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Then we have

Corollary 12.4.2 Let relations (1.2) and (4.1) hold. In addition, let

1 1
Лњ
О¶2p (A) в‰Ў max{ в€’ V+ , в€’ Vв€’ } > 0.
П€p (Vв€’ , 1) П€p (V+ , 1)

Then operator A represented by (1.1) is boundedly invertible. Moreover,

1
Aв€’1 в‰¤ .
Лњ
О¶2p (A)rl (D)

Indeed, taking into account Lemma 11.6.2 and repeating the arguments of
the proof of the previous lemma, we arrive at the required result.

12.4.2 Estimates for resolvents
Under (4.1) denote,
(p)
mв€’1 k
Оёk N2p (VВ± )
Лњ
Jp (VВ± , m, z) = (z > 0).
z k+1
k=0

Recall that ОЅВ± (О») в‰Ў ni ((D в€’ О»)в€’1 VВ± ) в‰¤ в€ћ. Due to Theorem 12.2.2 and
inequality (4.4), we get

Theorem 12.4.3 Under conditions (1.1), (1.2) and (4.1), for a О» в€€ Пѓ(D),
let
1
Лњ
О¶2p (О», A) в‰Ў max{ в€’ V+ ,
Лњ
Jp (Vв€’ , ОЅв€’ (О»), w(О», D))
12.5. Notes 179

1
в€’ Vв€’ } > 0.
Лњ
Jp (V+ , ОЅ+ (О»), w(О», D))
Then О» is a regular point of operator A, represented by (1.1). Moreover,
1
RО» (A) в‰¤ .
Лњ
ПЃl (D)О¶2p (О», A)
Лњ
As it was shown in Subsection 11.6.2, Jp (V, ОЅВ± , z) can be replaced by П€p (V, z)
for an arbitrary Volterra operator V в€€ C2p . Now Theorem 12.4.3 yields
Corollary 12.4.4 Under conditions (1.1), (1.2) and (4.1), for a О» в€€ Пѓ(D),
let
1
(1)
О¶2p (О», A) в‰Ў max{ в€’ V+ ,
П€p (Vв€’ , w(О», D))
1
в€’ Vв€’ } > 0.
П€p (V+ , w(О», D))
Then О» is a regular point of operator A, represented by (1.1). Moreover,
1
RО» (A) в‰¤ . (4.6)
(1)
ПЃl (D)О¶2p (О», A)

12.4.3 Spectrum localization
Лњ
Let П„ (A) and V be deп¬Ѓned by (3.1) and (3.2), respectively. Theorem 12.3.1
and Corollary 12.4.4 yield
Lemma 12.4.5 Under the conditions (1.1), (1.2) and
Лњ
V в€€ C2p , (4.7)
the equation
pв€’1 jЛњ Лњ
N2p (V ) 1 N2p (V )
П„ (A) exp [ (1 + )] = 1, (4.8)
z j+1 z 2p
2
j=0

has a unique positive root z2p (A). Moreover,
rsvD (A) в‰¤ z2p (A). (4.9)
By virtiue of Lemma 8.3.2, we can assert that z2p (A) в‰¤ П†p (A), where
Лњ
if N2p (V ) в‰¤ П„ (A)pe
peП„ (A)
П†p (A) := . (4.10)
[ln (N2p (V )/pП„ (A))]в€’1/2p
Лњ Лњ
if N2p (V ) > П„ (A)pe
Now the previous lemma yields
Corollary 12.4.6 Under conditions (1.1), (1.2) and (4.7), we have
rsvD (A) в‰¤ П†p (A). In particular,
rl (A) в‰Ґ rl (D) max{0, 1 в€’ П†p (A)}.
180 12. Relatively Compact Perturbations

12.5 Notes
The present chapter is based on the papers (GilвЂ™, 2001) and (GilвЂ™, 2002).

References
 GilвЂ™, M. I. (2001). On spectral variations under relatively bounded
perturbations, Arch. Math., 76, 458-466.
 GilвЂ™, M. I. (2002). Spectrum localization of inп¬Ѓnite matrices, Math-
ematical Physics, Analysis and Geometry, 4, 379-394 (2002).
13. Inп¬Ѓnite Matrices in
Hilbert Spaces
and Diп¬Ђerential Operators

The present chapter is concerned with applications of some results from Chap-
ters 7-12 to integro-diп¬Ђerential and diп¬Ђerential operators, as well as to inп¬Ѓnite
matrices in a Hilbert space. In particular, we suggest estimates for the spec-
tral radius of an inп¬Ѓnite matrix.

13.1 Matrices with Compact oп¬Ђ Diagonals
13.1.1 Upper bounds for the spectrum
Let {ek }в€ћ be an orthogonal normal basis in a separable Hilbert space H.
k=1
Let A be a linear operator in H represented by a matrix with the entries

ajk = (Aek , ej ) (j, k = 1, 2, ...), (1.1)

where (., .) is the scalar product in H. Then

A = D + V + + Vв€’ , (1.2)

where Vв€’ , V+ and D are the upper triangular, lower triangular, and diagonal
parts of A, respectively:

(V+ ek , ej ) = ajk for j < k, (V+ ek , ej ) = 0 for all j > k;

(Vв€’ ek , ej ) = ajk for j > k, (Vв€’ ek , ej ) = 0 for all j < k;
(Dek , ek ) = akk , (Dek , ej ) = 0 for j = k (j, k = 1, 2, ...). (1.3)

M.I. GilвЂ™: LNM 1830, pp. 181вЂ“188, 2003.
c Springer-Verlag Berlin Heidelberg 2003
182 13. Inп¬Ѓnite Matrices and Diп¬Ђerential Operators

Let {Pk }в€ћ be a maximal orthogonal resolution of the identity, where Pk
k=1
are deп¬Ѓned by
k
Pk = (., ej )ej (k = 1, 2, ...). (1.4)
j=1

Simple calculations show that
Pk V+ Pk = V+ Pk , Pk Vв€’ Pk = Pk Vв€’ (1.5)
and
Pk Dh = Pk Dh (h в€€ Dom (D)) (k = 1, 2, ...). (1.6)
в€ћ
|akk hk |2 < в€ћ; hk = (h, ek ); k = 1, 2, ....}
Dom (D) = {h = (hk ) в€€ H :
k=1

if D is unbounded. We restrict ourselves by the conditions
в€ћ jв€’1
в€ћ в€ћ
2 2 2
|ajk |2 < в€ћ.
|ajk | < в€ћ, N2 (Vв€’ )
N2 (V+ ) = = (1.7)
j=1 k=j+1 j=2 k=1

That is, V+ , Vв€’ are Hilbert-Schmidt matrices, but the results of Chapters 8
and 9 allow us to investigate considerably more general conditions than (1.7).
Without any loss of generality, assume that
N2 (Vв€’ ) в‰¤ N2 (V+ ). (1.8)
The case N2 (Vв€’ ) в‰Ґ N2 (V+ ) can be considered absolutely similarly. Put
if N2 (V+ ) в‰¤ eN2 (Vв€’ ),
eN2 (Vв€’ )
Лњ
П†1 = .
N2 (V+ )[ln (N2 (V+ )/N2 (Vв€’ ))]в€’1/2 if N2 (V+ ) > eN2 (Vв€’ )
(1.9)
Due to Corollary 9.4.3 and Remark 9.4.4 we get
Lemma 13.1.1 Under conditions (1.7), the spectrum of operator A = (ajk )в€ћ
j,k
is included in the set
Лњ
{z в€€ C : |akk в€’ z| в‰¤ П†1 , k = 1, 2, ..., }. (1.10)
In particular, the spectral radius of A satisп¬Ѓes the inequality
Лњ
rs (A) в‰¤ sup |akk | + П†1 , (1.11)
k

provided D is bounded: D в‰Ў supk |akk | < в€ћ. In addition,
Лњ
О±(A) := sup Re Пѓ(A) в‰¤ sup Re akk + П†1 ,
provided sup Re akk < в€ћ. So the considered matrix operator is stable (that
is, its spectrum is in the open left half plane), if
Лњ
Re akk + П†1 < 0 (k = 1, 2, ....).
13.1. Compact Oп¬Ђ-diagonals 183

13.1.2 Lower bounds for the spectrum
Now assume that under (1.7), DI = (D в€’ Dв€— )/2i is a Hilbert-Schmidt oper-
ator:
в€ћ
|ajj в€’ ajj |2 ]1/2 /2 < в€ћ.
N2 (DI ) = [ (1.12)
j=1

Then uder (1.7) AI = (A в€’ Aв€— )/2i is a Hilbert-Schmidt operator:
в€ћ в€ћ
|ajk в€’ akj |2 ]1/2 /2 < в€ћ.
N2 (AI ) = [
j=1 k=1

Recall that
в€ћ
2
|Im О»k (A)|2 ]1/2 .
в€’2
gI (A) = [2N2 (AI )
k=1
в€љ
One can replace gI (A) by 2N2 (AI ). Put

if gI (A) в‰¤ eN2 (Vв€’ )
eN2 (Vв€’ )
Лњ
в€†H (A) := .
gI (A)[ ln (gI (A)/N2 (Vв€’ )) ]в€’1/2 if gI (A) > eN2 (Vв€’ )
(1.13)
Due to Corollary 9.8.2, we get

Lemma 13.1.2 Under conditions (1.7), (1.12), for the matrix A = (ajk )в€ћ ,
j,k
the following relations are true:

Лњ
rs (A) в‰Ґ max{0, sup |akk | в€’ в€†H (A)}, (1.14)
k

Лњ Лњ
rl (A) в‰¤ inf |akk | + в€†H (A) and О±(A) в‰Ґ sup Re akk в€’ в€†H (A).
k k

Лњ
So A is unstable, provided, supk Re akk в€’ в€†H (A) в‰Ґ 0. Note that, according
to Corollary 9.6.2, in the case
в€ћ
|ajk |2 ]1/2 < в€ћ,
N2 (A) в‰Ў [
j,k=1

one can replace gI (A) by
в€ћ
2
|О»k (A)|2 ]1/2 в‰¤ [N2 (A) в€’ |T race A2 |]1/2 .
2
в€’
g(A) = [N2 (A)
k=1

Example 13.1.3 In the complex Hilbert space H = L2 [0, 1], let us consider
an operator A deп¬Ѓned by
184 13. Inп¬Ѓnite Matrices and Diп¬Ђerential Operators

1
K(x, s)u(s)ds (0 в‰¤ x в‰¤ 1),
(Au)(x) = u(x) + (1.15)
0
where K is a scalar Hilbert-Schmidt kernel. Take the orthogonal normal basis

ek (x) = e2ПЂikx (0 в‰¤ x в‰¤ 1; k = 0, В±1, В±2, ....). (1.16)

Let
в€ћ
K(x, s) = bjk ek (x)ej (s) (1.17)
j,k=в€’в€ћ

be the Fourier expansion of K with the Fourier coeп¬ѓcients bjk . Put

ajk = (Aek , ej ) = bjk (j = k), and ajj = (Aej , ej ) = 1 + bjj

for j, k = 0, В±1, В±2, ...). Here (., .) is the scalar product in L2 [0, 1]. Assume
that 1 + bjj = 0 for any integer j. According to (1.3), under consideration
we have
в€ћ в€ћ
2
|bjk |2 < в€ћ,
N (V+ ) =
j=в€’в€ћ k=j+1

and
jв€’1
в€ћ
2
|bjk |2 < в€ћ.
N (Vв€’ ) =
j=в€’в€ћ k=в€’в€ћ

Now relations (1.11) and (1.14) give us the bounds
Лњ
Лњ
|1 + bkk | в€’ в€†H (A)} в‰¤ rs (A) в‰¤ |1 + bkk | + П†1 .
max{0, sup sup
k=0,В±1,В±2,... k=0,В±1,В±2,...

13.2 Matrices with Relatively Compact Oп¬Ђ-
diagonals
Under (1.1), assume that

ПЃl (D) в‰Ў |akk | > 0
inf (2.1)
k=1,2,...

and the operators Dв€’1 VВ± are Hilbert-Schmidt ones: N2 (Dв€’1 VВ± ) = vВ± ,
where
в€ћ jв€’1 в€ћ в€ћ
|ajk |2 |ajk |2
2 2
< в€ћ; v+ = < в€ћ.
vв€’ = (2.2)
|ajj |2 |ajj |2
j=2 j=1
k=1 k=j+1

Note that the results of Section 12.4 allow us to investigate considerably more
general conditions than (2.2). Without any loss of generality assume that

vв€’ в‰¤ v+ . (2.3)
13.3. A Nonselfadjoint Diп¬Ђerential Operator 185

The case vв€’ в‰Ґ v+ can be similarly considered . Put

if v+ в‰¤ evв€’ ,
evв€’
Оґ(vв€’ , v+ ) := . (2.4)
v+ [ln (v+ /vв€’ )]в€’1/2 if v+ > evв€’

Due to Lemma 12.4.5 and Corollary 12.4.6, the spectrum of the matrix A =
(ajk )в€ћ , under conditions (2.2), (2.3) is included in the set
j,k

z
{z в€€ C : |1 в€’ | в‰¤ Оґ(vв€’ , v+ ), k = 1, 2, ..., }. (2.5)
akk
In particular, the lower spectral radius of A satisп¬Ѓes the inequality

rl (A) в‰Ґ max{0, 1 в€’ Оґ(vв€’ , v+ )} inf |akk |. (2.6)
k

In space H = L2 [0, 1], let us consider an operator A deп¬Ѓned by

1 d2 u(x) w(x) du(x)
(Au)(x) = в€’ + + l(x)u(x)
4 dx2 2 dx
(0 < x < 1, u в€€ Dom (A)) (3.1)
with the domain

Dom (A) = {h в€€ L2 [0, 1] : h в€€ L2 [0, 1], h(0) = h(1), h (0) = h (1)}. (3.2)

Here w(.), l(.) в€€ L2 [0, 1] are scalar functions. So the periodic boundary con-
ditions
u(0) = u(1), u (0) = u (1) (3.3)
are imposed. With the orthogonal normal basis (1.16), let
в€ћ в€ћ
Лњk ek and w = wk ek (wk = (w, ek ), Лњk = (l, ek ))
l= l Лњ Лњ l (3.4)
k=в€’в€ћ k=в€’в€ћ

be the Fourier expansions of l and w, respectively. Omitting simple calcula-
tions, we have
(Aek , ej ) = iПЂk wjв€’k + Лњjв€’k (k = j)
Лњ l
and
(Aek , ek ) = ПЂ 2 k 2 + iПЂk w0 + Лњ0 (j, k = 0, В±1, В±2, ...).
Лњ l
Here (., .) is the scalar product in L2 [0, 1]. Take Dom (D) = Dom (A) and
rewrite operator A as the matrix (ajk )в€ћ j,k=в€’в€ћ with the entries

akk = ПЂ 2 k 2 + iПЂk w0 + Лњ0
Лњ l
186 13. Inп¬Ѓnite Matrices and Diп¬Ђerential Operators

and
ajk = iПЂk wjв€’k + Лњjв€’k (j = k; j, k = 0, В±1, В±2, ...).
Лњ l
Assume that
rl (D) = inf |akk | > 0.
k

Then N2 (Dв€’1 V+ ) = vВ± , where
в€ћ kв€’1
2
|(ПЂ 2 k 2 + iПЂk w0 + Лњ0 )в€’1 (iПЂk wjв€’k + Лњjв€’k )|2 =
v+ = Лњ l Лњ l
k=в€’в€ћ j=в€’в€ћ

в€ћ в€’1
|(ПЂ 2 k 2 + iПЂk w0 + Лњ0 )в€’1 (iПЂk wm + Лњm )|2 в‰¤
Лњ l Лњ l
k=в€’в€ћ m=в€’в€ћ
в€ћ в€’1
22 в€’2 2
|wm |2 +
|ПЂ k + iПЂk w0 + Лњ0 |
2 Лњ l ПЂ|k| Лњ
m=в€’в€ћ
k=в€’в€ћ
в€ћ в€’1
22 в€’1 2
|Лњm |2 < в€ћ
|(ПЂ k + iПЂk w0 + Лњ0 ) |
2 Лњ l l
m=в€’в€ћ
k=в€’в€ћ

since w, l в€€ L2 . Similarly,
в€ћ в€ћ
2
|(ПЂ 2 k 2 + iПЂk w0 + Лњ0 )в€’1 (iПЂk wjв€’k + Лњjв€’k )|2 < в€ћ.
vв€’ = Лњ l Лњ l
k=в€’в€ћ j=k+1

Acccording to (2.5), the spectrum of the operator A deп¬Ѓned by (3.1) is in-
cluded in the set
z
{z в€€ C : |1 в€’ | в‰¤ Оґ(vв€’ , v+ ), k = 0, В±1, В±2, ..., },
ПЂ 2 k 2 + iПЂk w0 + Лњ0
Лњ l
where Оґ(vв€’ , v+ ) is deп¬Ѓned by (2.4). In particular, the lower spectral radius
of A satisп¬Ѓes the inequality

rl (A) в‰Ґ min |ПЂ 2 k 2 + iПЂk w0 + Лњ0 | max{0, 1 в€’ Оґ(vв€’ , v+ )}.
Лњ l
k

13.4 Integro-diп¬Ђerential Operators
In space H = L2 [0, 1] let us consider the operator
1
d2 u(x)
(Au)(x) = в€’ + w(x)u(x) + K(x, s)u(s)ds
4dx2 0

(u в€€ Dom (A), 0 < x < 1) (4.1)
13.5. Notes 187

with the domain Dom (A) deп¬Ѓned by (3.2). So the periodic boundary con-
ditions (3.3) hold. Here K is a Hilbert-Schmidt kernel and w(.) в€€ L2 [0, 1] is
a scalar-valued function. Take the orthonormal basis (1.16). Let (1.7) and
(3.4) be the Fourier expansions of K and of w, respectively. Obviously, for
all j, k = 0, В±1, В±2, ...,
ajk = (Aej , ek ) = wjв€’k + bjk (j = k) and
Лњ
akk = (Aek , ek ) = ПЂ 2 k 2 + w0 + bkk .
Лњ
Assume that
|ПЂ 2 k 2 + w0 + bkk | > 0.
rl (D) = inf Лњ (4.2)
k=0,В±1,В±2,...

Then we have N2 (Dв€’1 VВ± ) = vВ± with
в€ћ kв€’1
2
|(ПЂ 2 k 2 + w0 + bkk )в€’1 (wjв€’k + bjk )|2 < в€ћ,
v+ = Лњ Лњ
k=в€’в€ћ j=в€’в€ћ

and
в€ћ в€ћ
2
|(ПЂ 2 k 2 + w0 + bkk )в€’1 (wjв€’k + bjk )|2 < в€ћ.
vв€’ = Лњ Лњ
k=в€’в€ћ j=k+1

According to (2.5), the spectrum of the operator A deп¬Ѓned by (4.1) is included
in the set
z
{z в€€ C : |1 в€’ 2 2 | в‰¤ Оґ(vв€’ , v+ ), k = 0, В±1, В±2, ..., },
ПЂ k + w0 + bkk
Лњ
where Оґ(vв€’ , v+ ) is deп¬Ѓned by (2.4).

13.5 Notes
The results presented in this chapter are based on the paper (GilвЂ™, 2001). In
particular, inequality (1.11) is sharper than the well-known estimate
в€ћ
rs (A) в‰¤ sup |ajk |, (5.1)
j
k=1

cf. (KrasnoselвЂ™skij et al, 1989, inequality (16.2)), provided
в€ћ
Лњ
|ajk | > sup |akk | + П†1 (A).
sup
j k
k=1

For nonnegative matrices the following estimate is well-known, cf. (Kras-
noselвЂ™skij et al, 1989 inequality (16.15)):
в€ћ
rs (A) в‰Ґ rв€ћ (A) в‰Ў
Лњ min ajk (5.2)
j=1,...,в€ћ
k=1
188 13. Inп¬Ѓnite Matrices and Diп¬Ђerential Operators

Our relation (1.14) is sharper than estimate (5.2) in the case |ajk | = ajk (j, k =
1, 2, ...), provided
Лњ
max akk в€’ в€†H (A) > rв€ћ (A).
Лњ
k

That is, (1.11) improves estimate (5.1) and (1.14) improves estimate (5.2) for
matrices which are вЂќcloseвЂќ to triangular ones.
The results in Section 13.4 supplement the well-known results on diп¬Ђer-
ential operators, cf. (Edmunds and Evans, 1990), (Egorov and Kondratiev,
1996), (Locker, 1999) and references therein.

References
 Edmunds, D.E. and Evans V.D. (1990). Spectral Theory and Dif-
ferential Operators. Clarendon Press, Oxford.
 Egorov, Y. and Kondratiev, V. (1996). Spectral Theory of Elliptic
Operators. BirkhВЁuser Verlag, Basel.
a
 GilвЂ™, M.I. (2001). Spectrum localization of inп¬Ѓnite matrices, Math-
ematical Physics, Analysis and Geometry, 4, 379-394

 KrasnoselвЂ™skii, M.A., Lifshits, J. and A. Sobolev (1989). Positive
Linear Systems. The Method of Positive Operators, Heldermann
Verlag, Berlin.

 Locker, J. (1999). Spectral Theory of Nonself-Adjoint Two Point
Diп¬Ђerential Operators. Amer. Math. Soc, Mathematical Surveys
and Monographs, Volume 73, R.I.
14. Integral Operators in
Space L2

The present chapter is concerned with integral operators in L2 . In particular,
we suggest estimates for the spectral radius of an integral operator.

14.1 Scalar Integral Operators
Consider a scalar integral operator A deп¬Ѓned in H = L2 [0, 1] by
1
K(x, s)u(s)ds (u в€€ L2 [0, 1]; x в€€ [0, 1]),
(Au)(x) = a(x)u(x) + (1.1)
0

where a(.) is a real bounded measurable function, K is a real Hilbert-Schmidt
kernel. Deп¬Ѓne the maximal resolution of the identity P (t) (в€’ в‰¤ t в‰¤ 1; > 0)
by
if в€’ в‰¤ t < x,
0
(P (t)u)(x) =
u(x) if x в‰¤ t в‰¤ 1
with x в€€ [0, 1]. Then, the conditions (1.1) and (1.2) from Section 9.1 are
valid with
1
(Du)(x) = a(x)u(x), (V+ u)(x) = K(x, s)u(s)ds,
x

and x
K(x, s)u(s)ds (u в€€ L2 [0, 1]; x в€€ [0, 1]).
(Vв€’ u)(x) =
0
So
1 1
2
K 2 (x, s) ds dx
N2 (V+ ) =
0 x

M.I. GilвЂ™: LNM 1830, pp. 189вЂ“197, 2003.
c Springer-Verlag Berlin Heidelberg 2003
190 14. Integral Operators

and
1 x
2
K 2 (x, s)ds dx.
N2 (Vв€’ ) =
0 0
Without any loss of generality, assume that

N2 (Vв€’ ) в‰¤ N2 (V+ ). (1.2)

The case N2 (Vв€’ ) в‰Ґ N2 (V+ ) can be similarly considered. So according to
Лњ
relations (4.1) and (4.2) from Section 9.4, we have П„ (A) в‰¤ N2 (Vв€’ ) and V =
V+ . Put
if N2 (V+ ) в‰¤ eN2 (Vв€’ )
eN2 (Vв€’ )
Лњ
П†1 = .
N2 (V+ )[ln (N2 (V+ )/ N2 (Vв€’ )) ]в€’1/2
if N2 (V+ ) > eN2 (Vв€’ )
(1.3)
Due to Corollary 9.4.3 and Remark 9.4.4, the spectrum of operator A is
included in the set
Лњ
{z в€€ C : |a(x) в€’ z| в‰¤ П†1 , 0 в‰¤ x в‰¤ 1}.

Hence, the spectral radius of A satisп¬Ѓes the inequality
Лњ
rs (A) в‰¤ sup |a(x)| + П†1 .
xв€€[0,1]

In particular, if a(x) в‰Ў 0, then
Лњ
rs (A) в‰¤ П†1 (A). (1.4)

Let us derive the lower estimates for the spectrum. Clearly,
1 1
2 2 в€—
|K(x, s) в€’ K(s, x)|2 ds dx/4.
в‰Ў в€’ A )/2i) =
N2 (AI ) N2 ((A
0 0

Recall that
в€ћ
2
|Im О»k (A)|2 ]1/2
в€’2
gI (A) = [2N2 (AI )
k=1
в€љ
and one can replace gI (A) by 2N2 (AI ). Put
if gI (A) в‰¤ eN2 (Vв€’ )
eN2 (Vв€’ )
Лњ
в€†H := . (1.5)
gI (A)[ln (gI (A)/N2 (Vв€’ )) ]в€’1/2 if gI (A) > eN2 (Vв€’ )

Due to Corollary 9.8.2, for the integral operator deп¬Ѓned by (1.1), the following
relations are true:
Лњ
rs (A) в‰Ґ max{0, sup |a(x)| в€’ в€†H }, (1.6)
xв€€[0,1]

Лњ Лњ
rl (A) в‰¤ inf |a(x)| + в€†H and О±(A) в‰Ґ sup Re a(x) в€’ в€†H .
x xв€€[0,1]
14.2. Relatively Small Kernels 191

14.2 Matrix Integral Operators
with Relatively Small Kernels
Let П‰ вЉ† Rm be a set with a п¬Ѓnite Lebesgue measure, and H в‰Ў L2 (П‰, Cn )
be a Hilbert space of functions deп¬Ѓned on П‰ with values in Cn and equipped
with the scalar product

(f, h)H = (f (s), h(s))C n ds,
П‰

where (., .)C n is the scalar product in Cn . Consider in L2 (П‰, Cn ) the operator

K(x, s)h(s)ds (h в€€ L2 (П‰, Cn )),
(Ah)(x) = Q(x)h(x) + (2.1)
П‰

where Q(x), K(x, s) are matrix-valued functions deп¬Ѓned on П‰ and П‰ Г— П‰,
respectively. It is assumed that Q is bounded measurable and K is a Hilbert-
Schmidt kernel. So
Лњ Лњ
A = Q + K,
where
Лњ
(Qh)(x) = Q(x)h(x)
and
Лњ K(x, s)h(s)ds (x в€€ П‰).
(Kh)(x) =
П‰
Besides,
2 1/2
Лњ
N2 (K) = [ K(x, s) C n ds] ,
П‰ П‰
where . is the Euclidean norm.
Cn

Lemma 14.2.1 The spectrum of operator A deп¬Ѓned by (2.1) lies in the set

{О» в€€ C : N2 (K) sup (Q(x) в€’ IC n О»)в€’1
Лњ в‰Ґ 1}.
Cn
xв€€П‰

Proof: Since,

A в€’ О»I = Q + K в€’ О»I = (Q в€’ О»I)(I + (Q в€’ О»I)в€’1 K),
Лњ Лњ Лњ Лњ Лњ

if
(Q в€’ О»I)в€’1 K
Лњ Лњ < 1,
H

then О» is a regular point. So for any Вµ в€€ Пѓ(A),

1 в‰¤ (Q в€’ ВµI)в€’1 в‰¤ (Q в€’ ВµI)в€’1
Лњ Лњ Лњ Лњ
K H N2 (K).
H H

But
(Q в€’ ВµI)в€’1 в‰¤ sup (Q(x) в€’ IC n Вµ)в€’1
Лњ Cn .
H
xв€€П‰
192 14. Integral Operators

This proves the lemma. 2

Due to Corollary 2.1.2, for a п¬Ѓxed x we have
nв€’1
g k (Q(x))
в€’1
в€љ
(Q(x) в€’ IC n О») в‰¤ . (2.2)
Cn
k!ПЃk+1 (Q(x), О»)
k=0

Now Lemma 14.2.1 yields
Lemma 14.2.2 Let operator A be deп¬Ѓned by (2.1). Then its spectrum lies
in the set
nв€’1
g k (Q(x))
в€љ
Лњ
{О» в€€ C : N2 (K) в‰Ґ 1, x в€€ П‰}.
k!ПЃk+1 (Q(x), О»)
k=0

Corollary 14.2.3 Let operator A be deп¬Ѓned by (2.1). In addition, let
nв€’1
g k (Q(x))
в€љ
Лњ
N2 (K) sup < 1,
k!dk+1 (Q(x))
xв€€П‰ 0
k=0

where
min |О»k (Q(x))|.
d0 (Q(x)) = (2.3)
k=1,...,n

Then A is boundedly invertible in L2 (П‰, Cn ).
With a п¬Ѓxed x в€€ П‰, consider the algebraic equation
nв€’1
g k (Q(x))z nв€’kв€’1
n
в€љ
Лњ
z = N2 (K) . (2.4)
k!
k=0

Lemma 14.2.4 Let z0 (x) be the extreme right (unique positive) root of (2.4).
Then for any point Вµ в€€ Пѓ(A) there are x в€€ П‰ and an eigenvalue О»j (Q(x)) of
matrix Q(x), such that
|Вµ в€’ О»j (Q(x))| в‰¤ z0 (x). (2.5)
In particular,
rs (A) в‰¤ sup(rs (Q(x)) + z0 (x)).
x

Proof: Due to Lemma 14.2.2, for any point Вµ в€€ Пѓ(A) there is x в€€ П‰, such
that the inequality
nв€’1
g k (Q(x))
в€љ
Лњ в‰Ґ1
N2 (K)
k!ПЃk+1 (Q(x), Вµ)
k=0

is valid. Comparing this with (2.4), we have ПЃ(Q(x), Вµ) в‰¤ z0 (x). This proves
the required result. 2
14.3. Perturbations of Convolutions 193

Corollary 14.2.5 Let Q(x) be a normal matrix for all x в€€ П‰. Then for any
point Вµ в€€ Пѓ(A) there are x в€€ П‰ and О»j (Q(x))Пѓ(Q(x)), such that

Лњ
|Вµ в€’ О»j (Q(x))| в‰¤ N2 (K).

Лњ
In particular, rs (A) в‰¤ N2 (K) + supx (rs (Q(x))).
Лњ
Indeed, since Q(x) is normal, we have g(Q(x)) = 0 and z0 (x) = N2 (K). Now
the result is due to the latter theorem.
Put
nв€’1 k
g (Q(x))
в€љ
Лњ
b(x) := N2 (K) .
k!
k=0

Due to Corollary 1.6.2, z0 (x) в‰¤ Оґn (x), where

b(x) if b(x) в‰¤ 1 and Оґn (x) = b(x) if b(x) > 1.
n
Оґn (x) =

Now Theorem 14.2.4 implies

Theorem 14.2.6 Under condition (2.7), for any point Вµ в€€ Пѓ(A), there are
x в€€ П‰ and an eigenvalue О»j (Q(x)) of Q(x), such that

|Вµ в€’ О»j (Q(x))| в‰¤ Оґn (x).

In particular, rs (A) в‰¤ supx (rs (Q(x)) + Оґn (x)).

14.3 Perturbations of Matrix Convolutions
Consider in H = L2 ([в€’ПЂ, ПЂ], Cn ) the convolution operator
ПЂ
K0 (x в€’ s)h(s)ds (h в€€ L2 ([в€’ПЂ, ПЂ], Cn )),
(Ch)(x) = Q0 h(x) + (3.1)
в€’ПЂ

where Q0 is a constant matrix, K0 is a matrix-valued function deп¬Ѓned on
[в€’ПЂ, ПЂ] with
K0 C n в€€ L2 [в€’ПЂ, ПЂ],
having the Fourier expansion
в€ћ
Dk eikx
K0 (x) =
k=в€’в€ћ

with the matrix Fourier coeп¬ѓcients
ПЂ
1
K0 (s)eв€’iks ds.
Dk = в€љ
2ПЂ в€’ПЂ
194 14. Integral Operators

Put
Bk = Q0 + Dk .
We have
Ceikx = Bk eikx . (3.2)
Let djk be an eigenvector of Bk , corresponding to an eigenvalue О»j (Bk ) (j =
1, ...n). Then
ПЂ
ikx ikx
K0 (x в€’ s)djk eiks ds =
Ce djk = e Q0 djk +
в€’ПЂ

eikx Bk djk = eikx О»j (Bk )djk .
Since the set
{eikx }k=в€ћ
k=в€’в€ћ

is a basis in L2 [в€’ПЂ, ПЂ] we have the following result

Lemma 14.3.1 The spectrum of operator (3.1) consists of the points

О»j (Bk ) (k = 0, В±1, В±2, ... ; j = 1, ...n).

Let Pk be orthogonal projectors deп¬Ѓned by
ПЂ
1
(Pk h)(x) = eikx h(s)eв€’iks ds.
2ПЂ в€’ПЂ

Since
в€ћ
Pk = IH ,
k=в€’в€ћ

it can be directly checked by (3.2) that the equality
в€ћ
C= Bk Pk
k=в€’в€ћ

holds. Hence, the relation
в€ћ
в€’1
(Bk в€’ IC n О»)в€’1 Pk
(C в€’ IH О») =
k=в€’в€ћ

is valid for any regular О». Therefore,

(C в€’ IH О»)в€’1 (Bk в€’ IC n О»)в€’1
в‰¤ sup Cn .
H
k=0, В±1,...

Using Corollary 2.1.2, we get
14.3. Perturbations of Convolutions 195

Lemma 14.3.2 The resolvent of convolution C deп¬Ѓned by (3.1) satisп¬Ѓes the
inequality
nв€’1
g k (Bl )
в€’1
в€љ
(C в€’ О»I) в‰¤ sup .
H
k!ПЃk+1 (Bl , О»)
l=0, В±1,...
k=0

Consider now the operator
ПЂ
(Ah)(x) в‰Ў Q0 h(x) + K0 (x в€’ s)h(s)ds + (Zh)(x) (в€’ПЂ в‰¤ x в‰¤ ПЂ). (3.3)
в€’ПЂ

where Z is a bounded operator in L2 ([в€’ПЂ, ПЂ], Cn ). We easily have by the
previous lemma that the inequalities
nв€’1
g k (Bl )
в€’1
в€љ
(C в€’ О»I) в‰¤Z
Z sup <1
H H H
k!ПЃk+1 (Bl , О»)
l=0, В±1,...
k=0

imply that О» is a regular point. Hence we arrive at
Lemma 14.3.3 The spectrum of operator A deп¬Ѓned by (3.3) lie in the set
nв€’1
g k (Bl )
в€љ
{О» в€€ C : Z в‰Ґ 1}.
sup
H
k!ПЃk+1 (Bl , О»)
l=0, В±1,...
k=0

In other words, for any Вµ в€€ Пѓ(A), there are

l = 0, В±1, В±2, ... and j = 1, ..., n,

such that
nв€’1
g k (Bl )
в€љ в‰Ґ 1.
Z H
k!|Вµ в€’ О»j (Bl )|k+1
k=0

Corollary 14.3.4 Operator A deп¬Ѓned by (3.3) is invertible provided that
nв€’1
g k (Bk )
в€љ в‰¤ c0 < 1 (c0 = const)
Z H
k!|О»j (Bl )|k+1
k=0

for all
l = 0, В±1, В±2, ... and j = 1, ..., n.

Let zl be the extreme right (unique positive) root of the equation
nв€’1
z nв€’1в€’k g k (Bl )
n
в€љ
z=Z . (3.4)
H
k!
k=0

Since the function in the right part of (3.4) monotonically increases as z > 0
increases, Lemma 14.3.4 implies
196 14. Integral Operators

Theorem 14.3.5 For any point Вµ of the spectrum of operator (3.3), there
are indexes l = 0, В±1, В±2, ... and j = 1, ..., n, such that

|Вµ в€’ О»j (Bl )| в‰¤ zl , (3.5)

where zl is the extreme right (unique positive) root of the algebraic equation
(3.4). In particular,

rs (A) в‰¤ max rs (Bl ) + zl .
l=0,В±1,...

If all the matrices Bl are normal, then g(Bl ) в‰Ў 0, zl = Z H, and (3.5) takes
the form
|Вµ в€’ О»j (Bl )| в‰¤ Z H .
Assume that
nв€’1
g k (Bl )
в€љ в‰¤ 1 (l = 0, В±1, В±2, ...).
bl := Z (3.6)
H
k!
k=0

Then due to Lemma 1.6.1
zl в‰¤ n
bl .
Now Theorem 14.3.5 implies

Corollary 14.3.6 Let A be deп¬Ѓned by (3.3) and condition (3.6) hold. Then
for any Вµ в€€ Пѓ(A) there are l = 0, В±1, В±2, ... and j = 1, ..., n, such that

|Вµ в€’ О»j (Bl )| в‰¤ n
bl .

In particular,
rs (A) в‰¤ n
sup bl + rs (Bl ).
l=0,В±1,В±2,...

14.4 Notes
Inequality (1.4) improves the well-known estimate
1
Лњ
rs (A) в‰¤ Оґ0 (A) в‰Ў vrai sup |K(x, s)|ds,
x 0

cf. (KrasnoselвЂ™skii et al., 1989, Section 16.6) for operators which are вЂќcloseвЂќ
to Volterra ones.
The material in this chapter is taken from the papers (GilвЂ™, 2000), (GilвЂ™,
2003).
14.4. Notes 197

References
 GilвЂ™, M.I. (2000). Invertibility conditions and bounds for spectra of ma-
trix integral operators, Monatshefte fВЁr mathematik, 129, 15-24.
u
 GilвЂ™, M.I. (2003). Inner bounds for spectra of linear operators, Proceed-
ings of the American Mathematical Society , (to appear).
 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.
15. Operator Matrices

In the present chapter we consider the invertibility and spectrum of matrices,
whose entries are unbounded, in general, operators. In particular, under some
restrictions, we improve the Gershgorin-type bounds. Applications to matrix
diп¬Ђerential operators are also discussed.

15.1 Invertibility Conditions
Let H be an orthogonal sum of Hilbert spaces Ek (k = 1, ..., n < в€ћ) with
norms . Ek :
H в‰Ў E1 вЉ• E2 вЉ• ... вЉ• En .
Consider in H the operator matrix
пЈ« пЈ¶
A11 A12 . . . A1n
пЈ¬ A21 A22 . . . A2n пЈ·
A=пЈ¬ пЈ·, (1.1)
пЈ­. .пЈё
... .
An1 An2 . . . Ann

where Ajk are linear operators acting from Ek to Ej . In the present chapter,
invertibility conditions and bounds for the spectrum of operator (1.1) are
investigated under the assumption that we have an information about the
spectra of diagonal operators.
Let h = (hk в€€ Ek )n be an element of H. Everywhere in the present
k=1
chapter the norm in H is deп¬Ѓned by the relation
n
2 1/2
hв‰Ўh =[ hk Ek ] (1.2)
H
k=1

and I = IH is the unit operator in H.

M.I. GilвЂ™: LNM 1830, pp. 199вЂ“213, 2003.
c Springer-Verlag Berlin Heidelberg 2003
200 15. Operator Matrices

Denote by V, W and D the upper triangular, lower triangular, and diag-
onal parts of A, respectively. That is,
пЈ« пЈ¶
0 A12 . . . A1n
пЈ¬0 . . . A2n пЈ·
0
V =пЈ¬ пЈ·,
пЈ­ . ... .пЈё
.
0 0 ... 0
пЈ« пЈ¶
0 0 ... 0 0
пЈ¬ A21 0пЈ·
0 ... 0
W =пЈ¬ пЈ·
пЈ­. пЈё
... . .
An1 An2 ... An,nв€’1 0
and
D = diag [A11 , A22 , ..., Ann ].
Recall that, for a linear operator A, Dom (A) means the domain, Пѓ(A) is the
spectrum, О»k (A) (k = 1, 2, ...) are the eigenvalues with their multiplicities,
ПЃ(A, О») is the distance between the spectrum of A and a О» в€€ C.

Theorem 15.1.1 Let the diagonal operator D be invertible and the operators

VA в‰Ў Dв€’1 V, WA в‰Ў Dв€’1 W be bounded . (1.3)

nв€’1
j
(в€’1)k+j VA WA < 1
k
(1.4)
j,k=1

hold. Then operator A deп¬Ѓned by (1.1) is invertible.

Proof: We have

A = D + V + W = D(I + VA + WA ) = D[(I + VA )(I + WA ) в€’ VA WA ].

Simple calculations show that
n n
VA = WA = 0. (1.5)

So VA and WA are nilpotent operators and, consequently, the operators,
I + VA and I + WA are invertible. Thus,

A = D(I + VA )[I в€’ (I + VA )в€’1 VA WA (I + WA )в€’1 ](I + WA ).

Therefore, the condition

(I + VA )в€’1 VA WA (I + WA )в€’1 < 1
15.1. Invertibility Conditions 201

provides the invertibility of A. But according to (1.5),

nв€’1 nв€’1
в€’1 kв€’1 k в€’1
(в€’1)kв€’1 WA .
k
(I + VA ) VA = (в€’1) VA , WA (I + WA ) =
k=1 k=1

Hence, the required result follows. 2

Corollary 15.1.2 Let operator matrix A deп¬Ѓned by (1.1) be an upper trian-
gular one (W=0), D an invertible operator and Dв€’1 V a bounded one. Then
A is invertible.
Similarly, let operator matrix A be a lower triangular one (V=0) and
в€’1
D W a bounded operator. Then A is invertible.

Corollary 15.1.3 Let the diagonal operator D be invertible and the condi-
tions (1.3) and
nв€’2
k j
VA W A VA WA <1 (1.6)
j,k=0

hold. Then operator (1.1) is invertible.
In particular, let VA , WA = 1. Then (1.6) can be written in the form

(1 в€’ VA nв€’1 )(1 в€’ WA nв€’1 )
VA W A < 1. (1.7)
(1 в€’ VA )(1 в€’ WA )

Indeed, taking into account that
nв€’2 nв€’2
1 в€’ VA nв€’1 1 в€’ WA nв€’1
k k
VA = , WA =
1 в€’ VA 1 в€’ WA
k=0 k=0

and using Theorem 15.1.1, we arrive at the required result. 2

We need also the following

Lemma 15.1.4 Let

ajk в‰Ў Ajk < в€ћ (j, k = 1, ..., n).
Ek в†’Ej

Then the norm of operator A deп¬Ѓned by (1.1) is subject to the relation

Aв‰¤a
Лњ Cn ,

where a is the linear operator in the Euclidean space Cn , deп¬Ѓned by the matrix
Лњ
with the entries ajk and . C n is the Euclidean norm.
202 15. Operator Matrices

The proof is a simple application of relation (1.2) and it is left to the reader.
The latter corollary implies
n
2 2
в‰¤
A Ajk Ek в†’Ej . (1.8)
j,k=1

Consider the case n = 2:
A11 A12
A= . (1.9)
A21 A22

Clearly,

0 Aв€’1 A12 0 0
11
VA = and WA = .
Aв€’1 A21
0 0 0
22

Hence,
Aв€’1 A12 Aв€’1 A21 0
11 22
VA W A = .
0 0
Thus, due to Theorem 15.1.1, if

Aв€’1 A12 Aв€’1 A21 < 1,
11 22

then operator (1.9) is invertible.

15.2 Bounds for the Spectrum
Theorem 15.2.1 For any regular point О» of D, let

V (О») := (D в€’ IH О»)в€’1 V and W (О») := (D в€’ IH О»)в€’1 W be bounded operators .
Лњ Лњ
(2.1)
Then the spectrum of operator A deп¬Ѓned by (1.1) lies in the union of the sets
Пѓ(D) and
nв€’2
k j
Лњ Лњ Лњ Лњ
{О» в€€ C : V (О»)W (О») в‰Ґ 1}.
V (О») W (О»)
j,k=0

Indeed, if for some О» в€€ Пѓ(A),
nв€’2
k j
Лњ Лњ Лњ Лњ
V (О»)W (О») V (О») W (О») < 1, (2.2)
j,k=0

then due to Corollary 15.1.3, A в€’ О»I is invertible. This proves the required
result. 2
15.2. Bounds for the Spectrum 203

Corollary 15.2.2 Let operator matrix (1.1) be an upper triangular one, and
Лњ
V (О») be bounded for all regular О» of D. Then

Пѓ(A) = в€Єn Пѓ(Akk ) = Пѓ(D). (2.3)
k=1

Лњ
Similarly, let (1.1) be lower triangular and W (О») be bounded for all regular
О» of D. Then (2.3) holds.
Лњ
Indeed, let A be upper triangular. Then W (О») = 0. Now the result is due to
Theorem 15.2.1. The lower triangular case can be similarly considered.
This result shows that Theorem 15.2.1 is exact.

Lemma 15.2.3 Let W and V be bounded operators and the condition

(D в€’ IH О»)в€’1 в‰¤ О¦(ПЃв€’1 (D, О»)) (О» в€€ Пѓ(D))
/ (2.4)

hold, where О¦(y) is a continuous increasing function of y в‰Ґ 0 with the prop-
erties О¦(0) = 0 and О¦(в€ћ) = в€ћ. In addition, let z0 be the unique positive
root of the scalar equation
nв€’1
О¦k+j (y) V j k
W = 1. (2.5)
j,k=1

Then the spectral variation of operator A deп¬Ѓned by (1.1) with respect to D
satisп¬Ѓes the inequality
1
svD (A) в‰¤ .
z0
Proof: Due to (2.4)

V (О») в‰¤ V О¦(ПЃв€’1 (D, О»)), W (О») в‰¤ W О¦(ПЃв€’1 (D, О»)).
Лњ Лњ

For any О» в€€ Пѓ(A) and О» в€€ Пѓ(D), Theorem 15.2.1 implies
/
nв€’1
О¦k+j (ПЃв€’1 (D, О»)) V k j
в‰Ґ 1.
W
j,k=1

Taking into account that О¦ is increasing and comparing the latter inequality
with (2.5), we have
ПЃв€’1 (D, О») в‰Ґ z0
for any О» в€€ Пѓ(A). This proves the required result. 2

For instance, let n = 2. Then (2.5) takes the form

О¦2 (y) V W = 1. (2.6)
204 15. Operator Matrices

Hence it follows that
1
z0 = ОЁ( ),
V W
where ОЁ is the function inverse to О¦: О¦(ОЁ(y)) = y. Thus, in the case n = 2,
1
svD (A) в‰¤ .
ОЁ( в€љ 1
)
V W

15.3 Operator Matrices with Normal Entries
Assume that H is an orthogonal sum of the same Hilbert spaces Ek в‰Ў E
(k = 1, ..., n) with norm . E . Consider in H the operator matrix deп¬Ѓned by
(1.1), assuming that
Ajj = Sj (j = 1, ..., n), (3.1)
where Sj are normal, unbounded in general operators in Ej , and
Ajk = П†k (Sj ) (j = k; j, k = 1, ..., n), (3.2)
where П†k (s) are scalar-valued measurable functions of s в€€ C. In addition,
assume that
О±jk в‰Ў sup |П†k (t)tв€’1 | < в€ћ. (3.3)
tв€€Пѓ(Sj )

Then Aв€’1 Ajk are bounded normal operators with the norms
jj

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