math. Hungarica, 95 (1-2) 39-53.
 KВЁnig, H. (1986). Eigenvalue Distribution of Compact Operators,
o
BirkhВЁuser Verlag, Basel- Boston-Stuttgart.
a
 Locker, J. (1999). Spectral Theory of Non-Self-Adjoint Two Point Diп¬Ђer-
ential Operators., Amer. Math. Soc., Mathematical Surveys and Mono-
graphs, Volume 73.
 KrasnoselвЂ™skii, M. A., J. Lifshits, and A. Sobolev (1989). Positive Linear
Systems. The Method of Positive Operators, Heldermann Verlag, Berlin.
 Pietsch, A. (1987). Eigenvalues and s-Numbers, Cambridge University
Press, Cambridge.
 Zabreiko, P.P., Koshelev A.I., KrasnoselвЂ™skii, M. A., Mikhlin, S.G.,
Rakovshik, L.S. and B.Ya. Stetzenko (1968). Integral Equations, Nauka,
Moscow. In Russian
17. Integral Operators in
Space Lв€ћ

In the present chapter integral operators in space Lв€ћ [0, 1] are considered.
Invertibility conditions, estimates for the norm of the inverse operators and
positive invertibility conditions are established. In addition, bounds for the
erators and integro-diп¬Ђerential ones are also discussed.

17.1 Invertibility Conditions
Recall that Lв€ћ в‰Ў Lв€ћ [0, 1] is the space of scalar-valued functions deп¬Ѓned on
[0, 1] and equipped with the norm

|h|Lв€ћ = ess sup |h(x)| (h в€€ Lв€ћ ).
xв€€[0,1]

Everywhere in this chapter K is a linear operator in Lв€ћ deп¬Ѓned by
Лњ
1
K(x, s)h(s)ds (h в€€ Lв€ћ , x в€€ [0, 1]),
Лњ
(Kh)(x) = (1.1)
0

where K(x, s) is a scalar kernel deп¬Ѓned on [0, 1]2 and having the property
1
ess sup |K(x, s)|ds < в€ћ. (1.2)
xв€€[0,1]
0

Deп¬Ѓne the Volterra operators
x
(Vв€’ h)(x) = K(x, s)h(s)ds (1.3)
0

M.I. GilвЂ™: LNM 1830, pp. 227вЂ“234, 2003.
c Springer-Verlag Berlin Heidelberg 2003
17. Integral Operators in Lв€ћ
228

and
1
(V+ h)(x) = K(x, s)h(s)ds. (1.4)
x
Set
wв€’ (s) в‰Ў ess |K(x, s)|,
sup
0в‰¤sв‰¤xв‰¤1

w+ (s) в‰Ў ess |K(x, s)|
sup
0в‰¤xв‰¤sв‰¤1

and
1
Mв€ћ (VВ± ) в‰Ў wВ± (s)ds.
0
Now we are in a position to formulate the main result of the chapter.
Theorem 17.1.1 Let the conditions (1.2) and

eMв€ћ (Vв€’ )+Mв€ћ (V+ ) < eMв€ћ (V+ ) + eMв€ћ (Vв€’ ) (1.5)

hold. Then operator I в€’ K is boundedly invertible in Lв€ћ and the inverse
Лњ
operator satisп¬Ѓes the inequality

eMв€ћ (Vв€’ )+Mв€ћ (V+ )
|(I в€’ K)в€’1 |Lв€ћ в‰¤
Лњ . (1.6)
eMв€ћ (V+ ) + eMв€ћ (Vв€’ ) в€’ eMв€ћ (Vв€’ )+Mв€ћ (V+ )
The proof of this theorem is presented in the next section.
Note that condition (1.5) is equivalent to the following one:

Оё(K) в‰Ў (eMв€ћ (V+ ) в€’ 1)(eMв€ћ (Vв€’ ) в€’ 1) < 1. (1.7)

Besides (1.6) takes the form
Mв€ћ (Vв€’ )+Mв€ћ (V+ )
Лњ в€’1 |Lв€ћ в‰¤ e
|(I в€’ K) . (1.8)
1 в€’ Оё(K)

17.2 Proof of Theorem 17.1.1
Under condition (1.2), operators VВ± are quasinilpotent due to the well-known
Theorem V.6.2 (Zabreiko, et al., 1968, p. 153). Now Corollary 16.2.2 yields.
Lemma 17.2.1 With the notation
в€ћ
k
j(VВ± ) в‰Ў |VВ± |Lв€ћ ,
k=0
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