BOUNDS FOR THE SPECTRUM OF ANALYTIC QUASINORMAL OPERATOR PENCILS

January 23, 2003 11:33 WSPC/152-CCM 00090

Communications in Contemporary MathematicsVol. 5, No. 1 (2003) 101–118c© World Scientific Publishing Company

BOUNDS FOR THE SPECTRUM OF ANALYTIC

QUASINORMAL OPERATOR PENCILS

M. I. GIL’

Department of Mathematics, Ben Gurion University of the Negev,P.0. Box 653, Beer-Sheva 84105, Israel

[email protected]

Received 23 April 2001Revised 1 May 2002

We consider a class of pencils (operator valued functions of a complex argument) in aseparable Hilbert space. Bounds for the λ-nonlinear spectrum are suggested. Applica-tions to differential operators, integral operators with delay and infinite matrix pencilsare also discussed.

Keywords: Linear operators; pencils; spectrum; integral and differential operators;infinite matrices.

Mathematics Subjects Classification 2000: 47A55, 47A75, 47G10, 47G20

1. Introduction and Statement of the Main Result

Numerous mathematical and physical problems lead to analytic pencils (ana-

lytic operator-valued functions of a complex argument), cf. [1, 6, 14, 16, 26], and

references therein. A lot of papers and books are devoted to the spectrum of

analytic operator pencils functions, in particular, polynomial operator pencils, (see

for instance, [13, 18, 24, 25, 27]). Mainly, the completeness of the root vectors

and asymptotic distributions of the eigenvalues are considered. However, in many

applications, for example, in numerical mathematics and stability analysis, bounds

for the spectrum are very important. But the bounds are investigated considerably

less than the asymptotic distributions and the completeness. Below we establish

bounds for the spectrum of a class of operator pencils. We also discuss applications

of these bounds to differential operators, integral operators with delay and matrix

pencils. Our results are new even in the case of the usual (linear) spectral problems,

cf. [4, 5, 15, 21, 22].

Note also that our results below are intimately connected with the spectral

problems, considered in the books [11, Chap. X] and [3, Chap. V].

Let H be a separable Hilbert space with a scalar product (·, ·), the norm ‖ · ‖and the unit operator I. For a linear operator A, σ(A) is the spectrum. Recall that

101

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January 23, 2003 11:33 WSPC/152-CCM 00090

102 M. I. Gil’

a linear operator V is a Volterra one if it is quasinilpotent (that is, σ(V ) = {0})and compact, cf. Gohberg and Krein [12].

In the present paper we consider an operator-valued function A(λ) of the type

A(λ) = D(λ) + V+ + V−(λ ∈ C) (1.1)

where V± are Volterra operators in H with the properties pointed below and D(λ)

is a normal operator in H of the form

D(λ) =

∫ ∞−∞

e(λ, t)dP (t) (λ ∈ C)

where e(λ, t) for any real finite t is an entire function of λ, and for any λ ∈ C is

a continuous function of t ∈ R. Moreover, for each finite λ, D(λ) is bounded. In

addition, P (t)(−∞ ≤ t ≤ ∞) is a maximal resolution of the identity. That is, P (·)is a left-continuous orthogonal resolution of the identity defined on (−∞,∞) with

the property: any gap P (t0 + 0) − P (t0) of P (·) (if it exists) is one-dimensional,

cf. Brodskii [2], Gohberg and Krein [12], and Gil’ [7, p. 69]. In addition, we assume

that

P (t)V+P (t) = V+P (t) and P (t)V−P (t) = P (t)V− (t ∈ R) . (1.2)

Furthermore, λ is a regular point of A(·) ifA(λ) is boundedly invertible. The comple-

ment of the set of all regular points to the closed complex plane is the (λ-nonlinear)

spectrum of A(·) and is denoted by Σ(A(·)).So for a linear operator A0, σ(A0) =

∑(A(·)) with A(λ) = A0 − λI.

Clearly, D(λ) is invertible if and only if

ρ(D(λ)) ≡ inft∈γ(P )

|e(λ, t)| > 0 (1.3)

where γ(P ) denotes the set of all points of the growth of P (·). That is,

P (t2)− P (t1) = 0 (t1 < t2)

if and only if the segment (t1, t2] does not belong to γ(P ).

Let Y be an ideal of linear compact operators in H with a norm | · |Y , such that

|CB|Y ≤ ‖C‖|B|Y and |BC|Y ≤ ‖C‖|B|Y for an arbitrary bounded linear operator

C in H and a B ∈ Y . Assume that Y has the following property: any Volterra

operator V ∈ Y satisfies the inequalities

‖V k‖ ≤ θk|V |kY (k = 1, 2, . . .) (1.4)

where constants θk are independent of V and

k√θk → 0 (k →∞) .

For instance, if Y is the Hilbert–Schmidt ideal, then (1.4) holds with θk = 1/√k!

(see [7, Sec. 2.3]). We also will check that the von Neumann–Schatten ideal has

property (1.4). It is assumed that

V+, V− ∈ Y (1.5)

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Bounds for the Spectrum of Analytic Quasinormal Operator Pencils 103

and ideal Y has property (1.4). Furthermore, for a Volterra operator V ∈ Y , denote

JY (V,m, z) ≡m−1∑k=0

z−1−kθk|V |kY (z > 0) .

Set

W±(λ) ≡ D−1(λ)V± (λ /∈ Σ(D(·))) .

Due to [7, Lemma 3.2.4], and according to (1.2), the operators W±(λ) are

quasinilpotent for any λ /∈ Σ(D(·)).A natural number ni(V ) will be called the indicator of the nilpotentness of a

nilpotent operator V , if V ni(V ) = 0 but V ni(V )−1 6= 0. If V is quasinilpotent but

not nilpotent, then we put ni(V ) =∞. Set

ν±(λ) ≡ ni(W±(λ)) .

Everywhere below one can replace ν±(λ) by ∞. Now we are in a position to

formulate the main result of the paper

Theorem 1.1. Let A(λ) be given by (1.1). In addition, under conditions (1.2) and

(1.5), for a λ /∈ Σ(D(·)), let

ζ(A(λ)) ≡ max

{1

JY (V−, ν−(λ), ρ(D(λ)))− ‖V+‖,

1

JY (V+, ν+(λ), ρ(D(λ)))− ‖V−‖

}> 0 . (1.6)

Then λ is a regular point of operator-valued function A(·) represented by (1.1).

Moreover,

‖A−1(λ)‖ ≤ 1

ζ(A(λ))ρ(D(λ)). (1.7)

The proof of this theorem is presented in the next section.

2. Proof of Theorem 1.1

We need the following simple

Lemma 2.1. Let

A0 = I +W1 +W2 , (2.1)

where W1,W2 are bounded linear operators in H. In addition, W1 is a quasinilpotent

operator. Then under the condition

ψ0 ≡ ‖(I +W1)−1W2‖ < 1 , (2.2)

the operator A0 is boundedly invertible, and

‖A−10 ‖ ≤

‖(I +W1)−1‖1− ψ0

.

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104 M. I. Gil’

Proof. Since W1 is a quasinilpotent operator, I + W1 is invertible. According to

(2.1) we have

A0 = (I +W1)(I + (I +W1)−1W2) . (2.3)

Moreover, under condition (2.2), the operator I + (I +W1)−1W2 is also invertible

and

‖(I + (I +W1)−1W2)−1‖ ≤∞∑k=0

‖((I +W1)−1W2)k‖ ≤∞∑k=0

ψk0 = (1− ψ0)−1 .

Due to (2.3)

‖A−10 ‖ ≤ ‖(I + (I +W1)−1W2)−1‖‖(I +W1)−1‖ .

This proves the required result.

For the sake of brevity, put

jY (V ) = JY (V, ni(V ), 1) =

ni(V )−1∑k=0

θk|V |kY (2.4)

for a Volterra operator V from ideal Y with property (1.3). Everywhere below j−1Y (·)

means 1jY (·) .

Lemma 2.2. Let W1,W2 be Volterra operators from ideal Y with property (1.4).

Then under the condition

ζ(A0) ≡ max{j−1Y (W1)− ‖W2‖, j−1

Y (W2)− ‖W1‖} > 0 , (2.5)

the operator A0 represented by (2.1) is boundedly invertible. Moreover, the inverse

operator satisfies the inequality ‖A−10 ‖ ≤ ζ−1(A0).

Proof. Due to (1.4), we have

‖(I +W1)−1‖ ≤∞∑k=0

‖W k1 ‖ =

ni(W1)−1∑k=0

‖W k1 ‖

≤ni(W1)−1∑k=0

θk|W1|kY = jY (W1) .

Hence, ψ0 ≤ ‖W2‖jY (W1). So the condition

‖W2‖jY (W1) < 1 (2.6)

implies (2.2). Now Lemma 2.1 yields the invertibility of A0 with the estimate

‖A−10 ‖ ≤

jY (W1)

1− ‖W2‖jY (W1)=

1

j−1Y (W1)− ‖W2‖

. (2.7)

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But condition (2.6) is equivalent to the following one: j−1Y (W1) − ‖W2‖ > 0.

Exchanging W1 and W2, we can assert that the condition j−1Y (W2) − ‖W1‖ > 0

yields the invertibility and estimate

‖A−10 ‖ ≤

1

j−1Y (W2)− ‖W1‖

.

This, (2.5) and (2.7) imply the required result.

Proof of Theorem 1.1. Due to (1.1),

A(λ) = D(λ)(I +W+(λ) +W−(λ)) . (2.8)

Let us apply Lemma 2.2 to the operator-valued function I + W+(λ) + W−(λ).

According to (2.4), condition (2.5) means that at least one of the following

inequalities

j−1Y (W−(λ)) > ‖W+(λ)‖, or j−1

Y (W+(λ)) > ‖W−(λ)‖

holds. Equivalently,

jY (W−(λ))‖W+(λ)‖ < 1 or (and) jY (W+(λ))‖W−(λ)‖ < 1 . (2.9)

Since D(λ) is normal,

‖D−1(λ)‖ ≤ 1

ρ(D(λ)). (2.10)

Thus, ‖W+(λ)‖ ≤ ‖V+‖ρ−1(D(λ)) and

jY (W±(λ)) ≤ν(λ)−1∑k=0

θkρ−k(D(λ))|V±|kY .

Hence,

‖W+(λ)‖jY (W−(λ)) ≤ ‖V+‖ρ−1(D(λ))jY (W−(λ))

= ‖V+‖ν(λ)−1∑k=0

θk|V−|kYρk+1(D(λ))

= ‖V+‖JY (V−, ν−(λ), ρ(D(λ))) .

Similarly

‖W−(λ)‖jY (W+(λ)) ≤ ‖V−‖JY (V+, ν+(λ), ρ(D(λ))) .

Thus, condition (1.6) implies at least one of the inequalities (2.9). Now Lemma 2.2

yields the inequality

‖(I +W+(λ) +W−(λ))−1‖ ≤ ζ−1(A(λ)) .

Taking into account relations (2.8) and (2.10) we arrive at the required result.

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106 M. I. Gil’

3. Spectrum Localization

Theorem 1.1 implies

Theorem 3.1. Let A(λ) be given by (1.1). Then, under conditions (1.2) and (1.5),

for any µ ∈ Σ(A(·)), either µ ∈ Σ(D(·)), or the both inequalities

‖V+‖JY (V−, ν−(µ), ρ(D(µ)) ≥ 1 and ‖V−‖JY (V+, ν+(µ), ρ(D(µ)) ≥ 1 (3.1)

are true.

It is easy to show that the latter result is exact in the following sense: if either

V− = 0, or (and) V+ = 0, then due to the Corollary 3.1,

Σ(A(·)) = Σ(D(·)) . (3.2)

Now put

ν± ≡ supλ/∈Σ(D(·))

ν±(λ) = supλ/∈Σ(D(·))

ni(W±(λ)) .

In the sequel one can replace ν± by ∞.

Corollary 3.2. Let A(λ) be given by (1.1). In addition, let (1.2) and (1.5) hold

with V+ 6= 0 and V− 6= 0. Then each of the following equations:

‖V+‖JY (V−, ν−, z) = 1 and ‖V−‖JY (V+, ν+, z) = 1 (3.3)

has a unique positive root zup(Y ) and zlow(Y ), respectively. Moreover, any µ ∈Σ(A(·)) satisfies the inequality

ρ(D(µ)) ≤ min{zup(Y ), zlow(Y )} . (3.4)

Indeed, inequalities (3.1) imply

‖V+‖JY (V−, ν−, ρ(D(µ)) ≥ 1 and ‖V−‖JY (V+, ν+, ρ(D(µ)) ≥ 1 .

Comparing these inequalities with Eq. (3.3), we arrive at the required result.

To estimate the quantities zup(Y ), zlow(Y ), let us consider the equation

∞∑k=1

akzk = 1 (3.5)

where the coefficients ak are nonnegative and have the property

γ0 ≡ 2 maxk

k√ak <∞ .

Lemma 3.3. The unique nonnegative root z0 of Eq. (3.5) satisfies the estimate

z0 ≥ 1/γ0.

Proof. Set in (3.5) z = xγ−10 . We have

1 =∞∑k=1

akγ−k0 xk . (3.6)

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But∞∑k=1

akγ−k0 ≤

∞∑k=1

2−k = 1

and therefore, the unique positive root x0 of (3.6) satisfies the inequality x0 ≥ 1.

Hence, z0 = γ−10 x0 ≥ γ−1

0 .

Note that the latter lemma generalizes the well-known result for algebraic

equations, cf. Ostrowski [19, p. 277].

Lemma 3.3 gives us the inequalities

zup(Y ) ≤ 2 maxj=1,2,...

j

√θj−1|V−|j−1

Y ‖V+‖

and

zlow(Y ) ≤ 2 maxj=1,2,...

j

√θj−1|V+|j−1

Y ‖V−‖, with θ0 = 1 .

Now Theorem 3.2 implies

Corollary 3.4. Under conditions (1.1), (1.2) and (1.5) for any µ ∈ Σ(A(·)) the

estimate ρ(D(µ)) ≤ ψY (A) is valid, where

ψY (A) ≡ 2 min

{max

j=1,2,...

j

√θj−1|V−|j−1

Y ‖V+‖, maxj=1,2,...

j

√θj−1|V+|j−1

Y ‖V−‖}.

4. Pencils with Hilbert Schmidt Nilpotent Parts

Let N1(K) be the Hilbert–Schmidt norm of a Hilbert–Schmidt operator (HSO) K:

N1(K) ≡ [Trace K∗K]1/2 .

Throughout this section it is assumed that

W±(λ) = D−1(λ)V±(λ /∈ Σ(D(·))) are Volterra Hilbert–Schmidt operators .(4.1)

For a Volterra Hilbert–Schmidt operator V , put

J2(V,m, z) =m−1∑k=0

Nk1 (V )

zk+1√k!

(z > 0) .

Recall that ν±(λ) ≡ ni(W±(λ)) ≤ ∞.

Theorem 4.1. Under conditions (1.2) and (4.1), for a λ /∈ Σ(D(·)), let

ζH(A(λ)) ≡ max

{1

J2(V−, ν−(λ), ρ(D(λ)))− ‖V+‖ ,

1

J2(V+, ν+(λ), ρ(D(λ)))− ‖V−‖

}> 0 . (4.2)

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108 M. I. Gil’

Then λ is a regular point of the operator-valued function A(·), represented by (1.1).

Moreover,

‖A−1(λ)‖ ≤ 1

ρ(D(λ))ζH (A(λ)).

Proof. Due to [7, Lemma 2.3.1], we have

‖V j‖ ≤ N j1 (V )√j!

(j = 1, 2, . . .) (4.3)

for any Volterra HSO V . Now the required result follows from Theorem 1.1.

Theorem 4.1 yields

Corollary 4.2. Let A(λ) be given by (1.1). Then, under conditions (1.2) and (4.1),

for any µ ∈ Σ(A(·)) we have either µ ∈ Σ(D(·)), or the both inequalities

‖V+‖J2(V−, ν−(µ), ρ(D(µ))) ≥ 1 and ‖V−‖J2(V+, ν+(µ), ρ(D(µ))) ≥ 1

are true.

Furthermore, for a constant a > 0, the Schwarz inequality implies,( ∞∑k=0

ak√k!

)2

=

( ∞∑k=0

2k/2ak

2k/2√k!

)2

≤∞∑j=0

2−j∞∑k=0

2ka2k

k!= 2 exp[2a2] .

Hence

J2(V±,m, z) ≤√

2z−1ez−2N2

1 (V±) . (4.4)

Now Theorem 4.1 implies

Corollary 4.3. Let relations (1.2) and (4.1) hold. In addition, for a λ /∈ Σ(D(·)),let

ζH1(A(λ)) ≡ max

{√1/2ρ(D(λ)) exp

[− N2

1 (V+)

ρ2(D(λ))

]− ‖V−‖ ,

√1/2ρ(D(λ)) exp

[− N2

1 (V−)

ρ2(D(λ))

]− ‖V+‖

}> 0 .

Then λ is a regular point of operator-valued function A(·), represented by (1.1).

Moreover,

‖A−1(λ)‖ ≤ 1

ρ(D(λ))ζH1(A(λ)).

Theorem 4.4. Under conditions (1.1), (1.2) and (4.1), let V+ 6= 0 and V− 6= 0.

Then each of the following equations:

‖V+‖J2(V−, ν−, z) = 1 and ‖V−‖J2(V+, ν+, z) = 1 (4.5)

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has a unique positive root zup(HS) and zlow(HS), respectively. Moreover, for any

µ ∈ Σ(A(·)), the inequality

ρ(D(µ)) ≤ min{zup(HS), zlow(HS)} ≤ ψ2(A)

is true, where

ψ2(A) = 2 min

{max

j=1,2,...

j

√‖V+‖N j−1

1 (V−)√(j − 1)!

, maxj=1,2,...

j

√‖V−‖N j−1

1 (V+)√(j − 1)!

}. (4.6)

Proof. Theorem 3.2 yields

ρ(D(µ)) ≤ min{zup(HS), zlow(HS)} .

But due to Lemma 3.3, min{zup(HS), zlow(HS)} ≤ ψ2(A).

We need the following simple lemma.

Lemma 4.5. The unique positive root z0 of the equation

zez = a (a = const > 0) (4.7)

satisfies the estimate

z0 ≥ ln[1/2 +√

1/4 + a] . (4.8)

If, in addition, the condition a ≥ e holds, then

z0 ≥ ln a− ln ln a . (4.9)

Proof. Since z ≤ ez − 1(z ≥ 0), we arrive at the relation a ≤ e2z0 − ez0 . Hence,

ez0 ≥ r1,2, where r1,2 are the roots of the polynomial y2 − y − a. This proves

inequality (4.8).

Furthermore, if the condition a ≥ e holds, then z0ez0 ≥ e and z0 ≥ 1. Now (4.7)

yields ez0 ≤ a and z0 ≤ ln a. So

a = z0ez0 ≤ ez0 ln a .

Hence, inequality (4.9) follows.

According to (4.4) and (4.5), we have zup(HS) ≤ zup and zlow(HS) ≤ zlow

where zup and zlow are unique positive roots of equations√

2‖V+‖z−1 exp [z−2N21 (V−)] = 1 (4.10)

and√

2‖V−‖z−1 exp [z−2N21 (V+)] = 1 ,

respectively. Clearly, (4.10) is equivalent to the equation

2‖V+‖2z−2 exp [2z−2N21 (V−)] = 1 .

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110 M. I. Gil’

Substitute in the latter equation z2 = 2N21 (V−)x−1. Then with the notation

aup ≡N2

1 (V−)

‖V+‖2and alow ≡

N21 (V+)

‖V−‖2

we have xex = aup. Now Lemma 4.5 implies

z2up ≤

2N21 (V−)

ln [1/2 +√

1/4 + aup].

Similarly,

z2low ≤

2N21 (V+)

ln[1/2 +√

1/4 + alow].

Denote

δ(A) ≡√

2 min

{N1(V−)

ln1/2[1/2 +√

1/4 + ‖V+‖−2N21 (V−)]

,

N1(V+)

ln1/2[1/2 +√

1/4 + ‖V−‖−2N21 (V+)]

}. (4.11)

Then

min{zup(HS), zlow(HS)} ≤ min{zup, zlow} ≤ δ(A) .

Clearly, δ(A)→ 0, if either N1(V−)→ 0, or N1(V+)→ 0.

Thus, Theorem 4.4 implies

Corollary 4.6. Under conditions (1.1), (1.2) and (4.1), for any µ ∈ Σ(A(·)), the

inequality ρ(D(µ)) ≤ δ(A) is true, where δ(A) is given by (4.11).

5. Operators with von Neumann Schatten Nilpotent Parts

In this section it is assumed that V± are Volterra operators belonging to the

Neumann–Schatten ideal C2p with some integer p > 1. That is,

Np(V±) ≡ [Trace (V ∗±V±)p]1/2p <∞ . (5.1)

For a natural k, denote

θ(p)k =

1√[k/p]!

where [x] means the integer part of x > 0. For a Volterra operator V ∈ C2p(p > 1),

put

J2p(V,m, z) =m−1∑k=0

θ(p)k Nk

p (V )

zk+1(p > 1, z > 0) .

Recall that ν±(λ) ≡ ni(V±(λ)) ≤ ∞.

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Theorem 5.1. Under conditions (1.2) and (5.1), for a λ /∈ Σ(D(·)), let

ζ2p(A(λ)) ≡ max

{1

J2p(V−, ν−(λ), ρ(D(λ)))− ‖V+‖ ,

1

J2p(V+, ν+(λ), ρ(D(λ)))− ‖V−‖

}> 0 . (5.2)

Then λ is a regular point of operator-valued function A(·), represented by (1.1).

Moreover,

‖A−1(λ)‖ ≤ 1

ρ(D(λ))ζ2p(A(λ)).

Proof. For any Volterra operator V ∈ C2p we have V p ∈ C2. According to (4.3)

‖V jp‖ ≤ N j1 (V p)√j!

≤Npjp (V )√j!

(V ∈ C2p; j = 1, 2, . . .) .

Hence, for any k = m+ jp(m = 0, . . . , p− 1; j = 0, 1, 2, . . .),

‖V k‖ = ‖Vm+pj‖ ≤ ‖Vm‖N j

1 (V p)√j!

≤Nm+pjp (V )√j!

. (5.3)

Consequently,

‖V k‖ ≤ θ(p)k Nk

p (V ) (k = 1, 2, . . .) . (5.4)

Now the required result is due to Theorem 1.1.

Theorem 5.1 implies

Corollary 5.2. Under conditions (1.1), (1.2) and (5.1), for any µ ∈ Σ(A(·)), either

µ ∈ Σ(D(·)), or the both inequalities

‖V+‖J2p(V−, ν−(µ), ρ(D(µ))) ≥ 1 and ‖V−‖J2p(V+, ν+(µ), ρ(D(µ))) ≥ 1

are true.

Theorem 3.2 and relation (5.4) yield

Theorem 5.3. Let A(λ) be given by (1.1). In addition, under conditions (1.2) and

(5.1), let V+ 6= 0 and V− 6= 0. Then for any µ ∈ Σ(A(·)), the inequality

ρ(D(µ)) ≤ min{zup(C2p), zlow(C2p)}

is true, where zup(C2p) and zlow(C2p) are the unique positive roots of the equations

‖V+‖J2p(V−, ν−, z) = 1 and ‖V−‖J2p(V+, ν+, z) = 1 , (5.5)

respectively.

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112 M. I. Gil’

To estimate zup(C2p), zlow(C2p) we can apply Lemma 3.3. Note that according

to (5.3) one can replace Eq. (5.5) by the following ones:

‖V+‖p−1∑j=0

∞∑k=1

N j+pkp (V−)

zj+pk+1√k!

= 1 and ‖V−‖p−1∑j=0

∞∑k=1

N j+pkp (V+)

zj+pk+1√k!

= 1 . (5.6)

Denote

ψ2p(A) = 2 min

maxj=1,2,...,p−1;k=0,1,...

j+kp+1

√‖V+‖N j+kp

p (V−)√k!

,

maxj=1,2,...,p−1;k=0,1,...

j+kp+1

√‖V−‖N j+kp

p (V+)√k!

.

According to (5.6), Theorem 3.2 and Lemma 3.3 yield the inequality

ρ(D(µ)) ≤ ψ2p(A) for any µ ∈ σ(A(·))

provided relations (1.1), (1.2) and (5.1) hold.

6. A Matrix Differential Operator on R1

Let us consider in space L2(R1,Rn) the matrix differential operator T defined by

(Tu)(x) =d2u(x)

dx2+ θ

du(x)

dx+ ψu(x) (x ∈ R) (6.1)

with the domain {v ∈ L2(R1,Rn) :

d2v

dx2∈ L2(R1,Rn)

}where

θ = diag [θj ]nj=1, ψ = (ψjk)nj,k=1

are constant complex n × n-matrices. Applying the Fourier transformation to

operator (6.1), we arrive at the following operator pencil in Cn:

A(λ) = λ2I + λθ + ψ(λ ∈ C) (6.2)

where I is the unit matrix. In the considered case one can take

D(λ) = λ2I + λθ + diag[ψjj ] = diag[λ2 + λθj + ψjj ] .

In addition,

V+ =

0 ψ12 · · · ψ1n

0 0 · · · ψ2n

· 0 · · · ·0 0 · · · 0

and V− =

0 · · · 0 0

ψ21 · · · 0 0

· · · · · ·ψn1 · · · ψn,n−1 0

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are nilpotent n× n-matrices. So

N21 (V−) =

n∑j=1

j−1∑k=1

|ψjk|2; N21 (V+) =

n∑j=1

n∑k=j+1

|ψjk|2 (6.3)

and ν±(λ) = ni(D−1(λ)V±) ≤ n. Take the maximal spectral function P (t) =

{Qk}nk=1, where Qk are the projectors on the k first coordinates:

Qkh = (h1, . . . , hk, 0, . . . , 0)

for any vector h = (h1, . . . , hn) ∈ Cn. Clearly, conditions (1.2) hold. Obviously,

J2(V±, ν±(λ), z) ≡ν±(λ)−1∑k=0

Nk1 (V±)

zk+1√k!≤ J2(V±, n, z) ≡

n−1∑k=0

Nk1 (V±)

zk+1√k!

(z > 0) .

So instead of (4.5), one can consider the equations

‖V+‖CnJ2(V−, n, z) = 1, ‖V−‖CnJ2(V+, n, z) = 1 (6.4)

where ‖ · ‖Cn is the Euclidean norm. Denote by ylow the unique positive root of the

first equation in (6.4),and by yup the unique positive root of the second equation.

By virtue of Lemma 3.3, we can assert that

min{yup, ylow} ≤ min

{max

k=0,...n−1

k+1

√‖V+‖Nk

1 (V−), k+1

√max

k=0,...n−1‖V−‖Nk

1 (V+)

}.

Clearly, for any k < n,

k+1

√‖V+‖Nk

1 (V−) = Nk1 (V−)

k+1

√‖V+‖N1(V−)

≤ Nk1 (V−)

k+1

√N1(V+)

N1(V−).

Similarly,

k+1

√‖V−‖Nk

1 (V+) ≤ N1(V+)k+1

√N1(V−)

N1(V+).

If N1(V+) ≥ N1(V−), then

k+1

√‖V+‖Nk

1 (V−) ≤ N1(V+)n

√N1(V−)

N1(V+).

If N1(V+) ≤ N1(V−), then

k+1

√‖V−‖Nk

1 (V+) ≤ N1(V+)n

√N1(V+)

N1(V−).

Thus min{yup, ylow} ≤ qn(A), where

qn(A) =n

√min{N1(V+)Nn−1

1 (V−), N1(V−)Nn−11 (V+)} . (6.5)

Due to Corollary 3.4, for any µ ∈ Σ(A(·)), we have

ρ(D(µ)) = minj|µ2 + µθj + ψjj | ≤ qn(A) .

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114 M. I. Gil’

So the spectrum of the matrix pencil A(λ) = λ2 I + λθ + ψ lies in the set

{µ ∈ C : |µ2 + µθj + ψjj | ≤ qn(A), j = 1, 2, . . . , n} .Hence, it follows that if

| − t2 + itθj + ψjj | > qn(A) . (6.6)

for all t ∈ R and j = 1, 2, . . . , n, then A(it) is invertible, and thus operator T

defined by (6.1) is invertible. Assume that θj , ψjj are real and

ψjj < −qn(A) (j = 1, . . . , n) .

Then due to (6.6) operator T defined by (6.1) is invertible. Recall that qn(A) is

defined by (6.5) and (6.3).

7. Infinite Matrix Pencils

7.1. The general case

Let {ek}∞k=1 be an orthogonal normed basis in H, and

D(λ) = diag [ak(λ)]∞k=1 (7.1)

be the diagonal matrix in that basis, where ak(λ)(k = 1, 2, . . .) are entire functions.

In addition, V+ and V− are an upper triangular compact matrix and a lower

triangular compact one, 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 . (7.2)

Put as above A(λ) = D(λ) + V+ + V−. So

(A(λ)ej , ej) = aj(λ), (A(λ)ej , ek) = ajk = const (j 6= k) .

It is simple to see that the spectrum Σ(D(·)) of the diagonal matrix D(·) is the

closure of the set of all the roots of aj(λ), j = 1, 2, . . .. Take the maximal spectral

function P (t) = {Pk}∞k=1, where Pk are defined by

Pk =k∑j=1

(., ej)ej .

Clearly, conditions (1.2) hold. Moreover,

ρ(D(λ)) ≡ infk=1,2,...

|ak(λ)| .

In particular, let V± be Hilbert–Schmidt matrices. That is,

N21 (V−) =

∞∑j=1

j−1∑k=1

|ajk|2 <∞; N21 (V+) =

∞∑j=1

∞∑k=j+1

|ajk|2 <∞ . (7.3)

Then due to Corollary 4.6, the spectrum of A(λ) is included in the set

{z ∈ C : |aj(z)| ≤ δ(A); j = 1, 2, . . .} ,where δ(A) is defined by (4.11).

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7.2. A quadratic pencil

In many cases, the Cauchy problem for the second order differential equations leads

to the pencil

A(λ) = λ2I +Bλ+ C (7.4)

where B is a normal compact operator, C is an arbitrary compact one, cf. [27].

Let {ek} be the set of normed eigenvectors of B and λk(B) be the corresponding

eigenvalues. Let

cjk = (Cek, ej) (j, k = 1, 2, . . .)

be the matrix entries of C in the basis {ek}. Then A(λ) can be written in the form

(1.1) with the notation (7.1), (7.2), where

aj(λ) = λ2 + λj(B)λ + cjj

and ajk = cjk (j 6= k). Assume that V+, V− are Hilbert–Schmidt operators. So

condition (7.3) holds. Let s1j , s2j be the roots of the equations aj(λ) = λ2+λj(B)λ+

cjj = 0. Then

D(λ) = diag [(λ− s1j)(λ− s2j)] .

Hence, Σ(D(·)) is the closure of the sequences {s1j , s2j j = 1, 2, . . .}. Moreover,

ρ(D(λ)) = infj=1,2,...

|(λ− s1j)(λ− s2j)| .

For any µ ∈ Σ(A(·)) due to Corollary 4.6 we have

infj=1,2,...

|(µ− s1j)(µ− s2j)| ≤ δ(A)

where δ(A) is defined by (4.11). So Σ(A(·)) lies in the set

{z ∈ C : |(z − s1j)(z − s2j)| ≤ δ(A); j = 1, 2, . . .} .

8. Differential Delay Equations with Fredholm Operators

Let us consider in L2[0, 1] an integro-differential equation with delay

ut(x, t) + a(x) + b(x)u(x, t− 1) + (Fu)(x, t) = 0(t ≥ 0 x ∈ [0, 1]) (8.1)

where a(x), b(x) are real bounded measurable functions, defined on [0, 1], and

(Fu)(x, t) =

∫ 1

0

K(x, s)u(s, t)ds .

Here K(·, ·) is a real Hilbert–Schmidt kernel. Equations of the type (8.1) arise in

various applications, cf. Pao [20].

By virtue of application of the Laplace transformation to Eq. (8.1) we get the

operator pencil in L2[0, 1] defined by

[A(λ)h](x) = (λI + a(x) + b(x)e−λ)h(x)

+ (Fh)(x) (x ∈ [0, 1]; h ∈ L2[0, 1]; λ ∈ C) . (8.2)

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116 M. I. Gil’

Here I denotes the unit operator in L2[0, 1]. Put

D(λ) = Iλ+ a(x) + b(x)e−λ; (V+h)(x) =

∫ 1

x

K(x, s)h(s)ds

and

(V−h)(x) =

∫ x

0

K(x, s)h(s)ds (h ∈ L2[0, 1]) .

For 0 ≤ t ≤ 1, define the maximal spectral function P (t) by

(P (t)h)(x) = 0 for t < x ≤ 1 and (P (t)h)(x) = h(x) for 0 ≤ x ≤ t .

In addition, put

P (t) = I for t > 1 and P (t) = 0 for t < 0 .

Clearly, conditions (1.2) are fulfilled. Moreover,

N21 (V+) ≡

∫ 1

0

∫ 1

x

K2(x, s) ds dx <∞ ,

N21 (V−) ≡

∫ 1

0

∫ x

0

K2(x, s) ds dx <∞ .

For any µ ∈ Σ(A(·)) due to Corollary 4.6 we have

ρ(D(µ)) ≡ infx|µ+ a(x) + b(x)e−µ| ≤ δ(A) (8.3)

where δ(A) is defined by (4.11). So in the considered case Σ(A(·)) lies in the set

{z ∈ C : |z + a(x) + b(x)e−z | ≤ δ(A); 0 ≤ x ≤ 1} .

We will say that A(λ) is a stable pencil, if its spectrum lies in open left half-plane.

Let us derive conditions, that provide the stablility of the pencil defined by

(8.2). Assume that

a(x) > |b(x)|+ δ(A) (x ∈ [0, 1]) . (8.4)

Let µ = α+ iβ ∈ Σ(A(·)) with real α, β. Then due to (8.3),

infx|α+ a(x) + b(x)e−α cosβ| ≤ δ(A) .

If α ≥ 0, then

infxα+ a(x)− |b(x)| ≤ δ(A) .

This contradicts (8.4). So the inequality α ≥ 0 is impossible. Thus, the pencil,

defined by (8.2) is stable. This means that, under condition (8.4), all the solutions

of the Eq. (8.1) tend to zero in the norm L2[0, 1].

About other bounds for the spectrum of integral and integro-differential opera-

tors, and their applications to the stability of parabolic integrodifferential equations

see also [8–10].

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Acknowledgment

I am very grateful to the referee for his very helpful remarks. The research of the

author was supported by the Israel Ministry of Secience and Technology.

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Documents

BOUNDS FOR THE SPECTRUM OF ANALYTIC QUASINORMAL OPERATOR PENCILS