<<

. 8
( 11)



>>


Then we have

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

1 1
˜
ζ2p (A) ≡ max{ ’ V+ , ’ V’ } > 0.
ψp (V’ , 1) ψp (V+ , 1)

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

1
A’1 ¤ .
˜
ζ2p (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)
m’1 k
θk N2p (V± )
˜
Jp (V± , m, z) = (z > 0).
z k+1
k=0

Recall that ν± (») ≡ ni ((D ’ »)’1 V± ) ¤ ∞. 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 » ∈ σ(D),
let
1
˜
ζ2p (», A) ≡ max{ ’ V+ ,
˜
Jp (V’ , ν’ (»), w(», D))
12.5. Notes 179

1
’ V’ } > 0.
˜
Jp (V+ , ν+ (»), w(», D))
Then » is a regular point of operator A, represented by (1.1). Moreover,
1
R» (A) ¤ .
˜
ρl (D)ζ2p (», A)
˜
As it was shown in Subsection 11.6.2, Jp (V, ν± , z) can be replaced by ψp (V, z)
for an arbitrary Volterra operator V ∈ C2p . Now Theorem 12.4.3 yields
Corollary 12.4.4 Under conditions (1.1), (1.2) and (4.1), for a » ∈ σ(D),
let
1
(1)
ζ2p (», A) ≡ max{ ’ V+ ,
ψp (V’ , w(», D))
1
’ V’ } > 0.
ψp (V+ , w(», D))
Then » is a regular point of operator A, represented by (1.1). Moreover,
1
R» (A) ¤ . (4.6)
(1)
ρl (D)ζ2p (», A)

12.4.3 Spectrum localization
˜
Let „ (A) and V be de¬ned 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
˜
V ∈ C2p , (4.7)
the equation
p’1 j˜ ˜
N2p (V ) 1 N2p (V )
„ (A) exp [ (1 + )] = 1, (4.8)
z j+1 z 2p
2
j=0

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


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


References
[1] Gil™, M. I. (2001). On spectral variations under relatively bounded
perturbations, Arch. Math., 76, 458-466.
[2] Gil™, M. I. (2002). Spectrum localization of in¬nite matrices, Math-
ematical Physics, Analysis and Geometry, 4, 379-394 (2002).
13. In¬nite Matrices in
Hilbert Spaces
and Di¬erential Operators

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




13.1 Matrices with Compact o¬ Diagonals
13.1.1 Upper bounds for the spectrum
Let {ek }∞ 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 + + V’ , (1.2)

where V’ , 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;

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

M.I. Gil™: LNM 1830, pp. 181“188, 2003.
c Springer-Verlag Berlin Heidelberg 2003
182 13. In¬nite Matrices and Di¬erential Operators


Let {Pk }∞ be a maximal orthogonal resolution of the identity, where Pk
k=1
are de¬ned by
k
Pk = (., ej )ej (k = 1, 2, ...). (1.4)
j=1

Simple calculations show that
Pk V+ Pk = V+ Pk , Pk V’ Pk = Pk V’ (1.5)
and
Pk Dh = Pk Dh (h ∈ Dom (D)) (k = 1, 2, ...). (1.6)
In addition,

|akk hk |2 < ∞; hk = (h, ek ); k = 1, 2, ....}
Dom (D) = {h = (hk ) ∈ H :
k=1

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

That is, V+ , V’ 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 (V’ ) ¤ N2 (V+ ). (1.8)
The case N2 (V’ ) ≥ N2 (V+ ) can be considered absolutely similarly. Put
if N2 (V+ ) ¤ eN2 (V’ ),
eN2 (V’ )
˜
φ1 = .
N2 (V+ )[ln (N2 (V+ )/N2 (V’ ))]’1/2 if N2 (V+ ) > eN2 (V’ )
(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 )∞
j,k
is included in the set
˜
{z ∈ C : |akk ’ z| ¤ φ1 , k = 1, 2, ..., }. (1.10)
In particular, the spectral radius of A satis¬es the inequality
˜
rs (A) ¤ sup |akk | + φ1 , (1.11)
k

provided D is bounded: D ≡ supk |akk | < ∞. In addition,
˜
±(A) := sup Re σ(A) ¤ sup Re akk + φ1 ,
provided sup Re akk < ∞. So the considered matrix operator is stable (that
is, its spectrum is in the open left half plane), if
˜
Re akk + φ1 < 0 (k = 1, 2, ....).
13.1. Compact O¬-diagonals 183


13.1.2 Lower bounds for the spectrum
Now assume that under (1.7), DI = (D ’ D— )/2i is a Hilbert-Schmidt oper-
ator:

|ajj ’ ajj |2 ]1/2 /2 < ∞.
N2 (DI ) = [ (1.12)
j=1

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


Recall that

2
|Im »k (A)|2 ]1/2 .
’2
gI (A) = [2N2 (AI )
k=1

One can replace gI (A) by 2N2 (AI ). Put

if gI (A) ¤ eN2 (V’ )
eN2 (V’ )
˜
∆H (A) := .
gI (A)[ ln (gI (A)/N2 (V’ )) ]’1/2 if gI (A) > eN2 (V’ )
(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 )∞ ,
j,k
the following relations are true:

˜
rs (A) ≥ max{0, sup |akk | ’ ∆H (A)}, (1.14)
k


˜ ˜
rl (A) ¤ inf |akk | + ∆H (A) and ±(A) ≥ sup Re akk ’ ∆H (A).
k k

˜
So A is unstable, provided, supk Re akk ’ ∆H (A) ≥ 0. Note that, according
to Corollary 9.6.2, in the case

|ajk |2 ]1/2 < ∞,
N2 (A) ≡ [
j,k=1


one can replace gI (A) by

2
|»k (A)|2 ]1/2 ¤ [N2 (A) ’ |T race A2 |]1/2 .
2

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 de¬ned by
184 13. In¬nite Matrices and Di¬erential Operators

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

ek (x) = e2πikx (0 ¤ x ¤ 1; k = 0, ±1, ±2, ....). (1.16)

Let

K(x, s) = bjk ek (x)ej (s) (1.17)
j,k=’∞

be the Fourier expansion of K with the Fourier coe¬cients bjk . Put

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

for j, k = 0, ±1, ±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
∞ ∞
2
|bjk |2 < ∞,
N (V+ ) =
j=’∞ k=j+1

and
j’1

2
|bjk |2 < ∞.
N (V’ ) =
j=’∞ k=’∞

Now relations (1.11) and (1.14) give us the bounds
˜
˜
|1 + bkk | ’ ∆H (A)} ¤ rs (A) ¤ |1 + bkk | + φ1 .
max{0, sup sup
k=0,±1,±2,... k=0,±1,±2,...




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

ρl (D) ≡ |akk | > 0
inf (2.1)
k=1,2,...


and the operators D’1 V± are Hilbert-Schmidt ones: N2 (D’1 V± ) = v± ,
where
∞ j’1 ∞ ∞
|ajk |2 |ajk |2
2 2
< ∞; v+ = < ∞.
v’ = (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

v’ ¤ v+ . (2.3)
13.3. A Nonselfadjoint Di¬erential Operator 185


The case v’ ≥ v+ can be similarly considered . Put

if v+ ¤ ev’ ,
ev’
δ(v’ , v+ ) := . (2.4)
v+ [ln (v+ /v’ )]’1/2 if v+ > ev’

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

z
{z ∈ C : |1 ’ | ¤ δ(v’ , v+ ), k = 1, 2, ..., }. (2.5)
akk
In particular, the lower spectral radius of A satis¬es the inequality

rl (A) ≥ max{0, 1 ’ δ(v’ , v+ )} inf |akk |. (2.6)
k




13.3 A Nonselfadjoint Di¬erential Operator
In space H = L2 [0, 1], let us consider an operator A de¬ned by

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

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

Here w(.), l(.) ∈ 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
∞ ∞
˜k ek and w = wk ek (wk = (w, ek ), ˜k = (l, ek ))
l= l ˜ ˜ l (3.4)
k=’∞ k=’∞

be the Fourier expansions of l and w, respectively. Omitting simple calcula-
tions, we have
(Aek , ej ) = iπk wj’k + ˜j’k (k = j)
˜ l
and
(Aek , ek ) = π 2 k 2 + iπk w0 + ˜0 (j, k = 0, ±1, ±2, ...).
˜ l
Here (., .) is the scalar product in L2 [0, 1]. Take Dom (D) = Dom (A) and
rewrite operator A as the matrix (ajk )∞ j,k=’∞ with the entries


akk = π 2 k 2 + iπk w0 + ˜0
˜ l
186 13. In¬nite Matrices and Di¬erential Operators


and
ajk = iπk wj’k + ˜j’k (j = k; j, k = 0, ±1, ±2, ...).
˜ l
Assume that
rl (D) = inf |akk | > 0.
k

Then N2 (D’1 V+ ) = v± , where
∞ k’1
2
|(π 2 k 2 + iπk w0 + ˜0 )’1 (iπk wj’k + ˜j’k )|2 =
v+ = ˜ l ˜ l
k=’∞ j=’∞

∞ ’1
|(π 2 k 2 + iπk w0 + ˜0 )’1 (iπk wm + ˜m )|2 ¤
˜ l ˜ l
k=’∞ m=’∞
∞ ’1
22 ’2 2
|wm |2 +
|π k + iπk w0 + ˜0 |
2 ˜ l π|k| ˜
m=’∞
k=’∞
∞ ’1
22 ’1 2
|˜m |2 < ∞
|(π k + iπk w0 + ˜0 ) |
2 ˜ l l
m=’∞
k=’∞

since w, l ∈ L2 . Similarly,
∞ ∞
2
|(π 2 k 2 + iπk w0 + ˜0 )’1 (iπk wj’k + ˜j’k )|2 < ∞.
v’ = ˜ l ˜ l
k=’∞ j=k+1

Acccording to (2.5), the spectrum of the operator A de¬ned by (3.1) is in-
cluded in the set
z
{z ∈ C : |1 ’ | ¤ δ(v’ , v+ ), k = 0, ±1, ±2, ..., },
π 2 k 2 + iπk w0 + ˜0
˜ l
where δ(v’ , v+ ) is de¬ned by (2.4). In particular, the lower spectral radius
of A satis¬es the inequality

rl (A) ≥ min |π 2 k 2 + iπk w0 + ˜0 | max{0, 1 ’ δ(v’ , v+ )}.
˜ l
k




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

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


with the domain Dom (A) de¬ned by (3.2). So the periodic boundary con-
ditions (3.3) hold. Here K is a Hilbert-Schmidt kernel and w(.) ∈ 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, ±1, ±2, ...,
ajk = (Aej , ek ) = wj’k + bjk (j = k) and
˜
akk = (Aek , ek ) = π 2 k 2 + w0 + bkk .
˜
Assume that
|π 2 k 2 + w0 + bkk | > 0.
rl (D) = inf ˜ (4.2)
k=0,±1,±2,...

Then we have N2 (D’1 V± ) = v± with
∞ k’1
2
|(π 2 k 2 + w0 + bkk )’1 (wj’k + bjk )|2 < ∞,
v+ = ˜ ˜
k=’∞ j=’∞

and
∞ ∞
2
|(π 2 k 2 + w0 + bkk )’1 (wj’k + bjk )|2 < ∞.
v’ = ˜ ˜
k=’∞ j=k+1

According to (2.5), the spectrum of the operator A de¬ned by (4.1) is included
in the set
z
{z ∈ C : |1 ’ 2 2 | ¤ δ(v’ , v+ ), k = 0, ±1, ±2, ..., },
π k + w0 + bkk
˜
where δ(v’ , v+ ) is de¬ned by (2.4).


13.5 Notes
The results presented in this chapter are based on the paper (Gil™, 2001). In
particular, inequality (1.11) is sharper than the well-known estimate

rs (A) ¤ sup |ajk |, (5.1)
j
k=1

cf. (Krasnosel™skij et al, 1989, inequality (16.2)), provided

˜
|ajk | > sup |akk | + φ1 (A).
sup
j k
k=1

For nonnegative matrices the following estimate is well-known, cf. (Kras-
nosel™skij et al, 1989 inequality (16.15)):

rs (A) ≥ r∞ (A) ≡
˜ min ajk (5.2)
j=1,...,∞
k=1
188 13. In¬nite Matrices and Di¬erential Operators


Our relation (1.14) is sharper than estimate (5.2) in the case |ajk | = ajk (j, k =
1, 2, ...), provided
˜
max akk ’ ∆H (A) > r∞ (A).
˜
k

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


References
[1] Edmunds, D.E. and Evans V.D. (1990). Spectral Theory and Dif-
ferential Operators. Clarendon Press, Oxford.
[2] Egorov, Y. and Kondratiev, V. (1996). Spectral Theory of Elliptic
Operators. Birkh¨user Verlag, Basel.
a
[3] Gil™, M.I. (2001). Spectrum localization of in¬nite matrices, Math-
ematical Physics, Analysis and Geometry, 4, 379-394

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

[5] Locker, J. (1999). Spectral Theory of Nonself-Adjoint Two Point
Di¬erential 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 de¬ned in H = L2 [0, 1] by
1
K(x, s)u(s)ds (u ∈ L2 [0, 1]; x ∈ [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. De¬ne the maximal resolution of the identity P (t) (’ ¤ t ¤ 1; > 0)
by
if ’ ¤ t < x,
0
(P (t)u)(x) =
u(x) if x ¤ t ¤ 1
with x ∈ [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 ∈ L2 [0, 1]; x ∈ [0, 1]).
(V’ u)(x) =
0
So
1 1
2
K 2 (x, s) ds dx
N2 (V+ ) =
0 x


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


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

N2 (V’ ) ¤ N2 (V+ ). (1.2)

The case N2 (V’ ) ≥ N2 (V+ ) can be similarly considered. So according to
˜
relations (4.1) and (4.2) from Section 9.4, we have „ (A) ¤ N2 (V’ ) and V =
V+ . Put
if N2 (V+ ) ¤ eN2 (V’ )
eN2 (V’ )
˜
φ1 = .
N2 (V+ )[ln (N2 (V+ )/ N2 (V’ )) ]’1/2
if N2 (V+ ) > eN2 (V’ )
(1.3)
Due to Corollary 9.4.3 and Remark 9.4.4, the spectrum of operator A is
included in the set
˜
{z ∈ C : |a(x) ’ z| ¤ φ1 , 0 ¤ x ¤ 1}.

Hence, the spectral radius of A satis¬es the inequality
˜
rs (A) ¤ sup |a(x)| + φ1 .
x∈[0,1]

In particular, if a(x) ≡ 0, then
˜
rs (A) ¤ φ1 (A). (1.4)

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

Recall that

2
|Im »k (A)|2 ]1/2
’2
gI (A) = [2N2 (AI )
k=1

and one can replace gI (A) by 2N2 (AI ). Put
if gI (A) ¤ eN2 (V’ )
eN2 (V’ )
˜
∆H := . (1.5)
gI (A)[ln (gI (A)/N2 (V’ )) ]’1/2 if gI (A) > eN2 (V’ )

Due to Corollary 9.8.2, for the integral operator de¬ned by (1.1), the following
relations are true:
˜
rs (A) ≥ max{0, sup |a(x)| ’ ∆H }, (1.6)
x∈[0,1]

˜ ˜
rl (A) ¤ inf |a(x)| + ∆H and ±(A) ≥ sup Re a(x) ’ ∆H .
x x∈[0,1]
14.2. Relatively Small Kernels 191


14.2 Matrix Integral Operators
with Relatively Small Kernels
Let ω ⊆ Rm be a set with a ¬nite Lebesgue measure, and H ≡ L2 (ω, Cn )
be a Hilbert space of functions de¬ned on ω with values in Cn and equipped
with the scalar product

(f, h)H = (f (s), h(s))C n ds,
ω

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

K(x, s)h(s)ds (h ∈ L2 (ω, Cn )),
(Ah)(x) = Q(x)h(x) + (2.1)
ω

where Q(x), K(x, s) are matrix-valued functions de¬ned on ω and ω — ω,
respectively. It is assumed that Q is bounded measurable and K is a Hilbert-
Schmidt kernel. So
˜ ˜
A = Q + K,
where
˜
(Qh)(x) = Q(x)h(x)
and
˜ K(x, s)h(s)ds (x ∈ ω).
(Kh)(x) =
ω
Besides,
2 1/2
˜
N2 (K) = [ K(x, s) C n ds] ,
ω ω
where . is the Euclidean norm.
Cn

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

{» ∈ C : N2 (K) sup (Q(x) ’ IC n »)’1
˜ ≥ 1}.
Cn
x∈ω

Proof: Since,

A ’ »I = Q + K ’ »I = (Q ’ »I)(I + (Q ’ »I)’1 K),
˜ ˜ ˜ ˜ ˜

if
(Q ’ »I)’1 K
˜ ˜ < 1,
H

then » is a regular point. So for any µ ∈ σ(A),

1 ¤ (Q ’ µI)’1 ¤ (Q ’ µI)’1
˜ ˜ ˜ ˜
K H N2 (K).
H H

But
(Q ’ µI)’1 ¤ sup (Q(x) ’ IC n µ)’1
˜ Cn .
H
x∈ω
192 14. Integral Operators


This proves the lemma. 2

Due to Corollary 2.1.2, for a ¬xed x we have
n’1
g k (Q(x))
’1

(Q(x) ’ IC n ») ¤ . (2.2)
Cn
k!ρk+1 (Q(x), »)
k=0

Now Lemma 14.2.1 yields
Lemma 14.2.2 Let operator A be de¬ned by (2.1). Then its spectrum lies
in the set
n’1
g k (Q(x))

˜
{» ∈ C : N2 (K) ≥ 1, x ∈ ω}.
k!ρk+1 (Q(x), »)
k=0

Corollary 14.2.3 Let operator A be de¬ned by (2.1). In addition, let
n’1
g k (Q(x))

˜
N2 (K) sup < 1,
k!dk+1 (Q(x))
x∈ω 0
k=0

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

Then A is boundedly invertible in L2 (ω, Cn ).
With a ¬xed x ∈ ω, consider the algebraic equation
n’1
g k (Q(x))z n’k’1
n

˜
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 µ ∈ σ(A) there are x ∈ ω and an eigenvalue »j (Q(x)) of
matrix Q(x), such that
|µ ’ »j (Q(x))| ¤ z0 (x). (2.5)
In particular,
rs (A) ¤ sup(rs (Q(x)) + z0 (x)).
x

Proof: Due to Lemma 14.2.2, for any point µ ∈ σ(A) there is x ∈ ω, such
that the inequality
n’1
g k (Q(x))

˜ ≥1
N2 (K)
k!ρk+1 (Q(x), µ)
k=0

is valid. Comparing this with (2.4), we have ρ(Q(x), µ) ¤ 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 ∈ ω. Then for any
point µ ∈ σ(A) there are x ∈ ω and »j (Q(x))σ(Q(x)), such that

˜
|µ ’ »j (Q(x))| ¤ N2 (K).

˜
In particular, rs (A) ¤ N2 (K) + supx (rs (Q(x))).
˜
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
n’1 k
g (Q(x))

˜
b(x) := N2 (K) .
k!
k=0

Due to Corollary 1.6.2, z0 (x) ¤ δn (x), where

b(x) if b(x) ¤ 1 and δn (x) = b(x) if b(x) > 1.
n
δn (x) =

Now Theorem 14.2.4 implies

Theorem 14.2.6 Under condition (2.7), for any point µ ∈ σ(A), there are
x ∈ ω and an eigenvalue »j (Q(x)) of Q(x), such that

|µ ’ »j (Q(x))| ¤ δn (x).

In particular, rs (A) ¤ supx (rs (Q(x)) + δn (x)).



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

where Q0 is a constant matrix, K0 is a matrix-valued function de¬ned on
[’π, π] with
K0 C n ∈ L2 [’π, π],
having the Fourier expansion

Dk eikx
K0 (x) =
k=’∞

with the matrix Fourier coe¬cients
π
1
K0 (s)e’iks ds.
Dk = √
2π ’π
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 »j (Bk ) (j =
1, ...n). Then
π
ikx ikx
K0 (x ’ s)djk eiks ds =
Ce djk = e Q0 djk +
’π


eikx Bk djk = eikx »j (Bk )djk .
Since the set
{eikx }k=∞
k=’∞

is a basis in L2 [’π, π] we have the following result

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

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

Let Pk be orthogonal projectors de¬ned by
π
1
(Pk h)(x) = eikx h(s)e’iks ds.
2π ’π

Since

Pk = IH ,
k=’∞

it can be directly checked by (3.2) that the equality

C= Bk Pk
k=’∞

holds. Hence, the relation

’1
(Bk ’ IC n »)’1 Pk
(C ’ IH ») =
k=’∞

is valid for any regular ». Therefore,

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

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


Lemma 14.3.2 The resolvent of convolution C de¬ned by (3.1) satis¬es the
inequality
n’1
g k (Bl )
’1

(C ’ »I) ¤ sup .
H
k!ρk+1 (Bl , »)
l=0, ±1,...
k=0

Consider now the operator
π
(Ah)(x) ≡ Q0 h(x) + K0 (x ’ s)h(s)ds + (Zh)(x) (’π ¤ x ¤ π). (3.3)
’π

where Z is a bounded operator in L2 ([’π, π], Cn ). We easily have by the
previous lemma that the inequalities
n’1
g k (Bl )
’1

(C ’ »I) ¤Z
Z sup <1
H H H
k!ρk+1 (Bl , »)
l=0, ±1,...
k=0

imply that » is a regular point. Hence we arrive at
Lemma 14.3.3 The spectrum of operator A de¬ned by (3.3) lie in the set
n’1
g k (Bl )

{» ∈ C : Z ≥ 1}.
sup
H
k!ρk+1 (Bl , »)
l=0, ±1,...
k=0

In other words, for any µ ∈ σ(A), there are

l = 0, ±1, ±2, ... and j = 1, ..., n,

such that
n’1
g k (Bl )
√ ≥ 1.
Z H
k!|µ ’ »j (Bl )|k+1
k=0

Corollary 14.3.4 Operator A de¬ned by (3.3) is invertible provided that
n’1
g k (Bk )
√ ¤ c0 < 1 (c0 = const)
Z H
k!|»j (Bl )|k+1
k=0

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

Let zl be the extreme right (unique positive) root of the equation
n’1
z n’1’k g k (Bl )
n

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 µ of the spectrum of operator (3.3), there
are indexes l = 0, ±1, ±2, ... and j = 1, ..., n, such that

|µ ’ »j (Bl )| ¤ zl , (3.5)

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

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


If all the matrices Bl are normal, then g(Bl ) ≡ 0, zl = Z H, and (3.5) takes
the form
|µ ’ »j (Bl )| ¤ Z H .
Assume that
n’1
g k (Bl )
√ ¤ 1 (l = 0, ±1, ±2, ...).
bl := Z (3.6)
H
k!
k=0

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

Corollary 14.3.6 Let A be de¬ned by (3.3) and condition (3.6) hold. Then
for any µ ∈ σ(A) there are l = 0, ±1, ±2, ... and j = 1, ..., n, such that

|µ ’ »j (Bl )| ¤ n
bl .

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




14.4 Notes
Inequality (1.4) improves the well-known estimate
1
˜
rs (A) ¤ δ0 (A) ≡ vrai sup |K(x, s)|ds,
x 0

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


References
[1] Gil™, M.I. (2000). Invertibility conditions and bounds for spectra of ma-
trix integral operators, Monatshefte f¨r mathematik, 129, 15-24.
u
[2] Gil™, M.I. (2003). Inner bounds for spectra of linear operators, Proceed-
ings of the American Mathematical Society , (to appear).
[3] Krasnosel™skii, M. A., J. Lifshits, and A. Sobolev (1989). Positive Linear
Systems. The Method of Positive Operators, Heldermann Verlag, Berlin.

[4] 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
di¬erential operators are also discussed.


15.1 Invertibility Conditions
Let H be an orthogonal sum of Hilbert spaces Ek (k = 1, ..., n < ∞) with
norms . Ek :
H ≡ E1 • E2 • ... • En .
Consider in H the operator matrix
« 
A11 A12 . . . A1n
¬ A21 A22 . . . A2n ·
A=¬ ·, (1.1)
. .
... .
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 ∈ Ek )n be an element of H. Everywhere in the present
k=1
chapter the norm in H is de¬ned by the relation
n
2 1/2
h≡h =[ hk Ek ] (1.2)
H
k=1

and I = IH is the unit operator in H.

M.I. Gil™: LNM 1830, pp. 199“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,
« 
0 A12 . . . A1n
¬0 . . . A2n ·
0
V =¬ ·,
 . ... .
.
0 0 ... 0
« 
0 0 ... 0 0
¬ A21 0·
0 ... 0
W =¬ ·
. 
... . .
An1 An2 ... An,n’1 0
and
D = diag [A11 , A22 , ..., Ann ].
Recall that, for a linear operator A, Dom (A) means the domain, σ(A) is the
spectrum, »k (A) (k = 1, 2, ...) are the eigenvalues with their multiplicities,
ρ(A, ») is the distance between the spectrum of A and a » ∈ C.

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

VA ≡ D’1 V, WA ≡ D’1 W be bounded . (1.3)

In addition, let the condition
n’1
j
(’1)k+j VA WA < 1
k
(1.4)
j,k=1


hold. Then operator A de¬ned by (1.1) is invertible.

Proof: We have

A = D + V + W = D(I + VA + WA ) = D[(I + VA )(I + WA ) ’ 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 ’ (I + VA )’1 VA WA (I + WA )’1 ](I + WA ).

Therefore, the condition

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


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

n’1 n’1
’1 k’1 k ’1
(’1)k’1 WA .
k
(I + VA ) VA = (’1) VA , WA (I + WA ) =
k=1 k=1

Hence, the required result follows. 2


Corollary 15.1.2 Let operator matrix A de¬ned by (1.1) be an upper trian-
gular one (W=0), D an invertible operator and D’1 V a bounded one. Then
A is invertible.
Similarly, let operator matrix A be a lower triangular one (V=0) and
’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
n’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 ’ VA n’1 )(1 ’ WA n’1 )
VA W A < 1. (1.7)
(1 ’ VA )(1 ’ WA )

Indeed, taking into account that
n’2 n’2
1 ’ VA n’1 1 ’ WA n’1
k k
VA = , WA =
1 ’ VA 1 ’ 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 ≡ Ajk < ∞ (j, k = 1, ..., n).
Ek ’Ej


Then the norm of operator A de¬ned by (1.1) is subject to the relation

A¤a
˜ Cn ,


where a is the linear operator in the Euclidean space Cn , de¬ned by the matrix
˜
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
¤
A Ajk Ek ’Ej . (1.8)
j,k=1

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

Clearly,

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

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

A’1 A12 A’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 » of D, let

V (») := (D ’ IH »)’1 V and W (») := (D ’ IH »)’1 W be bounded operators .
˜ ˜
(2.1)
Then the spectrum of operator A de¬ned by (1.1) lies in the union of the sets
σ(D) and
n’2
k j
˜ ˜ ˜ ˜
{» ∈ C : V (»)W (») ≥ 1}.
V (») W (»)
j,k=0

Indeed, if for some » ∈ σ(A),
n’2
k j
˜ ˜ ˜ ˜
V (»)W (») V (») W (») < 1, (2.2)
j,k=0

then due to Corollary 15.1.3, A ’ »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
˜
V (») be bounded for all regular » of D. Then

σ(A) = ∪n σ(Akk ) = σ(D). (2.3)
k=1

˜
Similarly, let (1.1) be lower triangular and W (») be bounded for all regular
» of D. Then (2.3) holds.
˜
Indeed, let A be upper triangular. Then W (») = 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 ’ IH »)’1 ¤ ¦(ρ’1 (D, »)) (» ∈ σ(D))
/ (2.4)

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


Then the spectral variation of operator A de¬ned by (1.1) with respect to D
satis¬es the inequality
1
svD (A) ¤ .
z0
Proof: Due to (2.4)

V (») ¤ V ¦(ρ’1 (D, »)), W (») ¤ W ¦(ρ’1 (D, »)).
˜ ˜

For any » ∈ σ(A) and » ∈ σ(D), Theorem 15.2.1 implies
/
n’1
¦k+j (ρ’1 (D, »)) V k j
≥ 1.
W
j,k=1


Taking into account that ¦ is increasing and comparing the latter inequality
with (2.5), we have
ρ’1 (D, ») ≥ z0
for any » ∈ σ(A). This proves the required result. 2

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

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


Hence it follows that
1
z0 = Ψ( ),
V W
where Ψ is the function inverse to ¦: ¦(Ψ(y)) = y. Thus, in the case n = 2,
1
svD (A) ¤ .
Ψ( √ 1
)
V W




15.3 Operator Matrices with Normal Entries
Assume that H is an orthogonal sum of the same Hilbert spaces Ek ≡ E
(k = 1, ..., n) with norm . E . Consider in H the operator matrix de¬ned by
(1.1), assuming that
Ajj = Sj (j = 1, ..., n), (3.1)
where Sj are normal, unbounded in general operators in Ej , and
Ajk = φk (Sj ) (j = k; j, k = 1, ..., n), (3.2)
where φk (s) are scalar-valued measurable functions of s ∈ C. In addition,
assume that
±jk ≡ sup |φk (t)t’1 | < ∞. (3.3)
t∈σ(Sj )

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

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