<< ÒÚ. 10(‚ÒÂ„Ó 11)—Œƒ≈–∆¿Õ»≈ >>

let the conditions (1.2) and

j(V+ )j(V‚à’ ) < j(V+ ) + j(V‚à’ )
17.2. Proof of Theorem 17.1.1 229

be fulÔ¬Ålled. Then I ‚à’ K is boundedly invertible in L‚àû and the inverse operator
Àú
satisÔ¬Åes the inequality

j(V‚à’ )j(V+ )
|(I ‚à’ K)‚à’1 |L‚àû ‚â¤
Àú .
j(V‚à’ ) + j(V+ ) ‚à’ j(V‚à’ )j(V+ )

Lemma 17.2.2 Under condition (1.2), operator V‚à’ deÔ¬Åned by (1.3) satisÔ¬Åes
the inequality
k
M‚àû (V‚à’ )
k
|V‚à’ |L‚àû ‚â¤ (k = 1, 2, ...). (2.1)
k!
Proof: We have
x 1
|V‚à’ h|L‚àû = ess supx‚àà[0,1] | K(x, s)h(s)ds| ‚â¤ w‚à’ (s)|h(s)|ds.
0 0

Repeating these arguments, we arrive at the relation
1 s1 sk
k
|V‚à’ h|L‚àû ‚â¤ |h(sk )|dsk . . . ds2 ds1 .
w‚à’ (s1 ) w‚à’ (s2 ) . . .
0 0 0

Taking |h|L‚àû = 1, we get
1 s1 sk‚à’1
k
|V‚à’ |L‚àû ‚â¤ w‚à’ (s1 ) w‚à’ (s2 ) . . . dsk . . . ds2 ds1 . (2.2)
0 0 0

It is simple to see that
1 sk‚à’1
w‚à’ (s1 ) . . . w‚à’ (sk )dsk . . . ds1 =
0 0

¬µ
Àú z1 zk‚à’1
¬µk
Àú
... dzk dzk‚à’1 . . . dz1 = ,
k!
0 0 0

where sj
zj = zk (sj ) ‚â° w‚à’ (s)ds (j = 1, ..., k)
0

and
1
¬µ=
Àú w‚à’ (s)ds.
0

Thus (2.2) gives
1
w‚à’ (s)ds)k k
( M‚àû (V‚à’ )
k 0
|V‚à’ |L‚àû ‚â¤ = .
k! k!
As claimed. 2
17. Integral Operators in L‚àû
230

Similarly, the inequality
k
M‚àû (V+ )
k
|V+ |L‚àû ‚â¤ (k = 1, 2, ...) (2.3)
k!
can be proved.
Relations (2.1) and (2.3) imply

|(I ‚à’ V¬± )‚à’1 |L‚àû ‚â¤ j(V¬± ) ‚â¤ eM‚àû (V¬± ) . (2.4)

The assertion of Theorem 17.1.1 follows from Lemma 17.2.1 and relations
(2.4).

Clearly,
Œ»I ‚à’ K = Œ»(I ‚à’ Œ»‚à’1 K) (Œ» = 0).
Àú Àú
Consequently, if
‚à’1 ‚à’1 ‚à’1
e(M‚àû (V‚à’ )+M‚àû (V+ ))|Œ»| < e|Œ»| M‚àû (V+ )
+ e|Œ»| M‚àû (V‚à’ )
,
Àú
then due to Theorem 17.1.1, Œ»I ‚à’ K is boundedly invertible. We thus get
Lemma 17.3.1 Under condition (1.2), any point Œ» = 0 of the spectrum
Àú Àú
œÉ(K) of operator K satisÔ¬Åes the inequality
‚à’1 ‚à’1 ‚à’1
e(M‚àû (V‚à’ )+M‚àû (V+ ))|Œ»| ‚â• e|Œ»| M‚àû (V+ )
+ e|Œ»| M‚àû (V‚à’ )
. (3.1)
Àú Àú
Let rs (K) be the spectral radius of K. Then (3.1) yields
Àú Àú Àú
‚à’1 ‚à’1 ‚à’1
ers (K)(M‚àû (V‚à’ )+M‚àû (V+ ))
‚â• ers (K)M‚àû (V+ )
+ ers (K)M‚àû (V‚à’ )
. (3.2)
Àú
Clearly, if V+ = 0 or ( and ) V‚à’ = 0, then rs (K) = 0.
Theorem 17.3.2 Under condition (1.2), let V+ = 0, V‚à’ = 0. Then the
equation

e(M‚àû (V‚à’ )+M‚àû (V+ ))z = ezM‚àû (V+ ) + ezM‚àû (V‚à’ ) (z ‚â• 0) (3.3)

has a unique positive zero z(K). Moreover, the inequality rs (K) ‚â¤ z ‚à’1 (K)
Àú
is valid.
Proof: Equation (3.3) is equivalent to the following one:

(eM‚àû (V+ )z ‚à’ 1)(ezM‚àû (V‚à’ ) ‚à’ 1) = 1. (3.4)

In addition, (3.2) is equivalent to the relation
Àú Àú
‚à’1 ‚à’1
(ers (K)M‚àû (V+ )
‚à’ 1)(ers (K)M‚àû (V‚à’ )
‚à’ 1) ‚â• 1.
17.4. Nonnegative Invertibility 231

Hence, the result follows, since the left part of equation (3.4) monotonically
increases. 2
From (3.3) it follows that

e(M‚àû (V‚à’ )+M‚àû (V+ ))z ‚â• 2

and
ez(M‚àû (V+ )‚à’M‚àû (V‚à’ )) = eM‚àû (V+ )z ‚à’ 1 ‚â•
exp [ln 2 M‚àû (V+ )(M‚àû (V‚à’ ) + M‚àû (V+ ))‚à’1 ] ‚à’ 1.
Thus with the notation
M‚àû (V+ ) ‚à’ M‚àû (V‚à’ )
Œ¥‚àû (K) = (3.5)
ln [ exp ( M‚àû (V‚à’(V+ )ln (V+ ) ) ‚à’ 1]
M‚àû 2
)+M‚àû

we have
‚à’1
z(K) ‚â• Œ¥‚àû (K), (3.6)
provided
M‚àû (V+ ) < M‚àû (V‚à’ ). (3.7)
Clearly, in (3.5) we can exchang the places of V‚à’ and V+ . Now Theorem
17.3.2 yields
Àú
Corollary 17.3.3 Under conditions (1.2) and (3.7), the inequality rs (K) ‚â¤
Œ¥‚àû (K) is true.

17.4 Nonnegative Invertibility
We will say that h ‚àà L‚àû is nonnegative if h(t) is nonnegative for almost all
t ‚àà [0, 1]; a linear operator A in L‚àû is nonnegative if Ah is nonnegative for
each nonnegative h ‚àà L‚àû . Recall that I is the identity operator.

Theorem 17.4.1 Let the conditions (1.2), (1.5) and

K(t, s) ‚â• 0 (0 ‚â¤ t, s ‚â¤ 1) (4.1)
Àú
hold. Then operator I ‚à’ K is boundedly invertible and the inverse operator
is nonnegative. Moreover,

(I ‚à’ K)‚à’1 ‚â• I.
Àú (4.2)
Àú
Proof: Relation (2.9) from Section 16.2 with A = I ‚à’ K, W = V‚à’ and
V = V+ implies

(I ‚à’ K)‚à’1 = (I ‚à’ V+ )‚à’1 (I ‚à’ BK )‚à’1 (I ‚à’ V‚à’ )‚à’1
Àú (4.3)
17. Integral Operators in L‚àû
232

where
BK = (I ‚à’ V+ )‚à’1 V+ V‚à’ (I ‚à’ V‚à’ )‚à’1 .
Moreover, by (4.1) we have V¬± ‚â• 0. So (I ‚à’ V¬± )‚à’1 ‚â• 0 and BK ‚â• 0.
Relations (2.4) give us the inequalities

|(I ‚à’ V¬± )‚à’1 V¬± |L‚àû ‚â¤ eM‚àû (V¬± ) ‚à’ 1.

Consequently,

|BK |L‚àû ‚â¤ (eM‚àû (V+ ) ‚à’ 1)(eM‚àû (V‚à’ ) ‚à’ 1).

But (1.5) is equivalent to (1.7). We thus get |BK |L‚àû < 1. Consequently,
‚àû
‚à’1 k
(I ‚à’ BK ) BK ‚â• 0.
=
k=0

Now (4.3) implies the inequality (I ‚à’ K)‚à’1 ‚â• 0. In addition, since I ‚à’ K ‚â¤ I,
Àú Àú
we have inequality (4.2). 2

17.5 Applications
Consider a diÔ¬Äerential operator A deÔ¬Åned by

d2 h(x) dh(x)
(Ah)(x) = ‚à’ + m(x)h(x) (0 < x < 1, h ‚àà Dom (A) )
+ g(x)
2
dx dx
(5.1)
on the domain

Dom (A) = {h ‚àà L‚àû , h ‚àà L‚àû + some boundary conditions } (5.2)

the coeÔ¬Écients g, w ‚àà L‚àû and are complex, in general. (5.3)

Let an operator S be deÔ¬Åned on Dom (A) by

(Sh)(x) = ‚à’h (x), h ‚àà Dom (A).

It is assumed that S has the Green function G(t, s). So that,
1
‚à’1
h)(x) ‚â° G(x, s)h(s)ds ‚àà Dom (A)
(S
0
17.5. Applications 233

for any h ‚àà L‚àû , and the derivative of G in x satisÔ¬Åes the condition
1
sup |Gx (x, s)|ds < ‚àû. (5.4)
x
0

Put
1
sup |G(x, s)|ds.
b‚àû (S) :=
x
0
We have
Àú
A = (I ‚à’ K)S,
where
1 1
d
Àú
(Kh)(x) = ‚à’(g(x) + m(x)) G(x, s)h(s)ds = K(x, s)h(s)ds
dx 0 0

with
K(x, s) = ‚à’g(x)Gx (x, s) ‚à’ m(x)G(x, s). (5.5)
According to (5.3) and (5.4), condition (1.2) holds. Take into account that

|S ‚à’1 h|L‚àû ‚â¤ b‚àû (S)|h|L‚àû .

Since
A‚à’1 = S ‚à’1 (I ‚à’ K)‚à’1 ,
Àú
Theorem 17.1.1 immediately implies the following result:
Proposition 17.5.1 Under (5.3)-(5.5), let condition (1.5) hold. Then op-
erator A deÔ¬Åned by (5.1), (5.2) is boundedly invertible in L‚àû . In addition,

b‚àû (S)eM‚àû (V‚à’ )+M‚àû (V+ )
|A‚à’1 |L‚àû ‚â¤ .
eM‚àû (V+ ) + eM‚àû (V‚à’ ) ‚à’ eM‚àû (V‚à’ )+M‚àû (V+ )

17.5.2 An integro-diÔ¬Äerential operator
On domain (5.2), let us consider the operator
1
d2 u(x)
(Eu)(x) = ‚à’ K0 (x, s)u(s)ds (u ‚àà Dom (A), 0 < x < 1), (5.6)
+
dx2 0

where K0 is a kernel with the property
1
|K0 (x, s)| ds < ‚àû.
ess supx (5.7)
0

Let S and G be the same as in the previous subsection. Then we can write
Àú Àú
E = (I ‚à’ K)S where K is deÔ¬Åned by (1.1) with
1
K(x, s) = ‚à’ K0 (x, x1 )G(x1 , s)dx1 (5.8)
0
17. Integral Operators in L‚àû
234

Àú
So if I ‚à’ K is invertible, then E is invertible as well. Clearly, under (5.4) and
(5.7), condition (1.2) holds. Since

E ‚à’1 = S ‚à’1 (I ‚à’ K)‚à’1 ,
Àú

Theorems 17.1.1 and 17.4.1 yield
Proposition 17.5.2 Under (5.4), (5.7) and (5.8), let condition (1.5) hold.
Then operator E deÔ¬Åned by (5.6), (5.2) is boundedly invertible in L‚àû and

b‚àû (S)eM‚àû (V‚à’ )+M‚àû (V+ )
‚à’1
|E |L‚àû ‚â¤ M (V ) .
e ‚àû + + eM‚àû (V‚à’ ) ‚à’ eM‚àû (V‚à’ )+M‚àû (V+ )
If, in addition, G ‚â• 0 and K0 ‚â¤ 0, then E ‚à’1 is positive. Moreover,
1
‚à’1 ‚à’1
h)(x) ‚â• (S
(E h)(x) = G(x, s)h(s)ds
0

for any nonnegative h ‚àà L‚àû .

17.6 Notes
The present chapter is based on the paper (Gil‚Ä™, 2001).
About well-known results on the spectrum of integral operators on L‚àû ,
see, for instance, the books (Diestel et al., 1995), (K¬®nig, 1986), (Kras-
o
nosel‚Ä™skii et al., 1989), (Pietsch, 1987) and references therein.

References
 Diestel, D., Jarchow, H, Tonge, A. (1995), Absolutely Summing Opera-
tors, Cambridge University Press, Cambridge.
 Gil‚Ä™, M.I. (2001). Invertibility and positive invertibility conditions of
integral operators in L‚àû , J. of Integral Equations and Appl. 13 , 1-14.
 K¬®nig, H. (1986). Eigenvalue Distribution of Compact Operators,
o
Birkh¬®user Verlag, Basel- Boston-Stuttgart.
a
 Krasnosel‚Ä™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., A.I. Koshelev, M. A. Krasnosel‚Ä™skii, S.G. Mikhlin, L.S.
Rakovshik, B.Ya. Stetzenko (1968). Integral Equations, Nauka, Moscow.
In Russian
18. Hille - Tamarkin
Matrices

In the present chapter we investigate inÔ¬Ånite matrices, whose oÔ¬Ä diagonal
parts are the Hille-Tamarkin matrices. Invertibility conditions and estimates
for the norm of the inverse matrices are established. In addition, bounds for
the spectrum are suggested. In particular, estimates for the spectral radius
are derived.

18.1 Invertibility Conditions
Everywhere in this chapter

A = (ajk )‚àû
j,k=1

is an inÔ¬Ånite matrix with the entries ajk (j, k = 1, 2, ...). Besides, V+ , V‚à’ and
D denote the strictly upper triangular, strictly lower triangular, and diagonal
parts of A, respectively:
Ô£« Ô£∂
Ô£« Ô£∂ 0 0 0 0 ...
0 a12 a13 a14 ... Ô£¬ ... Ô£·
a21 0 0 0
Ô£¬0 0 ... Ô£· Ô£¬ Ô£·
a23 a24
V+ = Ô£¬ Ô£· , V‚à’ = Ô£¬ ... Ô£·
a31 a32 0 0
Ô£¬ Ô£·
Ô£≠0 0 ... Ô£∏
0 a34 Ô£≠ ... Ô£∏
a41 a42 a43 0
. . . . ...
. . . ... .
(1.1)
and
D = diag [a11 , a22 , a33 , ...].

M.I. Gil‚Ä™: LNM 1830, pp. 235‚Ä“241, 2003.
c Springer-Verlag Berlin Heidelberg 2003
236 18. Hille - Tamarkin Matrices

Throughout this chapter it is assumed that V‚à’ and V+ are the Hille-Tamarkin
matrices. That is, for some Ô¬Ånite p > 1,
‚àû ‚àû
|ajk |q ]p/q ]1/p < ‚àû
[ (1.2)
j=1 k=1, k=j

with
11
+ = 1.
pq
As usually lp (1 < p < ‚àû) is the Banach space of number sequences equipped
with the norm
‚àû
|hk |p ]1/p (h = (hk ) ‚àà lp ).
|h|lp = [
k=1

So under (1.2), A represents a linear operator in lp which is also denoted by
A. Clearly,
‚àû
p
|akk hk |p < ‚àû}.
Dom (A) = Dom (D) = {h = (hk ) ‚àà l :
k=1

Assume that
d0 ‚â° inf |akk | > 0 (1.3)
k
and introduce the notations
‚àû ‚àû
|a‚à’1 ajk |q ]p/q )1/p ,
+
‚â°(
Mp (A) [ jj
j=1 k=j+1

‚àû j‚à’1
|a‚à’1 ajk |q ]p/q )1/p ,
‚à’
Mp (A) =( [ jj
j=2 k=1

and
‚àû
(Mp (A))k
¬±
¬±
‚àö
Jp (A) = .
p
k!
k=0
Now we are in a position to formulate the main result of the chapter.
Theorem 18.1.1 Let the conditions (1.2), (1.3) and
‚à’ + + ‚à’
Jp (A)Jp (A) < Jp (A) + Jp (A) (1.4)

hold. Then A is boundedly invertible in lp and the inverse operator satisÔ¬Åes
the inequality
‚à’ +
Jp (A)Jp (A)
‚à’1
|A | ‚â¤ (1.5)
lp + ‚à’ ‚à’ +
(Jp (A) + Jp (A) ‚à’ Jp (A)Jp (A))d0
The proof of this theorem is presented in the next section.
18.2. Proof of Theorem 18.1.1 237

18.2 Proof of Theorem 18.1.1
Lemma 18.2.1 Under condition (1.2), for the strictly upper and lower tri-
angular matrices V+ and V‚à’ , the inequalities

(vp )m
¬±
m
‚â¤‚àö
|V¬± |lp (m = 1, 2, ...)
p
m!
are valid, where
‚àû ‚àû
+
|ajk |q ]p/q )1/p
vp =( [
j=1 k=j+1

and
‚àû j‚à’1
‚à’
|ajk |q ]p/q )1/p .
vp =( [
j=2 k=1

This result follows from Lemma 3.2.1 when n ‚Ü’ ‚àû, since
1
Œ≥n,m,p ‚â¤ ‚àö .
p
m!
So operators V¬± are quasinilpotent. The latter lemma yields

Corollary 18.2.2 Under conditions (1.2), (1.3), the inequalities

(Mp (A))m
¬±
‚à’1 m
‚àö
|(D V¬± ) | ‚â¤ (m = 1, 2, ...)
lp p
m!
are valid.

Proof of Theorem 18.1.1: We have

A = V+ + V‚à’ + D = D(D‚à’1 V+ + D‚à’1 V‚à’ + I)

Clearly,
|D‚à’1 |lp = d‚à’1 .
0

From Lemma 18.2.2, it follows that

j(D‚à’1 V¬± ) ‚â¤ Jp (A)
¬±

Now Corollary 16.2.2 and condition (1.4) yield the invertibility of the operator

D‚à’1 V+ + D‚à’1 V‚à’ + I

and the estimate (1.5). 2
238 18. Hille - Tamarkin Matrices

18.3 Localization of the Spectrum
Let œÉ(A) be the spectrum of A. For a Œ» ‚àà C, assume that

œÅ(D, Œ») ‚â° inf |Œ» ‚à’ amm | > 0,
m

and put
‚àû ‚àû
+
|(ajj ‚à’ Œ»)‚à’1 ajk |q ]p/q )1/p ,
Mp (A, Œ») =( [
j=1 k=j+1

‚àû j‚à’1
‚à’
|(ajj ‚à’ Œ»)‚à’1 ajk |q ]p/q )1/p ,
Mp (A, Œ») =( [
j=2 k=1

and
‚àû
(Mp (A, Œ»))k
¬±
¬±
‚àö
Jp (A, Œ») = .
p
k!
k=0

Clearly
¬± ¬± ¬± ¬±
Mp (A, 0) = Mp (A), Jp (A, 0) = Jp (A).
Lemma 18.3.1 Under condition (1.2), for any ¬µ ‚àà œÉ(A) we have either
¬µ = ajj for some natural j, or
‚à’ + + ‚à’
Jp (A, ¬µ)Jp (A, ¬µ) ‚â• Jp (A, ¬µ) + Jp (A, ¬µ). (3.1)

Proof: Assume that
‚à’ + + ‚à’
Jp (A, ¬µ)Jp (A, ¬µ) < Jp (A, ¬µ) + Jp (A, ¬µ)

for some ¬µ ‚àà œÉ(A). Then due to Theorem 18.1.1, A ‚à’ ¬µI is invertible. This
contradiction proves the required result. 2

¬±
Recall that vp are deÔ¬Åned in Section 18.2 and denote,
‚àû
(vp )k
¬±
¬±
‚àö (z > 0)
Fp (z) = (3.2)
z k p k!
k=0

Lemma 18.3.2 Under condition (1.2), for any ¬µ ‚àà œÉ(A), either there is an
integer m, such that, ¬µ = amm , or
‚à’ + ‚à’ +
Fp (œÅ(D, ¬µ))Fp (œÅ(D, ¬µ)) ‚â• Fp (œÅ(D, ¬µ)) + Fp (œÅ(D, ¬µ)). (3.3)

Proof: Let ¬µ = akk for all natural k. Then

Mp (A, ¬µ) ‚â¤ œÅ‚à’1 (D, ¬µ)vp .
¬± ¬±

Hence,
¬± ¬±
Jp (A, ¬µ) ‚â¤ Fp (œÅ(D, ¬µ)). (3.4)
18.3. Localization of the Spectrum 239

In addition, (3.1) is equivalent to the relation
‚à’ +
(Jp (A, ¬µ) ‚à’ 1)(Jp (A, ¬µ) ‚à’ 1) ‚â• 1.

Now (3.4) implies (3.3). 2

Theorem 18.3.3 Under condition (1.2), let V+ = 0, V‚à’ = 0. Then the
equation
‚à’ + ‚à’ +
Fp (z)Fp (z) = Fp (z) + Fp (z) (3.5)
has a unique positive root Œ∂(A). Moreover, œÅ(D, ¬µ) ‚â¤ Œ∂(A) for any ¬µ ‚àà œÉ(A).
In other words, œÉ(A) lies in the closure of the union of the discs

{Œ» ‚àà C : |Œ» ‚à’ akk | ‚â¤ Œ∂(A)} (k = 1, 2, ...).

Proof: Equation (3.5) is equivalent to the following one:
‚à’ +
(Fp (z) ‚à’ 1)(Fp (z) ‚à’ 1) = 1. (3.6)

The left part of this equation monotonically decreases as z > 0 increases;
so it has a unique positive root Œ∂(A). In addition, (3.3) is equivalent to the
relation
‚à’ +
(Fp (œÅ(D, ¬µ)) ‚à’ 1)(Fp (œÅ(D, ¬µ)) ‚à’ 1) ‚â• 1. (3.7)
Hence the result follows. 2

Rewrite (3.5) as
‚àû ‚àû
(vp )k
‚à’
(vp )j
+
‚àö ‚àö = 1.
z j p j!
z k p k! j=1
k=1

Or
‚àû k‚à’1
(vp )k‚à’j (vp )j
+ ‚à’
k
Bk z = 1 with Bk = (k = 2, 3, ...).
j!(k ‚à’ j)!
p
j=1
k=2

Due to the Lemma 8.3.1, with the notation

Œ¥p (A) ‚â° 2 sup j
Bj ,
j=2,3,...

we get Œ∂(A) ‚â¤ Œ¥p (A). Now Theorem 18.3.3 yields

Corollary 18.3.4 Under condition (1.2), let V+ = 0, V‚à’ = 0. Then for any
¬µ ‚àà œÉ(A), the inequality œÅ(¬µ, D) ‚â¤ Œ¥(A) is true.
In other words, œÉ(A) lies in the closure of the union of the sets

{Œ» ‚àà C : |Œ» ‚à’ akk | ‚â¤ Œ¥p (A)} (k = 1, 2, ...).
240 18. Hille - Tamarkin Matrices

Note that Theorem 18.3.3 is exact: if A is triangular: either V‚à’ = 0, or
V+ = 0, then we due to that lemma œÉ(A) is the closure of the set

{akk , k = 1, 2, ...}.

Moreover, Theorem 18.3.3 and Corollary 18.3.4 imply

rs (A) ‚â¤ sup |akk | + Œ∂(A) ‚â¤ sup |akk | + Œ¥p (A), (3.8)
k=1,2,... k=1,2,...

provided D is bounded. Furthermore, let the condition
‚àû
|ajk | < ‚àû
sup (3.9)
j=1,2,...
k=1

hold. Then the well-known estimate
‚àû
rs (A) ‚â¤ |ajk |
sup (3.10)
j=1,2,...
k=1

is valid, see (Krasnosel‚Ä™skii et al. 1989, Theorem 16.2). Under condition
(3.9), relations (3.8) improve (3.10), provided

‚àû
|ajk |
Œ∂(A) < sup
j=1,2,...
k=1,k=j

or
‚àû
|ajk |.
Œ¥p (A) < sup
j=1,2,...
k=1,k=j

In conclusion, note that Theorem 18.1.1 is exact: if A is upper or lower
triangular, then A is invertible, provided D is invertible.

18.4 Notes
The present chapter is based on the paper (Gil‚Ä™, 2002).
About other results on the spectrum of Hille-Tamarkin matrices see, for
instance, the books (Diestel et al., 1995), (K¬®nig, 1986), (Pietsch, 1987), and
o
references therein.
Note that Hille-Tamarkin matrices arise, in particular, in recent investi-
gations of discrete Volterra equations, see (Kolmanovskii et al, 2000), (Gil‚Ä™
and Medina, 2002), (Medina and Gil‚Ä™, 2003).
18.4. Notes 241

References
 Diestel, D., Jarchow, H, Tonge, A. (1995), Absolutely Summing Opera-
tors, Cambridge University Press, Cambridge.
 Gil‚Ä™, M.I. (2002), Invertibility and spectrum of Hille-Tamarkin matrices,
Mathematische Nachrichten, 244, 1-11
 Gil‚Ä™, M.I. and Medina, R. (2002). Boundedness of solutions of matrix
nonlinear Volterra diÔ¬Äerence equations. Discrete Dynamics in Nature
and Society, 7, No 1, 19-22

 Kolmanovskii, V.B., A.D. Myshkis and J.P. Richard (2000). Estimate
of solutions for some Volterra diÔ¬Äerence equations, Nonlinear Analysis,
TMA, 40, 345-363.
 K¬®nig, H. (1986). Eigenvalue Distribution of Compact Operators,
o
Birkh¬®user Verlag, Basel- Boston-Stuttgart.
a
 Krasnosel‚Ä™skii, M. A., J. Lifshits, and A. Sobolev (1989). Positive Linear
Systems. The Method of Positive Operators, Heldermann Verlag, Berlin.
 Medina, R. and Gil‚Ä™, M.I. (2003). Multidimensional Volterra diÔ¬Äerence
equations. In the book: New Progress in DiÔ¬Äerence Equations, Eds. S.
Elaydi, G. Ladas and B. Aulbach, Taylor and Francis, London and New
York, p. 499-504
 Pietsch, A. (1987). Eigenvalues and s-Numbers, Cambridge University
Press, Cambridge.
19. Zeros of Entire
Functions

The present chapter is devoted to applications of our abstract results to the
theory of Ô¬Ånite order entire functions. We consider the following problem: if
the Taylor coeÔ¬Écients of two entire functions are close, how close are their
zeros? In addition, we establish bounds for sums of the absolute values of
the zeros in the terms of the coeÔ¬Écients of its Taylor series. These bounds

19.1 Perturbations of Zeros
Consider the entire function
‚àû
ck Œ»k (Œ» ‚àà C; c0 = 1)
f (Œ») =
k=0

with complex, in general, coeÔ¬Écients ck , k = 1, 2, ... . Put

Mf (r) := max |f (z)| (r > 0).
|z|=r

Recall that
ln ln Mf (r)
œÅ (f ) := limr‚Ü’‚àû
ln r
is the order of f . Moreover, the relation
n ln n
œÅ (f ) = limn‚Ü’‚àû
ln (1/|cn |)

is true, cf. (Levin, 1996, p. 6).

M.I. Gil‚Ä™: LNM 1830, pp. 243‚Ä“252, 2003.
c Springer-Verlag Berlin Heidelberg 2003
244 19. Zeros of Entire Functions

Everywhere in the present chapter it is assumed that the set

{zk (f )}‚àû
k=1

of all the zeros of f taken with their multiplicities is inÔ¬Ånite.
‚à’1
Note that if f has a Ô¬Ånite number m of the zeros, we can put zk (f ) = 0
‚à’1
for k = m, m + 1, ... and aplly our arguments below. Here and below zk (f )
means zk1 ) .
(f
Rewrite function f in the form
‚àû
ak Œ»k
f (Œ») = (a0 = 1) (1.1a)
(k!)Œ≥
k=0

with a positive Œ≥, and consider the function
‚àû
bk Œ» k
h(Œ») = (b0 = 1). (1.1b)
(k!)Œ≥
k=0

Assume that
‚àû ‚àû
2
|bk |2 < ‚àû.
|ak | < ‚àû, (1.2)
k=0 k=0

Relations (1.1) and (1.2), imply that functions f and h have orders no more
than 1/Œ≥.

DeÔ¬Ånition 19.1.1 The quantity
‚à’1 ‚à’1
zvf (h) = max min |zk (f ) ‚à’ zj (h)|
j k

will be called the variation of zeros of function h with respect to function f .

For a natural p > 1/2Œ≥, put
‚àû
|ak |2 ]1/2 + 2 [Œ∂(2Œ≥p) ‚à’ 1]1/2p ,
wp (f ) := 2 [ (1.3)
k=1

where Œ∂ is the Riemann Zeta function, and
p‚à’1 2p
wk (f ) 1 wp (f )
œà(f, y) := exp [ + ] (y > 0). (1.4)
y k+1 2y 2p
2
k=0

Finally, denote
‚àû
|ak ‚à’ bk |2 ]1/2 .
q := [
k=1
19.1. Perturbations of Zeros 245

Theorem 19.1.2 Let conditions (1.1) and (1.2) be fulÔ¬Ålled. In addition, let
r(q, f ) be the unique positive (simple) root of the equation

qœà(f, y) = 1.

Then zvf (h) ‚â¤ r(q, f ). That is, for any zero z(h) of h there is a zero z(f )
of f , such that
|z(h) ‚à’ z(f )| ‚â¤ r(q, f )|z(h)z(f )|. (1.5)
The proof of Theorem 19.1.2 is presented in the next section. Substitute in
(1.5) the equality y = xwp (f ) and apply Lemma 8.3.2. Then we have

r(q, f ) ‚â¤ Œ¥(q, f ), (1.6)

where
if wp (f ) ‚â¤ epq,
epq
Œ¥(q, f ) := .
wp (f ) [ln (wp (f )/qp)]‚à’1/2p if wp (f ) > epq

Theorem 19.1.2 and inequality (1.6) yield
Corollary 19.1.3 Let conditions (1.1) and (1.2) be fulÔ¬Ålled. Then zvf (h) ‚â¤
Œ¥(q, f ). That is, for any zero z(h) of h, there is a zero z(f ) of f , such that

|z(h) ‚à’ z(f )| ‚â¤ Œ¥(q, f )|z(h)z(f )|. (1.7)

Relations (1.5) and (1.7) imply the inequalities

|z(f )| ‚à’ |z(h)| ‚â¤ r(q, f )|z(h)||z(f )| ‚â¤ Œ¥(q, f )|z(h)||z(f )|.

Hence,

|z(h)| ‚â• (r(q, f )|z(f )| + 1)‚à’1 |z(f )| ‚â• (Œ¥(q, f )|z(f )| + 1)‚à’1 |z(f )|.

This inequality yields the following result
Corollary 19.1.4 Under conditions (1.1) and (1.2), for a positive number
R0 , let f have no zeros in the disc {z ‚àà C : |z| ‚â¤ R0 }. Then h has no zeros
in the disc {z ‚àà C : |z| ‚â¤ R1 } with

R0 R0
R1 = or R1 = .
Œ¥(q, f )R0 + 1 r(q, f )R0 + 1

Let us assume that under (1.1), there is a constant d0 ‚àà (0, 1), such that

|ak | < 1/d0
k
limk‚Ü’‚àû (1.8)

and
|bk | < 1/d0
k
limk‚Ü’‚àû
246 19. Zeros of Entire Functions

and consider the functions
‚àû
ak (d0 Œ»)k
Àú
f (Œ») = (1.9)
(k!)Œ≥
k=0

and
‚àû
bk (d0 Œ»)k
Àú
h(Œ») = .
(k!)Œ≥
k=0

Àú Àú Àú Àú
That is, f (Œ») ‚â° f (d0 Œ») and h(Œ») ‚â° h(d0 Œ»). So functions f (Œ») and h(Œ»)
satisfy conditions (1.2). Moreover,
‚àû
d2k |ak |2 ]1/2 + 2[Œ∂(2Œ≥p) ‚à’ 1]1/2p .
Àú
wp (f ) = 2[ 0
k=1

Àú Àú
Thus, we can apply Theorem 19.1.2 and its corollaries to functions f (Œ»), h(Œ»)
and take into account that
Àú Àú
d0 zk (f ) = zk (f ), d0 zk (h) = zk (h). (1.10)

19.2 Proof of Theorem 19.1.2
For a Ô¬Ånite integer n, consider the polynomials
n n
ak Œ»n‚à’k bk Œ»n‚à’k
F (Œ») = and Q(Œ») = (a0 = b0 = 1). (2.1)
(k!)Œ≥ (k!)Œ≥
k=0 k=0

Put
n n n
2 1/2 2Œ≥p 1/2p
|ak ‚à’bk |2 ]1/2 .
|ak | ]
wp (F ) := 2 [ +2 [ 1/k ] and q(F, Q) := [
k=1 k=2 k=1

In addition, {zk (F )}n n
k=1 and {zk (Q)}k=1 are the sets of all the zeros of F
and Q, respectively taken with their multiplicities. DeÔ¬Åne œà(F, y) according
to (1.4).

Lemma 19.2.1 For any zero z(Q) of Q, there is a zero z(F ) of F , such that

|z(F ) ‚à’ z(Q)| ‚â¤ r(Q, F ),

where r(Q, F ) be the unique positive (simple) root of the equation

q(F, Q)œà(F, y) = 1. (2.2)
19.2. Proofs of Theorem 19.1.2 247

Proof: In a Euclidean space Cn with the Euclidean norm . , introduce
operators An and Bn by virtue of the n √— n-matrices
Ô£« Ô£∂
‚à’a1 ‚à’a2 ... ‚à’an‚à’1 ‚à’an
Ô£¬ 1/2Œ≥ 0Ô£·
0 ... 0
Ô£¬ Ô£·
An = Ô£¬ 0 0Ô£·
1/3Œ≥ ... 0
Ô£¬ Ô£·
Ô£≠. .Ô£∏
. ... .
... 1/nŒ≥
0 0 0

Ô£« Ô£∂
and
‚à’b1 ‚à’b2 ... ‚à’bn‚à’1 ‚à’bn
Ô£¬ Ô£·
1/2Œ≥ 0 ... 0 0
Ô£¬ Ô£·
Bn = Ô£¬ Ô£·.
1/3Œ≥
0 ... 0 0
Ô£¬ Ô£·
Ô£≠ Ô£∏
. . ... . .
... 1/nŒ≥
0 0 0
It is simple to see that
F (Œ») = det (Œ»I ‚à’ An )
and Q(Œ») = det (Œ»I ‚à’ Bn ), where I is the unit matrix. So

Œ»k (An ) = zk (F ), Œ»k (Bn ) = zk (Q) (k = 1, 2, ..., n), (2.3)

where Œ»k (.), k = 1, ..., n are the eigenvalues with their multiplicities. Clearly,

An ‚à’ Bn = q(F, Q).

Due to Theorem 8.5.4, for any Œ»j (Bn ) there is an Œ»i (An ), such that

|Œ»j (Bn ) ‚à’ Œ»i (An )| ‚â¤ yp (An , Bn ), (2.4)

where yp (An , Bn ) is the unique positive (simple) root of the equation
p‚à’1
(2N2p (An ))k (2N2p (An ))2p
q(F, Q) exp [(1 + )/2] = 1,
y k+1 y 2p
k=0

where N2p (A) := [T race(AA‚à— )p ]1/2p is the Neumann-Schatten norm and the
asterisk means the adjointness. But An = M + C, where
Ô£« Ô£∂
‚à’a1 ‚à’a2 ... ‚à’an‚à’1 ‚à’an
Ô£¬0 0Ô£·
0 ... 0
M =Ô£¬ Ô£·
Ô£≠. .Ô£∏
. ... .
0 0 ... 0 0

Ô£« Ô£∂
and
0 0 ... 0 0
Ô£¬ Ô£·
1/2Œ≥ 0 ... 0 0
Ô£¬ Ô£·
C=Ô£¬ Ô£·.
1/3Œ≥
0 ... 0 0
Ô£¬ Ô£·
Ô£≠ Ô£∏
. . ... . .
... 1/nŒ≥
0 0 0
248 19. Zeros of Entire Functions

Therefore, with
n
|ak |2 ,
c=
k=1

Ô£« Ô£∂
we have
c 0 ... 0 0
Ô£¬0 0 ... 0 0 Ô£·
MM‚à— = Ô£¬ Ô£·
Ô£≠. . ... . . Ô£∏
0 0 ... 0 0
Ô£« Ô£∂
and
0 0 ... 0 0
Ô£¬ Ô£·
1/22Œ≥
0 ... 0 0
Ô£¬ Ô£·
CC ‚à— = Ô£¬ Ô£·.
0 0 ... 0 0
Ô£¬ Ô£·
Ô£≠ Ô£∏
. . ... . .
0 1/n2Œ≥
0 0 ...
Hence,
n
‚àö
1/k 2Œ≥p ]1/2p .
N2p (An ) ‚â¤ N2p (M ) + N2p (C) = c+[
k=2

Consequently yp (An , Bn ) ‚â¤ r(Q, F ). Therefore (2.3) and (2.4) imply (2.2),
as claimed. 2

Proof of Theorem 19.1.2: Consider the polynomials
n n
ak Œ»k bk Œ» k
fn (Œ») = and hn (Œ») = . (2.5)
(k!)Œ≥ (k!)Œ≥
k=0 k=0

Clearly, Œ»n fn (1/Œ») = F (Œ») and hn (1/Œ»)Œ»n = Q(Œ»). So

zk (F ) = 1/zk (fn ); zk (Q) = 1/zk (hn ). (2.6)

Take into account that the roots continuously depend on coeÔ¬Écients, we have
the required result, letting in the previous lemma n ‚Ü’ ‚àû. 2

19.3 Bounds for Sums of Zeros
Again consider an entire function f of the form (1.1a) and assume that the
condition
‚àû
|ak |2 ]1/2 < ‚àû
Œ∏f := [ (3.1)
k=1
holds. Let the zeros of f be numerated in the increasing way:

|zk (f )| ‚â¤ |zk+1 (f )| (k = 1, 2, ...). (3.2)
19.3. Bounds for Sums of Zeros 249

Theorem 19.3.1 Let f be an entire function of the form (1.1a). Then under
conditions (3.1) and (3.2), the inequalities
j j
‚à’1
(k + 1)‚à’Œ≥ (j = 1, 2, ...)
|zk (f )| ‚â¤ Œ∏f +
k=1 k=1

are valid.
The proof of this theorem is presented in this section below. Note that
under condition (1.8) we can omit condition (3.1) due to (1.9) and (1.10).
To prove Theorem 19.3.1, again consider the polynomial F (Œ») deÔ¬Åned in
(2.1) with the zeros ordered in the following way:

|zk (F )| ‚â• |zk+1 (F )| (k = 1, ..., n ‚à’ 1).

Set
n
|ak |2 ]1/2 .
Œ∏(F ) := [
k=1

Lemma 19.3.2 The zeros of F satisfy the inequalities
j j
(k + 1)‚à’Œ≥ (j = 1, ..., n ‚à’ 1)
|zk (F )| ‚â¤ Œ∏(F ) +
k=1 k=1

and
n n‚à’1
(k + 1)‚à’Œ≥ .
|zk (F )| ‚â¤ Œ∏(F ) +
k=1 k=1

Proof: Take into account that according to (2.3),
j j
|Œ»k (An )| ‚â¤ sk (An ) (j = 1, ..., n), (3.3)
k=1 k=1

where sk (An ), k = 1, 2, ... are the singular numbers of An ordered in the
decreasing way (Marcus and Minc, 1964, Section II.4.2). But An = M + C,
where M and C are introduced in Section 19.2. We can write

s1 (M ) = Œ∏(F ), sk (M ) = 0 (k = 2, ..., n).

sk (C) = 1/(k + 1)Œ≥ (k = 1, ..., n ‚à’ 1), sn (C) = 0.

Take into account that
j j j j
sk (M + C) ‚â¤
sk (An ) = sk (M ) + sk (C),
k=1 k=1 k=1 k=1
250 19. Zeros of Entire Functions

cf. (Gohberg and Krein, 1969, Lemma II.4.2). So

j j
(k + 1)‚à’Œ≥ (j = 1, ..., n ‚à’ 1)
sk (An ) ‚â¤ Œ∏(F ) +
k=1 k=1

and
n n‚à’1
(k + 1)‚à’Œ≥ .
sk (An ) ‚â¤ Œ∏(F ) +
k=1 k=1

Now (2.3) and (3.3) yield the required result. 2

Proof of Theorem 19.3.1: Again consider the polynomial fn (z) de-
Ô¬Åned as in (2.5). Now Lemma 19.3.2 and (2.6) yield the inequalities

j j
‚à’1
(k + 1)‚à’Œ≥ (j = 1, ..., n ‚à’ 1).
|zk (fn )| ‚â¤ Œ∏f + (3.4)
k=1 k=1

But the zeros of entire functions continuously depend on its coeÔ¬Écients. So
for any j = 1, 2, ...,
j j
‚à’1
|zk (f )|‚à’1
|zk (fn )| ‚Ü’
k=1 k=1

as n ‚Ü’ ‚àû. Now (3.4) implies the required result. 2

19.4 Applications of Theorem 19.3.1
Put
œ„1 = Œ∏f + 2‚à’Œ≥ and œ„k = (k + 1)‚à’Œ≥ (k = 2, 3, ...).

Corollary 19.4.1 Let œÜ(t) (0 ‚â¤ t < ‚àû) be a convex scalar-valued function,
such that œÜ(0) = 0. Then under conditions (1.1a), (3.1) and (3.2), the
inequalities
j j
‚à’1
)‚â¤
œÜ(|zk (f )| œÜ(œ„k ) (j = 1, 2, ...)
k=1 k=1

are valid. In particular, for any r ‚â• 2,
j j j
‚à’r r ‚à’Œ≥ r
(k + 1)‚à’rŒ≥ (j = 2, 3, ...). (4.1)
|zk (f )| ‚â¤ œ„k = (Œ∏f + 2 )+
k=1 k=1 k=2
19.4. Applications of Theorem 19.3.1 251

Indeed, this result is due to the well-known Lemma II.3.4 (Gohberg and
Krein, 1969) and Theorem 19.3.1.
Furthermore, assume that
rŒ≥ > 1, r ‚â• 2. (4.2)
Then the series
‚àû ‚àû
r ‚à’Œ≥ r
(k + 1)‚à’rŒ≥ = (Œ∏f + 2‚à’Œ≥ )r + Œ∂(Œ≥r) ‚à’ 1 ‚à’ 2‚à’rŒ≥
œ„k = (Œ∏f + 2 )+
k=1 k=2

converges. Here Œ∂(.) is the Riemann Zeta function, again. Now relation (4.1)
yields
Corollary 19.4.2 Under the conditions (1.1a), (3.1) and (4.2), the inequal-
ity
‚àû
|zk (f )|‚à’r ‚â¤ (Œ∏f + 2‚à’Œ≥ )r + Œ∂(Œ≥r) ‚à’ 1 ‚à’ 2‚à’Œ≥r (4.3)
k=1
is valid. In particular, if Œ≥ > 1, then due to (3.4)
‚àû
|zk (f )|‚à’1 ‚â¤ Œ∏f + Œ∂(Œ≥) ‚à’ 1. (4.4)
k=1

Consider now a positive scalar-valued function Œ¦(t1 , t2 , ..., tj ) with an integer
j, deÔ¬Åned on the domain
0 ‚â¤ tj ‚â¤ tj‚à’1 ‚â¤ t2 ‚â¤ t1 < ‚àû
and satisfying
‚à‚Œ¦ ‚à‚Œ¦ ‚à‚Œ¦
> > ... > > 0 for t1 > t2 > ... > tj . (4.5)
‚à‚t1 ‚à‚t2 ‚à‚tj
Corollary 19.4.3 Under conditions (1.1a), (3.1), (3.2) and (4.5),
Œ¦(|z1 (f )|‚à’1 , |z2 (f )|‚à’1 , ..., |zj (f )|‚à’1 ) ‚â¤ Œ¦(œ„1 , œ„2 , ..., œ„j ).
Indeed, this result is due to Theorem 19.3.1 and the well-known Lemma II.3.5
(Gohberg and Krein, 1969).
In particular, let {dk }‚àû be a decreasing sequence of non-negative num-
k=1
bers. Take
j
Œ¦(t1 , t2 , ..., tj ) = d k tk .
k=1
Then Corollary 19.4.3 yields
j j j
‚à’1
dk (k + 1)‚à’Œ≥
dk |zk (f )| ‚â¤ œ„k d k = d 1 Œ∏ f +
k=1 k=1 k=1

(j = 2, 3, ...).
252 19. Zeros of Entire Functions

19.5 Notes
The variation of the zeros of general analytic functions under perturbations
was investigated, in particular, by P. Rosenbloom (1969). He established
the perturbation result that provides the existence of a zero of a perturbed
function in a given domain. In the present chapter a new approach to the
problem is proposed.
The material in the present chapter is taken from the papers (Gil‚Ä™, 2000a,
2000b, 2000c and 2001). Corollary 19.4.2 supplements the classical Hadamard
theorem (Levin, 1996, p. 18), since it not only asserts the convergence of the
series of the zeros, but also gives us the estimate for the sums of the zeros.

References
 Gil‚Ä™, M.I. (2000a). Inequalities for imaginary parts of zeros of entire
functions. Results in Mathematics, 37, 331-334
 Gil‚Ä™, M.I. (2000b). Perturbations of zeros of a class of entire functions,
Complex Variables, 42, 97-106

 Gil‚Ä™, M.I. (2000c). Approximations of zeros of entire functions by zeros
of polynomials. J. of Approximation Theory, 106, 66-76

 Gil‚Ä™, M.I. (2001). Inequalities for zeros of entire functions, Journal of
Inequalities, 6 463-471.
 Gohberg, I. C. and Krein, M. G. (1969). Introduction to the Theory of
Linear Nonselfadjoint Operators, Trans. of Math. Monographs, v. 18,
Amer. Math. Soc., R.I.

 Levin, B. Ya. (1996). Lectures on Entire Functions, Trans. of Math.
Monographs, v. 150. Amer. Math. Soc., R. I.

 Marcus, M. and Minc, H. (1964). A Survey of Matrix Theory and Matrix
Inequalities, Allyn and Bacon, Boston.
 Rosenbloom, P.C. (1969). Perturbation of zeros of analytic functions. I.
Journal of Approximation Theory, 2, 111-126.
List of Main Symbols

A operator norm of an operator A
(., .) scalar product
|A| matrix whose elements are absolute values of A
‚à’1
A inverse to A
‚à—
A conjugate to A
AI = (A ‚à’ A‚à— )/2i
AR = (A + A‚à— )/2
C1 Trace class
C2 Hilbert-Schmidt ideal
C‚àû the set of all compact operator
Cp Neumann-Schatten ideal
n
C complex Euclidean space
det (A) determinant of A
det2 (A) generalized determinant of A 93
Dom (A) domain of A
g(A) 11, 83
gI (A) 106
H separable Hilbert space
I = IH identity operator (in a space H)
m.r.i. -maximal orthogonal resolution of identity 98
ni(A) nilpotency index of A 102
Np (A) Neumann-Schatten norm of A
N (A) = N2 (A) Hilbert-Schmidt (Frobenius) norm of A
n
R real Euclidean space
RŒ» (A) resolvent of A
rsvA (B) relative spectral variation of B with respect to A 167
RPTO 164
rs (A) spectral radius of A
rl (A) lower spectral radius of A
sj (A) s-number (singular number) of A
svA (B) spectral variation of B with respect to A
T r A = T race A trace of A
w(Œ», A) 165
Œ±(A) = sup Re œÉ(A)
Àú
Œ≤p , Œ≤p 108
Œ≥n,k 12
Œ»k (A) eigenvalue of A
œÉ(A) spectrum of A
(p)
Œ∏k = ‚àö 1 where [x] is the integer part of x
[k/p]!
œÅ(A, Œ») distance between a point lambda and the spectrum of A
Index

Hille-Tamarkin integral
Carleman inequality 33, 93
operator 215
Hille-Tamarkin matrix 235
diagonal part of
compact operator 82
Lidskij‚Ä™s theorem 79
matrix 8
noncompact operator 99
matrix-valued function 4
maximal resolution
estimate for norm of function of
of identity (m.r.i) 98
Hilbert-Schmidt operator 91
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