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