E
jj
Moreover, due to Lemma 15.1.4, with the notation
j’1
n
vA := max ±jk and wA := max ±jk
j j
k=j+1 k=1
we have
D’1 V ¤ vA and D’1 W ¤ wA .
Now Corollary 15.1.3 implies
Lemma 15.3.1 Let the conditions (3.1)(3.3) and
n’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
n’1 n’1
(1 ’ vA )(1 ’ wA )
vA wA < 1. (3.5)
(1 ’ vA )(1 ’ wA )
15.4. Matrices with Bounded o¬ Diagonals 205
Furthermore, under (3.1), (3.2) assume that for all regular » of D, the rela
tions
n
sup φk (t)(» ’ t)’1  < ∞ and
v (») := max
˜
j
k=j+1 t∈σ(Sj )
j’1
sup φk (t)(» ’ t)’1  < ∞.
w(») := max
˜ (3.6)
j
k=1 t∈σ(Sj )
are ful¬lled. Due to Lemma 15.1.4 with the notation from (2.1), we have
˜ ˜
W (») ¤ w(») and V (») ¤ v (»).
˜ ˜
Now Theorem 15.2.1 gives.
Lemma 15.3.2 Under conditions (3.1), (3.2) and (3.6), the spectrum of
operator A de¬ned by (1.1) lies in the union of the sets σ(D) and
n’1
v k (»)wj (») ≥ 1}.
{» ∈ C : ˜ ˜
j,k=1
15.4 Operator Matrices with Bounded
o¬ Diagonal Entries
Again assume that H is an orthogonal sum of the same Hilbert spaces Ek ≡ E
(k = 1, ..., n). In addition, Ajk (j = k) are arbitrary bounded operators:
v0 ≡ V < ∞, w0 ≡ W < ∞. (4.1)
Lemma 15.4.1 Let the conditions (3.1),(4.1) and
n’1
kj
ρ’k’j (D, 0)v0 w0 < 1
j,k=1
hold. Then operator (1.1) is invertible.
Proof: Since D is normal,
(D ’ »IH )’1 = ρ’1 (D, »). (4.2)
Therefore D’1 = ρ’1 (D, 0). Due to (4.1)
D’1 V ¤ D’1 ¤ ρ’1 (D, 0)v0 .
V
Similarly, D’1 W ¤ ρ’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
n’2
n’j’1 n’k’1
z k+j v0 = z 2(n’1) .
w0 (4.3)
j,k=0
Then the spectral variation of A with respect to D satis¬es the inequality
svD (A) ¤ z1 .
Proof: With the substitution y = 1/z equation (4.3) takes the form
n’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
n’1
jk
p(A) := v0 w0 < 1, (4.5)
j,k=1
2(n’1)
¤ p(A). So under (4.5), the previous lemma gives the inequality
then z1
svD (A) ¤ 2(n’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 HilbertSchmidt norm.
Clearly,
2 2
N2 (V ) = N2 (Ajk ),
1¤j<k¤n
and
2 2
N2 (W ) = N2 (Ajk ).
1¤k<j¤n
Lemma 15.4.3 Under condition (3.1), let V and W be HilbertSchmidt op
erators and the inequality
n’1
j
(k!j!)’1/2 N2 (V )N2 (W )ρ’k’j (D, 0) < 1
k
(4.7)
j,k=1
hold. Then operator A de¬ned by (1.1) is invertible.
15.5. HilbertSchmidt Diagonals 207
Proof: Due to (4.7) D’1 is bounded, D’1 V and D’1 W are Hilbert Schmidt
operators. In addition, according to (1.5), they are nilpotent. Due to Corol
lary 6.9.2,
(D’1 V )k ¤ N2 (D’1 V )(k!)’1/2 , (D’1 W )k ¤ N2 (D’1 W )(k!)’1/2
k k
(4.8)
for any natural k < n. But
N2 (D’1 V ) ¤ D’1 N2 (V ) = ρ’1 (D, 0)N2 (V ).
Similarly,
N2 (D’1 W ) ¤ ρ’1 (D, 0)N2 (W ).
Now Theorem 15.1.1 implies the required result. 2
15.5 Operator Matrices with HilbertSchmidt
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 HilbertSchmidt operators .
(5.1)
Due to Theorem 6.4.1, for any HilbertSchmidt operator K in H, we can
write
∞
g k (K)
’1
√
(K ’»I) ¤ G(K, ρ(», K)) ≡ for all regular », (5.2)
k!ρk+1 (K, »)
k=0
where
∞
g k (K)
√
G(K, y) ≡ (y > 0)
k!y k+1
k=0
and
∞
2
»k (K)2 )1/2 (K ∈ C2 ).
’
g(K) = (N2 (K)
k=1
If K is a normal operator: KK — = K — K, then g(K) = 0. The following
relations are true:
g 2 (K) ¤ N2 (K) ’ T race(K 2 )
2
and
12 —
g 2 (K) ¤ N (K ’ K) (K ∈ C2 )
22
208 15. Operator Matrices
(see Section 6.3). So due to (5.1) and (5.2)
(Kj ’ »I)’1 ¤ G(Kj , ρ(Kj , »)) (5.3)
Ej
and therefore with » = ’1 we get
D’1 = max (IE + Kj )’1 ¤ b0 (D), (5.4)
Ej
j
where
∞
g k (Kj )
√
b0 (D) = max G(Kj , ρ(Kj , ’1)) = max .
k!ρk+1 (Kj , ’1)
j j
k=0
Lemma 15.5.1 Let the conditions (4.1), (5.1) and
n’1
bk+j (D)v0 w0 < 1
kj
(5.5)
0
j,k=1
be ful¬lled. Then operator A de¬ned by (1.1) is invertible.
Proof: Due to (4.1) and (5.4)
D’1 V ¤ D’1 v0 ¤ b0 (D)v0 .
Similarly,
D’1 W ¤ 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 de¬ned by (1.1), lies in the set
∪n „¦j (»),
j=1
where
n’1
Gs+k (Kj , ρ(Kj , » ’ 1))v0 w0 ≥ 1}.
ks
„¦j (») = {» ∈ C :
s,k=1
In other words, for any µ ∈ σ(A), there are an integer j and a
»(Kj ) ∈ σ(Kj ),
such that
n’1
Gs+k (Kj , »(Kj ) + 1 ’ µ)v0 w0 ≥ 1.
ks
s,k=1
15.6. Example 209
Put
∞
g0 y k+1
k
√
h(y, D) = ,
k!
k=0
where g0 = maxj g(Kj ). Let us consider the scalar equation
n’1
hs+l (y, D)v0 w0 = 1.
ls
(5.6)
s,l=1
Thanks to (5.3), one can write
(D ’ »IH )’1 = max (IE + Kj ’ »IE )’1 ¤ h(ρ’1 (D, »), 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 de¬ned by (1.1) with
respect to D satis¬es the inequality
svD (A) ¤ x’1 .
0
15.6 Example
Let H = L2 ([0, π], Cn ) be the Hilbert space of functions de¬ned on [0, π] with
values in a Euclidean space Cn and the scalar product
π
(h, w)H = (h(x), w(x))C n dx,
0
where (., .)C n is the scalar product in Cn . Consider the operator A de¬ned
by the expression
d du(x)
Au(x) = ’ + B0 (x)u(x) (u ∈ Dom (A), 0 < x < π) (6.1)
d0 (x)
dx dx
on the domain
Dom (A) = {u ∈ H, u ∈ H, u(0) = u(π) = 0 }, (6.2)
with continuous real n — nmatrices
d0 (x) = diag [a1 (x), ..., an (x)], B0 (x) = (bjk (x))n
j,k=1 ,
where functions aj (x) are di¬erentiable and positive:
˜
dj ≡ min aj (x) > 0. (6.3)
x
210 15. Operator Matrices
Take E = L2 ([0, π], C1 ) and de¬ne operators Ajk by
d dv(x)
Ajj v(x) = ’ aj (x) + bjj (x)v(x)
dx dx
(v ∈ Dom (Ajj ), 0 < x < π),
where
Dom (Ajj ) = {v ∈ E : v ∈ E, v(0) = v(π) = 0 }
and
Ajk v(x) = bjk (x)v(x) (v ∈ E; 0 < x < π, j = k).
Assume that
˜
βj ≡ inf bjj (x) + dj > 0 (j = 1, ..., n). (6.4)
x
Omitting simple calculations, we have
˜
(Ajj v(x), v)E = (aj v , v )E + (bjj v, v)E ≥ dj (v , v )E + (bjj v, v)E ≥
˜
dj (v, v)E + (bjj v, v)E = βj (v, v)E .
Consequently,
A’1 ’1
¤ βj (j = 1, ..., n).
E
jj
Clearly,
¤ max bjk (x).
Ajk E
x
So
A’1 Ajk ’1
¤ βj max bjk (x).
E
jj x
With the notation
n’1 n
’2
βj max bjk (x)2 )1/2 .
vA ≡ (
˜
x
j=1 k=j+1
and
n j’1
’2
βj max bjk (x)2 )1/2
wA ≡ (
˜
x
j=2 k=1
we have
D’1 V ¤ vA ,
˜
and
D’1 W ¤ wA .
˜
Now Corollary 15.1.3 yields
15.6. Example 211
Proposition 15.6.1 Let the conditions (6.3), (6.4) and
n’1
˜k ˜ j
vA wA < 1 (6.5)
j,k=1
hold. Then operator A de¬ned by (6.1), (6.2) is invertible. In particular, let
vA , wA = 1.
˜˜
Then (6.5) can be written in the form
˜n’1 ˜ n’1
(1 ’ vA )(1 ’ wA )
vA wA
˜˜ < 1.
(1 ’ vA )(1 ’ wA )
˜ ˜
In the case n = 2 one can write
’1 ’1
vA = β1 max b12 (x), wA = β2 max b21 (x).
˜ ˜
x x
Inequality (6.5) takes the form
max b12 (x) max b21 (x) < β1 β2 .
x x
To investigate the spectrum of operator (6.1) assume for simplicity that
aj ≡ const > 0, bjj ≡ const (j = 1, 2, ...). (6.6)
Then it is simple to check that the eigenvalues of Ajj are
»k (Ajj ) = aj k 2 + bjj (k = 1, 2, ...).
Denote
n’1 n
max bjk (x)2 )1/2
vb ≡ (
x
j=1 k=j+1
and
n j’1
max bjk (x)2 )1/2 .
wb ≡ (
x
j=1 k=2
Clearly,
¤ vb , W ¤ wb .
V
Now Lemma 15.4.2 yields
Proposition 15.6.2 Let z2 be the unique nonnegative root of the algebraic
equation
n’2
n’j’1 n’k’1
z k+j vb = z 2(n’1) .
wb (6.7)
j,k=0
Then the spectral variation of A with respect to D satis¬es the inequality
svD (A) ¤ z2 .
212 15. Operator Matrices
In other words for any µ ∈ σ(A), there are natural k = 1, 2, ... and j ¤ n,
such that
µ ’ aj k 2 ’ bjj  ¤ z2 .
In particular, if n = 2, then
√
z2 = vb wb = [max b12 (x) max b21 (x)]1/2 .
x x
Certainly, instead of the ordinary di¬erential 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 wellknown results for blockmatrices
(Varga, 1965), (Levinger and Varga, 1966). But the Gershgorintype 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 di¬erential
operator supplements the well known results on di¬erential operators, cf.
(Edmunds and Evans, 1990), (Egorov and Kondratiev, 1996) and references
therein.
The material in this chapter is taken from the paper (Gil™, 2001).
References
[1] Davis, C. C. (1958). Separation of two subspaces. Acta Sci. Math.
(Szeged) 19, 172187.
[2] Edmunds, D.E. and Evans W.D. (1990). Spectral Theory and Di¬erential
Operators. Clarendon Press, Oxford.
[3] Egorov, Y and Kondratiev, V. (1996). Spectral Theory of Elliptic Oper
ators. Birkh¨user Verlag, Basel.
a
[4] Gaur A.K. and Kovarik, Z. V. (1991). Norms, states and numerical
ranges on direct sums, Analysis 11, 155164.
[5] Gil™, M. I. (2001). Invertibility conditions and bounds for spectra of
operator matrices. Acta Sci. Math, 67/1, 353368
15.7. Notes 213
[6] Kato, T. (1966). Perturbation Theory for Linear Operators, Springer
Verlag. New York.
[7] Kovarik, Z. V. (1975). Spectrum localization in Banach spaces II, Linear
Algebra and Appl. 12, 223229 .
[8] Kovarik, Z. V. (1977). Similarity and interpolation between projectors,
Acta Sci. Math. (Szeged), 39, 341351
[9] Kovarik, Z. V. (1980). Manifolds of frames of projectors, Linear Algebra
and Appl. 31, 151158.
[10] Kovarik, Z. V. and Sherif, N. (1985). Perturbation of invariant subspaces,
Linear Algebra and Appl. 64, 93113.
[11] Levinger, B.W. and Varga, R.S. (1966). Minimal Gershgorin sets II,
Paci¬c. J. Math., 17, 199210.
[12] Stampli, J. (1964), Sums of projectors, Duke Math. J., 31, 455461.
[13] Varga, R.S. (1965). Minimal Gershgorin sets, Paci¬c. J. Math., 15, 719
729.
16. Hille  Tamarkin
Integral Operators
In the present chapter, the HilleTamarkin 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,
bounds for the spectral radius are suggested. Applications to nonselfadjoint
di¬erential operators and integrodi¬erential ones are also discussed.
16.1 Invertibility Conditions
Recall that Lp ≡ Lp [0, 1] (1 < p < ∞) is the space of scalarvalued functions
de¬ned on [0, 1] and equipped with the norm
1
h(s)p ds]1/p .
hLp = [
0
Everywhere below K is a linear operator in Lp de¬ned by
˜
1
K(x, s)h(s)ds (h ∈ Lp , x ∈ [0, 1]),
˜
(Kh)(x) = (1.1)
0
where K(x, s) is a scalar kernel de¬ned on [0, 1]2 and having the property
1 1
K(x, s)q ds]p/q dx]1/p < ∞ (p’1 + q ’1 = 1).
Mp (K) ≡ [ [ (1.2)
0 0
˜
That is, K is a HilleTamarkin operator (Pietsch, 1987, p. 245). De¬ne the
Volterra operators
x
(V’ h)(x) = K(x, s)h(s)ds (1.3)
0
M.I. Gil™: LNM 1830, pp. 215“226, 2003.
c SpringerVerlag 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 (V’ ) ≡ [ (
0 0
1 1
K(t, s)q ds)p/q dt]1/p
Mp (V+ ) ≡ [ (
0 t
and
∞
k
Mp (V± )
±
√
≡
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
+’ + ’
Jp Jp < Jp + Jp (1.5)
hold. Then operator I ’ K is boundedly invertible in Lp and the inverse
˜
operator satis¬es the inequality
’+
Jp Jp
(I ’ K)’1 Lp ¤
˜ . (1.6)
+ ’ +’
Jp + Jp ’ 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:
+ ’
θ(K) ≡ (Jp ’ 1)(Jp ’ 1) < 1. (1.7)
Besides (1.6) takes the form
’+
Jp Jp
(I ’ K)’1 Lp ¤
˜ . (1.8)
1 ’ θ(K)
Due to H¨lder™s inequality, for arbitrary a > 1
o
∞ k
Mp (V± )
√ ¤
p
k!
k=0
∞ ∞
apk Mp (V± ) 1/p
kp
p p
’qk 1/q
= (1 ’ a’q )’1/q ea Mp (V± )/p .
[ a ] [ ]
k!
k=0 k=0
Take a = 21/p . Then p
Jp ¤ mp e2Mp (V± )/p
±
(1.9)
where
mp = (1 ’ 2’q/p )’1/q .
16.2. Preliminaries 217
Since,
p p p
Mp (K) = Mp (V’ ) + Mp (V+ ), (1.10)
we have
p
Jp Jp ¤ m2 e2Mp (K)/p .
’+
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 (V’ )/p + e2Mp (V+ )/p
hold. Then operator I ’ K is boundedly invertible in Lp and the inverse
˜
operator satis¬es the inequality
p
mp e2Mp (K)/p
’1
˜
(I ’ K) Lp ¤ 2M p (V )/p .
p p
e p ’ + e2Mp (V+ )/p ’ mp e2Mp (K)/p
16.2 Preliminaries
˜
Let X be a Banach space with a norm . . Recall that a linear operator V
in X is called a quasinilpotent one if
n
˜
V n = 0.
lim
n’∞
˜
For a quasinilpotent operator V in X, put
∞
Vk .
˜ ˜
j(V ) ≡
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
∞
(’1)k+j V k W j < 1
θA ≡ (2.2)
j,k=1
is ful¬lled, then operator A is boundedly invertible and the inverse operator
satis¬es the inequality
j(V )j(W )
A’1 ¤ .
1 ’ θA
218 16. Hille  Tamarkin Integral Operators
Proof: We have
A = I + V + W = (I + V )(I + W ) ’ V W. (2.3)
Since W and V are quasinilpotent, the operators, I + V and I + W are
invertible:
∞ ∞
’1 k k ’1
(’1)k W k .
(I + V ) = (’1) V , (I + W ) = (2.4)
k=0 k=0
Thus,
A = I + V + W = (I + V )[I ’ (I + V )’1 V W (I + W )’1 ](I + W ) =
(I + V )(I ’ BA )(I + W ) (2.5)
where
BA = (I + V )’1 V W (I + W )’1 . (2.6)
But according to (2.4)
∞ ∞
’1 k’1 k ’1
(’1)k’1 W k .
V (I + V ) = (’1) V , (I + W ) = (2.7)
k=1 k=1
So
∞
(’1)k+j V k W j .
BA = (2.8)
j,k=1
If (2.2) holds, then BA < 1 and
(I ’ BA )’1 ¤ (1 ’ θA )’1 .
So due to (2.5) I + V + W is invertible. Moreover,
A’1 = (I + W )’1 (I ’ BA )’1 (I + V )’1 . (2.9)
But (2.4) implies
(I + W )’1 ¤ j(W ), (I + V )’1 ¤ j(V ).
Now the required inequality for A’1 follows from (2.9). 2
Furthermore, take into account that by (2.7)
∞
’1
V k ¤ j(V ) ’ 1.
¤
V (I + V ) (2.10)
k=1
Similarly,
W (I + W )’1 ¤ j(W ) ’ 1. (2.11)
16.3. Powers of Volterra Operators 219
Thus
θA ¤ (j(W ) ’ 1)(j(V ) ’ 1).
So condition (2.2) is provided by the inequality
(j(W ) ’ 1)(j(V ) ’ 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
¬lled. Then operator A de¬ned by (2.1) is boundedly invertible and the in
verse operator satis¬es the inequality
j(V )j(W )
A’1 ¤ .
j(W ) + j(V ) ’ j(W )j(V )
˜
Let us turn now to integral operator K. Under condition (1.2), operators
V± are quasinilpotent due to the wellknown 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(V’ ) < j(V+ ) + j(V’ )
˜
be ful¬lled. Then I ’ K is boundedly invertible and the inverse operator
satis¬es the inequality
j(V’ )j(V+ )
(I ’ K)’1 Lp ¤
˜ .
j(V’ ) + j(V+ ) ’ j(V’ )j(V+ )
16.3 Powers of Volterra Operators
Lemma 16.3.1 Under condition (1.2), operator V’ de¬ned by (1.3) satis¬es
the inequality
k
Mp (V’ )
k
V’ Lp ¤ √ (k = 1, 2, ...). (3.1)
p
k!
Proof: Employing H¨lder™s inequality, we have
o
1 t
V’ hp p K(t, s)h(s)dsp dt ¤

=
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
V’ hp p h(s2 )p ds2 ds1 .
¤ w(s1 )
L
0 0
Using this inequality, we obtain
1 s1
V’ hp p k’1
k
V’ h(s2 )p ds2 ds1 .
¤ w(s1 )
L
0 0
Once more apply H¨lder™s inequality :
o
1 s1 s2
V’ hp p k’2
k
V’ h(s3 )p ds3 ds2 ds1 .
¤ w(s1 ) w(s2 )
L
0 0 0
Repeating these arguments, we arrive at the relation
1 s1 sk
V’ hp p
k
h(sk+1 )p dsk+1 . . . ds2 ds1 .
¤ w(s1 ) w(s2 ) . . .
L
0 0 0
Taking
1
hp p = h(s)p ds = 1,
L
0
we get
1 s1 sk’1
V’ p p ¤
k
w(s1 ) w(s2 ) . . . dsk . . . ds2 ds1 . (3.3)
L
0 0 0
It is simple to see that
1 sk’1
w(s1 ) . . . w(sk )dsk . . . ds1 =
0 0
µ
˜ z1 zk’1
µk
˜
... dzk dzk’1 . . . dz1 = ,
k!
0 0 0
where sk
zk = zk (sk ) ≡ w(s)ds
0
and
1
µ=
˜ w(s)ds.
0
Thus (3.3) gives
1
w(s)ds)k
(
V’ p p
k 0
¤ .
L
k!
16.4. Spectral Radius 221
But according to (3.2)
1
p
µ=
˜ w(s)ds = Mp (V’ ).
0
Therefore,
M pk (V’ )
V’ p p
k
¤ .
L
k!
As claimed. 2
Similarly, the inequality
k
Mp (V+ )
k
√
V+ Lp ¤ (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).
16.4 Spectral Radius of
a Hille  Tamarkin Operator
Set
∞
z k Mp (V± )
k
±
√
≡ (z ≥ 0).
Jp (z) p
k!
k=0
± ±
So Jp = Jp (1). Clearly,
»I ’ K = »(I ’ »’1 K) (» = 0).
˜ ˜
Consequently, if
Jp (»’1 )Jp (»’1 ) < Jp (»’1 ) + Jp (»’1 ),
+ ’ + ’
˜
then due to Theorem 16.1.1, »I ’ K is boundedly invertible. We thus get
Lemma 16.4.1 Under condition (1.2), any point » = 0 of the spectrum
˜ ˜
σ(K) of operator K satis¬es the inequality
Jp (»’1 )Jp (»’1 ) ≥ Jp (»’1 ) + Jp (»’1 ).
+ ’ + ’
(4.1)
˜ ˜ ˜
Let rs (K) = sup σ(K) be the spectral radius of K. Then (4.1) yields
+ ’1 ˜ ’ ’1 ˜ + ’1 ˜ ’ ’1 ˜
Jp (rs (K))Jp (rs (K)) ≥ 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 (V’ ) + Mp (V+ )) 2Mp (V’ ) 2Mp (V+ )
] ≥ exp [ p
mp exp [ ] + exp [ p ].
p˜ ˜ ˜
rs (K)p rs (K)p rs (K)p
222 16. Hille  Tamarkin Integral Operators
Theorem 16.4.2 Under condition (1.2), let V’ = 0 and V+ = 0. Then the
equation
+ ’ + ’
Jp (z)Jp (z) = Jp (z) + Jp (z) (4.3)
has a unique positive zero z(K). Moreover, the inequality rs (K) ¤ z ’1 (K)
˜
is valid.
Proof: Equation (4.3) is equivalent to the following one:
+ ’
(Jp (z) ’ 1)(Jp (z) ’ 1) = 1. (4.4)
Clearly, this equation has a unique positive root. In addition, (4.2) is equiv
alent to the relation
+ ’1 ˜ ’ ’1 ˜
(Jp (rs (K)) ’ 1)(Jp (rs (K)) ’ 1) ≥ 1.
Hence the result follows, since the left part of equation (4.4) monotonically
increases. 2
Rewrite (4.4) as
∞
z k Mp (V’ ) ∞ z j Mp (V+ )
k j
√ √ =1
p p
j!
k! j=1
k=1
Or
∞
bk z k = 1
k=2
with
k’1 k’j j
Mp (V’ )Mp (V+ )
bk = (k = 2, 3, ...).
j!(k ’ j)!
p
j=1
Due to Lemma 8.3.1, with the notation
j
δ(K) = 2 max bj ,
j=2,3,...
we get z(K) ≥ δ ’1 (K). Now Theorem 16.4.2 yields
˜
Corollary 16.4.3 Under condition (1.2), the inequality rs (K) ¤ δ(K) is
true.
Theorem 16.4.2 and Corollary 16.4.3 are exact: if either V’ ’ 0, or V+ ’ 0,
then z(K) ’ ∞, δ(K) ’ 0.
16.5 Nonnegative Invertibility
We will say that h ∈ Lp is nonnegative if h(t) is nonnegative for almost all
t ∈ [0, 1]; a linear operator A in Lp is nonnegative if Ah is nonnegative for
each nonnegative h ∈ Lp .
16.6. Applications 223
Theorem 16.5.1 Let the conditions (1.2), (1.5) and
K(t, s) ≥ 0 (0 ¤ t, s ¤ 1) (5.1)
˜
hold. Then operator I ’ K is boundedly invertible and the inverse operator
is nonnegative. Moreover,
(I ’ K)’1 ≥ I.
˜ (5.2)
˜
Relation (2.9) with A = I ’ K, W = V’ and V = V+ implies
Proof:
(I ’ K)’1 = (I ’ V+ )’1 (I ’ BK )’1 (I ’ V’ )’1
˜ (5.3)
where
BK = (I ’ V+ )’1 V+ V’ (I ’ V’ )’1 .
Moreover, by (5.1) we have V± ≥ 0. So due to (2.4), (I ’ V± )’1 ≥ 0 and
BK ≥ 0. Relations (2.7) and (2.8) according to (2.10) and (2.11) imply
BK Lp ¤ (Jp (V’ ) ’ 1)(Jp (V+ ) ’ 1),
since j(V± ) ¤ Jp (V± ). But (1.5) is equivalent to (1.7). We thus get BK Lp <
1. Consequently,
∞
’1 k
(I ’ BK ) BK ≥ 0.
=
k=0
Now (5.3) implies the inequality (I ’ K)’1 ≥ 0. Since I ’ K ¤ I, we have
˜ ˜
inequality (5.2). 2
16.6 Applications
16.6.1 A nonselfadjoint di¬erential operator
Consider a di¬erential operator A de¬ned by
d2 h(x) dh(x)
(Ah)(x) = ’ + w(x)h(x) (0 < x < 1, h ∈ Dom (A))
+ g(x)
dx2 dx
(6.1)
on the domain
Dom (A) = {h ∈ Lp : h ∈ Lp + boundary conditions }. (6.2)
In addition,
the coe¬cients g, w ∈ Lp and are complex, in general. (6.3)
224 16. Hille  Tamarkin Integral Operators
Let an operator S be de¬ned on Dom (A) by
(Sh)(x) = ’h (x), h ∈ Dom (A).
It is ssumed that S has the Green function G(t, s). So that,
1
’1
h)(x) ≡ G(x, s)h(s)ds ∈ Dom (A)
(S
0
for any h ∈ Lp . Besides, the derivative of the Green function in x satis¬es
the inequality
1
Gx (x, s)q ds < ∞.
vrai supx (6.4)
0
˜
Thus, A = (I ’ K)S, where
1 1
d
˜
(Kh)(x) = ’(g(x) + w(x)) G(x, s)h(s)ds = K(x, s)h(s)ds
dx 0 0
with
K(x, s) = ’g(x)Gx (x, s) ’ 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 < ∞.
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 < ∞.
w(x) [
0 0
Thus, condition (1.2) holds. Take into account that by H¨lder™s inequality
o
1 1
’1
G(x, s)h(s)dsp dx]1/p ¤ bp (S)hLp
S 
hLp = [
0 0
where
1 1
G(x, s)q ds)p/q dx]1/p .
bp (S) = [ (
0 0
Since
A’1 = S ’1 (I ’ K)’1 ,
˜
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 de¬ned by (6.1), (6.2) is boundedly invertible in Lp . In addition,
’+
bp (S)Jp Jp
’1
A Lp ¤ .
+ ’ +’
Jp + Jp ’ Jp Jp
16.7. Notes 225
16.6.2 An integrodi¬erential operator
On domain (6.2), let us consider the operator
1
d2 u(x)
(Eu)(x) = ’ K0 (x, s)u(s)ds (u ∈ 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 < ∞. (6.7)
0 0
Let S be the same as in the previous subsection. Then we can write E =
˜ ˜
(I ’ K)S where K is de¬ned by (1.1) with
1
K(x, s) = ’ K0 (x, x1 )G(x1 , s)dx1 (6.8)
0
˜
So if I ’ K is invertible, then E is invertible as well. By H¨lder™s inequality
o
1 1 1
K0 (x, x1 )G(x1 , s)dx1 q ds]p/q dx ¤

[
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 ’1 = S ’1 (I ’ K)’1 ,
˜
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 de¬ned by (6.6), (6.2) is boundedly invertible in Lp and
’+
bp (S)Jp Jp
’1
E Lp ¤ .
+ ’ +’
Jp + Jp ’ Jp Jp
If, in addition, G ≥ 0 and K0 ¤ 0, then E ’1 is positive. Moreover,
1
’1 ’1
h)(x) ≥ (S
(E h)(x) = G(x, s)h(s)ds
0
for any nonnegative h ∈ Lp .
226 16. Hille  Tamarkin Integral Operators
16.7 Notes
A lot of papers and books are devoted to the spectrum of HilleTamarkin
integral operators. Mainly, the distributions of the eigenvalues are consid
ered, cf. (Diestel et al., 1995), (K¨nig, 1986), (Pietsch, 1987) and references
o
therein.
Theorem 16.4.2 and Corollary 16.4.3 improve the wellknown estimate
1
˜
rs (K) ¤ sup K(x, s)ds
x 0
(Krasnosel™skii et al, 1989, Theorem 16.2) for operators, which are close to
Volterra ones.
The results of Section 16.6 supplement the wellknown results on the
spectra of di¬erential 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 (Gil™, 2002).
References
[1] Diestel, D., Jarchow, H, Tonge, A. (1995), Absolutely Summing Opera
tors, Cambridge University Press, Cambridge.
[2] Edmunds, D.E. and Evans W.D. (1990). Spectral Theory and Di¬erential
Operators. Clarendon Press, Oxford.
[3] Egorov, Y. and Kondratiev, V. (1996). Spectral Theory of Elliptic Oper
ators. Birkh¨user Verlag, Basel.
a
[4] Gil™, M.I. (2002). Invertibility and positive invertibility of HilleTamarkin
integral operators, Acta Math. Hungarica, 95 (12) 3953.
[5] K¨nig, H. (1986). Eigenvalue Distribution of Compact Operators,
o
Birkh¨user Verlag, Basel BostonStuttgart.
a
[6] Locker, J. (1999). Spectral Theory of NonSelfAdjoint Two Point Di¬er
ential Operators., Amer. Math. Soc., Mathematical Surveys and Mono
graphs, Volume 73.
[7] Krasnosel™skii, M. A., J. Lifshits, and A. Sobolev (1989). Positive Linear
Systems. The Method of Positive Operators, Heldermann Verlag, Berlin.
[8] Pietsch, A. (1987). Eigenvalues and sNumbers, Cambridge University
Press, Cambridge.
[9] Zabreiko, P.P., Koshelev A.I., Krasnosel™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 L∞
In the present chapter integral operators in space L∞ [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
spectral radius are suggested. Applications to nonselfadjoint di¬erential op
erators and integrodi¬erential ones are also discussed.
17.1 Invertibility Conditions
Recall that L∞ ≡ L∞ [0, 1] is the space of scalarvalued functions de¬ned on
[0, 1] and equipped with the norm
hL∞ = ess sup h(x) (h ∈ L∞ ).
x∈[0,1]
Everywhere in this chapter K is a linear operator in L∞ de¬ned by
˜
1
K(x, s)h(s)ds (h ∈ L∞ , x ∈ [0, 1]),
˜
(Kh)(x) = (1.1)
0
where K(x, s) is a scalar kernel de¬ned on [0, 1]2 and having the property
1
ess sup K(x, s)ds < ∞. (1.2)
x∈[0,1]
0
De¬ne the Volterra operators
x
(V’ h)(x) = K(x, s)h(s)ds (1.3)
0
M.I. Gil™: LNM 1830, pp. 227“234, 2003.
c SpringerVerlag Berlin Heidelberg 2003
17. Integral Operators in L∞
228
and
1
(V+ h)(x) = K(x, s)h(s)ds. (1.4)
x
Set
w’ (s) ≡ ess K(x, s),
sup
0¤s¤x¤1
w+ (s) ≡ ess K(x, s)
sup
0¤x¤s¤1
and
1
M∞ (V± ) ≡ w± (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
eM∞ (V’ )+M∞ (V+ ) < eM∞ (V+ ) + eM∞ (V’ ) (1.5)
hold. Then operator I ’ K is boundedly invertible in L∞ and the inverse
˜
operator satis¬es the inequality
eM∞ (V’ )+M∞ (V+ )
(I ’ K)’1 L∞ ¤
˜ . (1.6)
eM∞ (V+ ) + eM∞ (V’ ) ’ eM∞ (V’ )+M∞ (V+ )
The proof of this theorem is presented in the next section.
Note that condition (1.5) is equivalent to the following one:
θ(K) ≡ (eM∞ (V+ ) ’ 1)(eM∞ (V’ ) ’ 1) < 1. (1.7)
Besides (1.6) takes the form
M∞ (V’ )+M∞ (V+ )
˜ ’1 L∞ ¤ e
(I ’ K) . (1.8)
1 ’ θ(K)
17.2 Proof of Theorem 17.1.1
Under condition (1.2), operators V± are quasinilpotent due to the wellknown
Theorem V.6.2 (Zabreiko, et al., 1968, p. 153). Now Corollary 16.2.2 yields.
Lemma 17.2.1 With the notation
∞
k
j(V± ) ≡ V± L∞ ,
k=0