quasihermitian operator, Studia Mathematica, 103(1), 17-24.

[10] Gil™, M. I. (1993). Estimates for Norm of matrix-valued and

operator-value functions, Acta Applicandae Mathematicae 32, 59-

87.

[11] Gil™, M. I. (1995). Norm Estimations for Operator-Valued Func-

tions and Applications. Marcel Dekker, Inc, New York.

[12] Gohberg, I. C. and Krein, M. G. (1969). Introduction to the Theory

of Linear Nonselfadjoint Operators, Trans. Mathem. Monographs,

v. 18, Amer. Math. Soc., R.I.

[13] Gohberg, I. C. and Krein, M. G. (1970). Theory and Applications of

Volterra Operators in Hilbert Space, Trans. Mathem. Monographs,

v. 24, Amer. Math. Soc., Providence, R. I.

8. Bounded Perturbations

of Nonselfadjoint

Operators

In the present chapter we consider the operators of the kind A + B, where A

is a P -triangular operator and B is a bounded operator. We investigate the

invertibility conditions and bounds for the spectra of such operators. In par-

ticular, we consider perturbations of the von Neumann - Schatten operators

and operators with von Neumann - Schatten Hermitian components.

8.1 Invertibility of Boundedly Perturbed

P -Triangular Operators

Throughout the present chapter, A is a P -triangular operator in a separable

Hilbert space H with the nilpotent part V and diagonal part D. According to

De¬nition 7.2.2 and Lemma 7.5.1 this means that

A = D + V and σ(D) = σ(A). (1.1)

Besides, D is a normal operator and V is a Volterra one. Moreover, D and V

have the same maximal resolution of the identity. In addition, assume that

B is a bounded linear operator in H. (1.2)

In this section we suppose that D is boundedly invertible:

rl (D) = inf |σ(D)| > 0. (1.3)

M.I. Gil™: LNM 1830, pp. 123“134, 2003.

c Springer-Verlag Berlin Heidelberg 2003

8. Bounded Perturbations of P -Triangular Operators

124

Then due to Lemma 7.3.3, the operator

W := D’1 V

is a Volterra one. Let V belong to a norm ideal Y with a norm |·|Y , introduced

in Section 7.4. Namely, there are positive numbers θk (k ∈ N), with

1/k

’ 0 as k ’ ∞,

θk

such that

V k ¤ θk |V |k . (1.4)

Y

Put

ni(W )’1

θk |W |k .

JY (W ) = Y

k=0

Recall that ni(W ) is the nilpotency index of W . Clearly,

∞

θk |V |k

Y

JY (W ) ¤ .

k+1

rl (D)

k=0

Theorem 8.1.1 Under conditions (1.1)-(1.4), let

’1

rl (D) B JY (W ) < 1.

Then the operator A + B is invertible. Moreover,

JY (W )

(A + B)’1 ¤ .

rl (D) ’ B JY (W )

To prove this theorem we need the following simple result

Lemma 8.1.2 Let A1 , A2 be linear operators in H. In addition, let A1 be

invertible and

B1 A’1 < 1,

1

where B1 = A2 ’ A1 . Then A2 is also invertible, with

A’1

A’1 1

¤ .

2

1 ’ B1 A’1

1

Clearly A2 = A1 + B1 = (I + B1 A’1 )A1 . Hence

Proof: 1

∞

A’1 A’1 (’A’1 B1 )k .

=

2 1 1

k=0

This proves the result. 2

8.1. Invertibility 125

Proof of Theorem 8.1.1: From (1.1) and (1.4) it follows

ni(W )’1

’1 ’1 ’1 ’1

W k D’1

A = (I + D V) D =

k=0

’1

So A’1 ¤ rl (D)JY (W ). Now the required result is due to the previous

lemma. 2

Assume now that the nilpotent part of A belongs to a Neumann-Schatten

ideal:

N2p (V ) := [T race (V — V )p ]1/2p < ∞ (1.5)

for some integer p ≥ 1. Put

ni(W )’1

(p)

θk |W |k ,

Jp (W ) = Y

k=0

(p)

where θk are de¬ned in Section 6.7. Lemma 8.1.2 and Corollary 7.5.4 imply

Corollary 8.1.3 Under conditions (1.1)-(1.3) and (1.5), let

B Jp (W ) < rl (D).

Then operator A + B is invertible. Moreover,

Jp (W )

(A + B)’1 ¤ .

rl (D) ’ B Jp (W )

Note that Theorem 7.5.5 implies under (1.5) the inequality

A’1 ¤ ψp (V, rl (D)), (1.6)

where

p’1 j 2p

N2p (V ) b0 N2p (V )

ψp (V, rl (D)) = a0 exp [ ]. (1.7)

2p

j+1

rl (D)

rl (D)

j=0

Besides, the constants a0 , b0 can be taken as in the relations

√

c c

a0 = and b0 = for any c > 1, in particular, a0 = 2 and b0 = 1,

c’1 2

(1.8)

or as in the relations

a0 = e1/2 and b0 = 1/2. (1.9)

Lemma 8.1.2 and (1.6) yield

8. Bounded Perturbations of P -Triangular Operators

126

Corollary 8.1.4 Under the conditions (1.1)-(1.3) and (1.5), let

B ψp (V, rl (D)) < 1.

Then operator A + B is invertible. Moreover,

ψp (V, rl (D))

(A + B)’1 ¤ .

1 ’ B ψp (V, rl (D))

8.2 Resolvents of Boundedly Perturbed

P -Triangular Operators

We need the following result, which immediately follows from Lemma 8.1.2:

Lemma 8.2.1 Let A1 , A2 be linear operators in H. In addition, let » be a

regular point of A1 and

B1 (A1 ’ »I)’1 < 1 (B1 = A1 ’ A2 ).

Then » is regular also for A2 , and

(A1 ’ »I)’1

’1

(A2 ’ »I) ¤ .

1 ’ B1 (A1 ’ »I)’1

Again put ν(») = ni (V R» (D)). Recall that one can replace ν(») by ∞.

Under (1.4), let

m’1

θk |V |k

Y

JY (V, m, z) = (z > 0).

k+1

z

k=0

Then the previous lemma and Theorem 7.5.3 imply

Theorem 8.2.2 Under conditions (1.1), (1.2), (1.4), let

B JY (V, ν(»), ρ(D, »)) < 1.

Then » is a regular point of A + B. Moreover,

JY (V, ν(»), ρ(D, »))

(A + B ’ »I)’1 ¤ .

1 ’ B JY (V, ν(»), ρ(D, »))

Under (1.5) denote

m’1 k

N2p (V )

˜

Jp (V, m, z) = (z > 0).

z k+1

k=0

Then Lemma 8.2.1 and Corollary 7.5.4 yield

8.3. Roots of Functional Equations 127

Corollary 8.2.3 Under conditions (1.1), (1.2) and (1.5), let

˜

B Jp (V, ν(»), ρ(D, »)) < 1.

Then » is a regular point of A + B. Moreover,

˜

Jp (V, ν(»), ρ(D, »))

’1

(A + B ’ »I) ¤ .

˜

1 ’ B Jp (V, ν(»), ρ(D, »))

Furthermore, put

p’1 2p

k b0 N2p (V )

N2p (V )

] (z > 0, V ∈ C2p ),

ψp (V, z) = a0 exp [

z k+1 z 2p

k=0

where a0 , b0 do not depend on z and can be taken as in (1.8) or as in (1.9).

Now Theorem 7.5.5 and Lemma 8.2.1 yield

Corollary 8.2.4 Under conditions (1.1), (1.2) and (1.5), let

B ψp (V, ρ(D, »)) < 1.

Then » is a regular point of A + B. Moreover,

ψp (V, ρ(D, »))

(A + B ’ »I)’1 ¤ .

1 ’ B ψp (V, ρ(D, »))

8.3 Roots of Scalar Equations

Consider the scalar equation

∞

ak z k = 1 (3.1)

k=1

where the coe¬cients ak , k = 1, 2, ... have the property

γ0 ≡ 2 max |ak | < ∞.

k

k

We will need the following

Lemma 8.3.1 Any root z0 of equation (3.1) satis¬es the estimate |z0 | ≥

1/γ0 .

’1

Proof: Set in (3.1) z0 = xγ0 . We have

∞ ∞

’k ’k

ak γ0 xk |ak |γ0 |x|k .

¤

1= (3.2)

k=1 k=1

8. Bounded Perturbations of P -Triangular Operators

128

But

∞ ∞

’k

2’k = 1

|ak |γ0 ¤

k=1 k=1

’1 ’1

and therefore, |x| ≥ 1. Hence, |z0 | = γ0 |x| ≥ γ0 . As claimed. 2

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

equations, cf. the book (Ostrowski, 1973, p. 277).

Lemma 8.3.2 The extreme right (unique positive) root za of the equation

p’1

1 1 1

exp [ (1 + 2p )] = a (a ≡ const > 0) (3.3)

y j+1 2 y

j=0

satis¬es the inequality za ¤ δp (a), where

if a ¤ pe,

pe/a

δp (a) := . (3.4)

[ln (a/p)]’1/2p if a > pe

Proof: Assume that

pe ≥ a. (3.5)

Since the function

p’1

1 1 1

f (y) ≡ exp [ (1 + 2p )]

y j+1 2 y

j=0

is nonincreasing and f (1) = pe, we have za ≥ 1. But due to (3.3),

p’1

’j ’2p

za exp [(1 + za )/2] ¤ pe/a.

za = 1/a

j=0

So in the case (3.5), the lemma is proved. Let now

pe < a. (3.6)

Then za ¤ 1. But

p’1

xj+1 ¤ pxp ¤ p exp [xp ’ 1] ¤ p exp [(x2p + 1)/2 ’ 1]

j=0

= p exp [x2p /2 ’ 1/2] (x ≥ 1).

So

p’1

1 1 1 1

exp [ (1 + 2p )] ¤ p exp [ 2p ] (y ¤ 1).

f (y) =

y j+1 2 y y

j=0

8.4. Spectral Variations 129

But za ¤ 1 under (3.6). We thus have

1

a = f (za ) ¤ p exp [ 2p ].

za

Or

2p

za ¤ 1/ln (a/p).

This ¬nishes the proof. 2

8.4 Spectral Variations

De¬nition 8.4.1 Let A and B be linear operators in H. Then the quantity

inf |µ ’ »|

svA (B) := sup

µ∈σ(B) »∈σ(A)

is called the spectral variation of a B with respect to A. In addition,

hd(A, B) := max{svA (B), svB (A)}

is the Hausdor¬ distance between the spectra of A and B.

First, we will prove the following technical lemma

Lemma 8.4.2 Let A1 and A2 be linear operators in H with the same domain

and

q ≡ A1 ’ A2 < ∞.

In addition, let

R» (A1 ) ¤ F (ρ’1 (A1 , »)) (» ∈ σ(A1 )) (4.1)

where F (x) is a monotonically increasing non-negative function of a non-

negative variable x, such that F (0) = 0 and F (∞) = ∞. Then

svA1 (A2 ) ¤ z(F, q),

where z(F, q) is the extreme right-hand (positive) root of the equation

1 = qF (1/z). (4.2)

Proof: Due to (4.1) and Lemma 8.2.1,

1 ¤ qF (ρ’1 (A, µ)) for all µ ∈ σ(B). (4.3)

Compare this inequality with (4.2). Since F (x) monotonically increases,

z(F, q) is a unique positive root of (4.2), and ρ(A, µ) ¤ z(F, q). This proves

the required result. 2

The previous lemma and Theorem 7.5.3 imply

8. Bounded Perturbations of P -Triangular Operators

130

Theorem 8.4.3 Let conditions (1.1), (1.2) and (1.4) hold. Then

svD (A + B) ¤ zY (V, B)

where zY (V, B) is the extreme right-hand (positive) root of the equation

ν0 ’1

θk |V |k

Y

1 = B JY (ν0 , V, z) ≡ B k+1

z

k=0

and

ν0 = sup ni (V R» (D)).

»∈σ(D)

Note that to estimate the root zY (V, B), one can use Lemma 8.3.1.

Moreover, Lemma 8.4.2 and Theorem 7.5.5 imply

Theorem 8.4.4 Let conditions (1.1), (1.2) and (1.5) hold. Then svD (A+B)

¤ zp (V, B), where zp (V, B) is the extreme right-hand (positive) root of the

equation

p’1 j 2p

N2p (V ) 1 N2p (V )

1= B exp [ + ]. (4.4)

z j+1 2z 2p

2

j=0

Substitute in (4.4) the equality z = xN2p (V ). Then we have equation (3.3)

with

N2p (V )

a= .

B

Thanks to Lemma 8.3.2, we get yp (V, B) ¤ δp (V, B), where

N2p (V )

δp (V, B) := N2p (V )δp ( )

B

and δp (.) is de¬ned by (3.4). We thus have derived

Corollary 8.4.5 Under the hypothesis of Theorem 8.4.4, svD (A + B) ¤

δp (V, B).

8.5 Perturbations of Compact Operators

First assume that

A ∈ C2 . (5.1)

By virtue of Lemma 8.4.2 and Theorem 6.4.2 we arrive at the following result.

8.5. Compact Operators 131

Theorem 8.5.1 Let condition (5.1) hold and B be a bounded operator in H.

Then

svA (A + B) ¤ z1 (A, B),

˜

where z1 (A, B) is the extreme right-hand (positive) root of the equation

˜

1 g 2 (A)

B

1= exp [ + ]. (5.2)

2z 2

z 2

Furthermore, substitute in (5.2) the equality z = xg(A). Then we arrive at

the equation

1 1 1 g(A)

exp [ + 2 ] = .

z 2 2z B

˜

Applying Lemma 8.3.2 to this equation, we get z1 (A, B) ¤ ∆1 (A, B), where

˜

˜

∆1 (A, B) := g(A)δ1 (g(A)/ B )

where δ1 is de¬ned by (3.4) with p = 1. So

if g(A) ¤ e B ,

eB

˜

∆1 (A, B) := . (5.3)

g(A) [ln (g(A)/ B )]’1/2 if g(A) > e B

Now Theorem 8.5.1 yields

Corollary 8.5.2 Under the hypothesis of Theorem 8.5.1, for any µ ∈ σ(A +

˜

B), there is a µ0 ∈ σ(A), such that |µ ’ µ0 | ¤ δ1 (A, B). In particular,

˜

rs (A + B) ¤ rs (A) + ∆1 (A, B), (5.4)

˜ ˜

rl (A + B) ≥ max{rl (A) ’ ∆1 (A, B), 0} and ±(A + B) ¤ ±(A) + ∆1 (A, B).

(5.5)

Remark 8.5.3 According to Lemma 6.5.2, in Theorem 8.5.1 and its corol-

√

lary, one can replace g(A) by 2N2 (AI ).

Now let

A ∈ C2p (p = 2, 3, ...). (5.6)

Then by virtue of Lemma 8.4.2 and Theorem 6.7.3 we arrive at the following

result.

Theorem 8.5.4 Let condition (5.6) hold and B be a bounded operator in

H. Then svA (A + B) ¤ yp (A, B), where yp (A, B) is the extreme right-hand

˜ ˜

(positive) root of the equation

p’1

(2N2p (A))m 1 (2N2p (A))2p

1= B exp [ + ]. (5.7)

z m+1 2z 2p

2

m=0

8. Bounded Perturbations of P -Triangular Operators

132

Furthermore, substitute in (5.7) the equality z = x2N2p (A) and apply Lemma

8.3.2. Then we get yp (A, B) ¤ ∆p (A, B), where

˜

∆p (A, B) := 2N2p (A)δp (2N2p (A)/ B ).

Recall that δp (.) is de¬ned by (3.4). So

if 2N2p (A) ¤ ep B ,

pe B

∆p (A, B) := .

2N2p (A) [ln (2N2p (A)/p B )]’1/2p if 2N2p (A) > ep B

(5.8)

Now Theorem 8.5.4 yields

Corollary 8.5.5 Under the hypothesis of Theorem 8.5.4, for any µ ∈ σ(A +

B), there is a µ0 ∈ σ(A), such that |µ ’ µ0 | ¤ ∆p (A, B). In particular,

˜

relations (5.4) and (5.5) hold with ∆p (A, B) instead of ∆1 (A, B).

Remark 8.5.6 According to Theorem 7.9.1, in Theorem 8.5.4 and its corol-

˜

lary one can replace 2N2p (A) by βp N2p (AI ).

8.6 Perturbations of Operators with Compact

Hermitian Components

First, let Dom (A) = Dom (A— ) and

A ’ A— ∈ C2 . (6.1)

Then by virtue of Lemma 8.4.2 and Theorem 7.7.1 we arrive at the following

result.

Theorem 8.6.1 Let condition (6.1) hold and B be a bounded operator in

H. Then svA (A + B) ¤ x1 (A, B), where x1 (A, B) is the extreme right-hand

(positive) root of the equation

∞k

gI (A)

√

1= B . (6.2)

k!z k+1

k=0

√

Recall that gI (A) ¤ 2N2 (AI ). According to Theorem 7.7.1, one can replace

(6.2) by the equation

2

B 1 gI (A)

1= exp [ + ]. (6.3)

2z 2

z 2

Furthermore, substitute in (6.3) the equality z = xgI (A) and apply Lemma

8.3.2. Then we can assert that extreme right-hand root of equation (6.3) is

less than „1 (A, B), where

„1 (A, B) = gI (A)δ1 (gI (A)/ B ).

8.6. Operators with Compact Components 133

So, according to (3.4),

if gI (A) ¤ e B ,

eB

„1 (A, B) = .

gI (A) [ln (gI (A)/ B )]’1/2 if gI (A) > e B

Hence, Theorem 8.6.1 yields

Corollary 8.6.2 Let condition (6.1) hold and B be a bounded operator in

H. Then for any µ ∈ σ(A + B), there is a µ0 ∈ σ(A), such that |µ ’ µ0 |

¤ „1 (A, B). In particular,

rs (A + B) ¤ rs (A) + „1 (A, B),

rl (A + B) ≥ max{rl (A) ’ „1 (A, B), 0} and ±(A + B) ¤ ±(A) + „1 (A, B).

Now let

A ’ A— ∈ C2p (p = 2, 3, ...). (6.4)

Then by virtue of Lemma 8.4.2 and Theorem 7.9.1 we arrive at the following

result.

Theorem 8.6.3 Let condition (6.4) hold and B be a bounded operator in

H. Then svD (A + B) ¤ xp (A, B), where xp (A, B) is the extreme right-hand

˜ ˜

(positive) root of the equation

p’1 ∞ pk+m

wp (A)

√

1= B . (6.5)

k!z pk+m+1

m=0 k=0

where

˜

wp (A) = βp N2p (AI ).

According to Theorem 7.9.1, one can replace (6.5) by the equation

p’1

(wp (A))m 1 (wp (A))2p

1= B exp [ + ]. (6.6)

z m+1 2z 2p

2

m=0

Furthermore, substitute in (6.6) the equality z = xwp (A) and apply Lemma

8.3.2. Then we can assert that the extreme right-hand root of equation (6.6)

is less than

mp (A, B) := wp (A)δp (wp (A)/ B ),

where δp (a) is de¬ned by (3.4). That is,

if wp (A) ¤ ep B

pe B

mp (A, B) = .

wp (A) [ln (wp (A)/p B )]’1/2p if wp (A) > ep B

Now Theorem 8.6.3 implies

Corollary 8.6.4 Let conditions (1.2) and (6.6) hold. Then for any µ ∈

σ(A + B), there is a µ0 ∈ σ(A), such that |µ ’ µ0 | ¤ mp (A, B). In particular,

rs (A + B) ¤ rs (A) + mp (A, B),

rl (A + B) ≥ max{rl (A) ’ „p (A, B), 0} and ±(A + B) ¤ ±(A) + „p (A, B).

˜ ˜

8. Bounded Perturbations of P -Triangular Operators

134

8.7 Notes

The material of this chapter is based on the papers (Gil™, 2002) and (Gil™,

2003). About the well-known perturbations results see (Kato, 1966),

(Baumg´rtel, 1985) and references therein.

a

References

[1] Baumg´rtel, H. (1985). Analytic Perturbation Theory for Matri-

a

ces and Operators. Operator Theory, Advances and Appl., 52.

Birkh´user Verlag, Basel, Boston, Stuttgart.

a

[2] Gil™, M. I. (2002). Invertibility and spectrum localization of non-

selfadjoint operators, Advances in Applied Mathematics, 28, 40-58.

[3] Gil™, M. I. (2003). Inner bounds for spectra of linear operators,

Proceedings of the American Mathematical Society , (to appear)

[4] Kato, T. (1966). Perturbation Theory for Linear Operators,

Springer-Verlag. New York.

[5] Ostrowski, A. M. (1973). Solution of Equations in Euclidean and

Banach spaces. Academic Press, New York - London.

9. Spectrum Localization

of Nonself-adjoint

Operators

In the present chapter we consider operators of the form A = D + V+ + V’ ,

where D is a normal operator and V± are Volterra (quasinilpotent compact)

operators. Numerous integral, integro-di¬erential operators and in¬nite ma-

trices can be represented in such a form. We investigate the invertibility

conditions and bounds for the spectra of the mentioned operators.

9.1 Invertibility Conditions

Let P (t) (a ¤ t ¤ b; ’∞ ¤ a < b ¤ ∞) be a maximal resolution of the

identity in a separable Hilbert space H (see Section 7.2). Let

A = D + V + + V’ , (1.1)

where D is a normal operator and V± are Volterra operators satisfying the

conditions

P (t)V+ P (t) = V+ P (t); P (t)V’ P (t) = P (t)V’ ,

P (t)Dh = DP (t)h (a ¤ t ¤ b; h ∈ Dom (D)). (1.2)

As above, Y is a norm ideal of compact operators in H, which is complete in

an auxiliary norm | · |Y . In addition, there are positive numbers θk (k ∈ N)

1/k

with θk ’ 0 (k ’ ∞), for which, for arbitrary Volterra operators V ∈ Y ,

V k ¤ θk |V |k (k = 1, 2, ...). It is assumed that V± ∈ Y . That is,

Y

V± ¤ θk |V± |k (k = 1, 2, ...).

k

(1.3)

Y

M.I. Gil™: LNM 1830, pp. 135“149, 2003.

c Springer-Verlag Berlin Heidelberg 2003

136 9. Spectrum Localization

In addition, in this section it is assumed that D is boundedly invertible:

rl (D) = inf |σ(D)| > 0. (1.4)

Since D is normal and V± are Volterra operators, due to Lemma 7.3.3 con-

ditions (1.2) are enough to guarantee that

W± := D’1 V±

are also Volterra operators. Under (1.2)-(1.4), put

ni(W± )’1

θk |W± |k ,

JY (W± ) ≡ (1.5)

Y

k=0

where ni(V ) again denotes the “nilpotency index” of a quasinilpotent oper-

ator V (see Section 7.5). With this notation we have

Theorem 9.1.1 Under conditions (1.1)-(1.4), let

1 1

ζY (A) := max { ’ W+ , ’ W’ } > 0. (1.6)

JY (W’ ) JY (W+ )

Then A is boundedly invertible, and

1

A’1 ¤ . (1.7)

rl (D)ζY (A)

The proof of Theorem 9.1.1 is presented in the next section. Note that in

Theorem 9.1.1, one can replace JY (W± ) by

∞

θk |W± |k .

IY (W± ) := Y

k=0

Consider operators, whose o¬-diagonal parts belong to the Neumann-Schatten

ideal C2p with some integer p ≥ 1:

V± ∈ C2p . (1.8)

Again put

1

(p)

θj = ,

[j/p]!

where [x] means the integer part of a real number x. For a Volterra operator

V ∈ C2p denote

ni(V )’1

(p)k

Jp (V ) = θk N2p (V ).

k=0

Now Theorem 9.1.1 and Corollary 6.9.4 imply

9.2. Proof of Theorems 9.1.1 and 9.1.3 137

Corollary 9.1.2 Let relations (1.2), (1.4) and (1.8) hold. In addition, let

1 1

ζ2p (A) ≡ max{ ’ W+ , ’ W’ } > 0. (1.9)

Jp (W’ ) Jp (W+ )

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

1

A’1 ¤ .

ζ2p (A)rl (D)

It is simple to see that, in this corollary, one can replace Jp (W± ) by I2p (W± ),

where

p’1 ∞ j+pk

N2p (W± )

√

I2p (W± ) := .

k!

j=0 k=0

Moreover, put

p’1

j 2p

ψp (W± ) := a0 N2p (W± ) exp [b0 N2p (W± )].

j=0

Besides, the constants a0 , b0 can be taken as in the relations

√

c c

a0 = and b0 = for any c > 1, in particular, a0 = 2 and b0 = 1,

c’1 2

(1.10)

or as in the relations

a0 = e1/2 and b0 = 1/2. (1.11)

In the next section we also prove

Theorem 9.1.3 Let relations (1.2), (1.4) and (1.8) hold. In addition, let

1 1

˜ ’ W+ , ’ W’ } > 0.

ζp (A) := max{

ψp (W’ ) ψp (W+ )

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

1

A’1 ¤ .

˜

ζp (A)rl (D)

9.2 Proofs of Theorems 9.1.1 and 9.1.3

We need the following simple

Lemma 9.2.1 Under conditions (1.2) and (1.4), let

θ0 := (D + V’ )’1 V+ < 1. (2.1)

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

(D + V’ )’1

A’1 ¤ . (2.2)

1 ’ θ0

138 9. Spectrum Localization

Proof: According to (1.1) we have

A = (D + V’ )(I + (D + V’ )’1 V+ ). (2.3)

Thanks to (1.2) and Lemma 7.5.1,

σ(D + V± ) = σ(D). (2.4)

So according to (1.4), D + V± is invertible. Moreover, under condition (2.1),

the operator

I + (D + V’ )’1 V+

is invertible and

∞

’1 ’1

((D + V’ )’1 V+ )k ¤

¤

(I + (D + V’ ) V+ )

k=0

∞

θ0 = (1 ’ θ0 )’1 .

k

k=0

So due to (2.3)

A’1 ¤ (I + (D + V’ )’1 V+ )’1 (D + V’ )’1 .

This proves the required result. 2

Proof of Theorem 9.1.1: Since V± ∈ Y ,

(D + V’ )’1 V+ = (I + D’1 V’ )’1 D’1 V+ = (I + W’ )’1 W+ ¤

ni(W’ )’1

∞

k k

W’ ¤

W+ W’ = W+

k=0 k=0

ni(W’ )’1

θk |W’ |k = W+ JY (W’ ).

W+ Y

k=0

Hence

θ0 ¤ W+ JY (W’ ).

But condition (1.6) implies that at least one of the following inequalities

W+ JY (W’ ) < 1 (2.5)

or

W’ JY (W+ ) < 1 (2.6)

9.3. Resolvents of Quasinormal Operators 139

are valid. If condition (2.5) holds, then (2.1) is valid. Moreover, since D is a

normal operator, D’1 = rl (D)’1 . Thus,

∞

’1 ’1 ’1 ’1 k

¤D

(D + V’ ) = (I + W’ ) D W’ =

k=0

ni(W’ )’1 ni(W’ )’1

’1 k ’1

θk |W’ |k = rl (D)’1 JY (W’ ).

¤ rl (D)

rl (D) W’ Y

k=0 k=0

Thus, under (2.5), Lemma 9.2.1 yields the inequality

JY (W’ ) 1

A’1 ¤ = . (2.7)

0 ’1

rl (D)(1 ’ W+ JY (W’ )) rl (D)(JY (W’ ) ’ W+ )

Interchanging W’ and W+ , under condition (2.6), we get

1

A’1 ¤ .

0 ’1

rl (D)(JY (W+ ) ’ W’ )

This relation and (2.7) yield the required result. 2

Proof of Theorem 9.1.3: Due to Theorem 6.7.3,

(D + V± )’1 = (I + W± )’1 D’1 ¤ ψp (W± )ρ’1 (D).

l

In addition,

(D + V’ )’1 V+ = (I + W’ )’1 W+ ¤ ψp (W’ ) W+

and

(D + V+ )’1 V’ ¤ ψp (W+ ) W’ .

Now the required result is due to Lemma 9.2.1. 2

9.3 Resolvents of Quasinormal Operators

For a V ∈ Y , denote

m’1

z ’1’k θk |V |k (z > 0).

JY (V, m, z) := Y

k=0

Due to Lemma 7.3.4, (D ’ »I)’1 V± is a quasinilpotent operator for any

» ∈ σ(D). Put

ν± (») ≡ ni((D ’ »I)’1 V± ).

Everywhere below we can replace ν± (») by ∞.

Again, R» (A) is the resolvent and ρ(D, ») = inf z∈σ(D) |s ’ z|.

140 9. Spectrum Localization

Lemma 9.3.1 Under conditions (1.2), (1.3), for a » ∈ σ(D), let

1

ζ(A, ») ≡ max{ ’ V+ ,

JY (V’ , ν’ (»), ρ(D, »))

1

’ V’ } > 0. (3.1)

JY (V+ , ν+ (»), ρ(D, »))

Then » is a regular point of operator A represented by (1.1). Moreover,

1

R» (A) ¤ . (3.2)

ζ(A, »)ρ(D, »)

Since, D is a normal operator, (D ’ »I)’1 = ρ’1 (», D). Thus,

Proof:

|(D ’ »I)’1 V± |Y ¤ (D ’ »I)’1 |V± |Y =

ρ’1 (», D)|V± |Y .

Hence,

ν’ (»)’1

(D ’ »I)’1 V+ θk |(D ’ »I)’1 V’ |k ¤

Y

k=0

ν’ (»)’1

θk ρ’1’k (», D)|V’ |k = V+ JY (V’ , ν’ (»), ρ(D, »)).

V+ Y

k=0

Similarly,

ν+ (»)’1

’1

θk |(D ’ »I)’1 V+ |k ¤ V’ JY (V+ , ν+ (»), ρ(D, »)).

(D ’ »I) V’ Y

k=0

Now Theorem 9.1.1 with

A ’ »I = D + V+ + V’ ’ »I

instead of A, yields the required result. 2

Furthermore, Lemma 9.3.1 implies

Corollary 9.3.2 Under conditions (1.1), (1.2) and (1.3), for any µ ∈ σ(A),

there is a µ0 ∈ σ(D), such that, either µ = µ0 , or both the inequalities

V+ JY (V’ , ν’ (µ), |µ ’ µ0 |) ≥ 1 and

V’ JY (V+ , ν+ (µ), |µ ’ µ0 |) ≥ 1 (3.3)

are true.

9.4. Upper Bounds for Spectra 141

This result is exact in the following sense: if either V’ = 0, or (and) V+ = 0,

then due to the latter corollary,

σ(A) = σ(D). (3.4)

Now let condition (1.8) hold. Put

(p)

m’1 k

θk N2p (V± )

˜

Jp (V± , m, z) = (z > 0).

z k+1

k=0

Theorem 6.7.1 and Lemma 9.3.1 imply

Corollary 9.3.3 Under conditions (1.2) and (1.8), for a » ∈ σ(D), let

1

ζ2p (», A) ≡ max{ ’ V+ ,

˜

Jp (V’ , ν’ (»), ρ(D, »))

1

’ V’ } > 0.

˜

Jp (V+ , ν+ (»), ρ(D, »))

Then » is a regular point of operator A, represented by (1.1). Moreover,

1

R» (A) ¤ .

ρ(D, »)ζ2p (», A)

Furthermore, thanks to Theorem 9.1.3 with A’» instead of A, we can replace

˜

Jp by the function

p’1 j 2p

N2p (V± ) b0 N2p (V± )

ψp (V± , z) = a0 exp [ ],

z j+1 z 2p

j=0

where a0 , b0 can be taken from (1.10) or from (1.11). Then we have

Corollary 9.3.4 Under conditions (1.2) and (1.8), for a » ∈ σ(D), let

1

˜

ζ2p (», A) ≡ max{ ’ V+ ,

ψp (V’ , ρ(D, »))

1

’ V’ } > 0.

ψp (V+ , ρ(D, »))

Then » is a regular point of operator A, represented by (1.1). Moreover,

1

R» (A) ¤ .

ρ(D, »)ζ2p (», A)

142 9. Spectrum Localization

9.4 Upper Bounds for Spectra

Recall that svD (A) denotes the spectral variatin of A with respect to D. Put

„ (A) := min{ V’ , V+ }, (4.1)

if V+ ≥ V’

V+

˜

V := (4.2)

V’ if V’ > V+

and

ni ((D ’ »I)’1 V ).

˜

ν0 = sup

˜

»∈σ(D)

In the sequel one can replace ν0 by ∞.

˜

˜

Theorem 9.4.1 Under conditions (1.1), (1.2), let V ∈ Y and

V k ¤ θk |V |k (k = 1, 2, ...).

˜ ˜Y (4.3)

Then the equation

˜˜

„ (A)JY (V , ν0 , z) = 1 (4.4)

has a unique positive root zY (A). Moreover, svD (A) ¤ zY (A).

Proof: Due to Corollary 9.3.2,

˜˜

„ (A)JY (V , ν0 , ρ(D, µ)) ≥ 1

for any µ ∈ σ(A). Comparing this inequality with (4.4), we have ρ(D, µ) ¤

zY (A). This inequality proves the theorem. 2

Lemma 9.4.2 Under the conditions (1.1), (1.2) and

˜

V ∈ C2p (p = 1, 2, ...), (4.5)

the equation

p’1 j˜ 2p ˜

N2p (V ) N2p (V )

1

„ (A) exp [ (1 + )] = 1. (4.6)

z j+1 z 2p

2

j=0

has a unique positive root z2p (A). Moreover,

svD (A) ¤ z2p (A). (4.7)

Proof: Due to Corollary 9.3.4,

p’1 j˜ 2p ˜

N2p (V ) N2p (V )

1

)] ≥ 1

„ (A) exp [ (1 + 2p

ρj+1 (D, µ) 2 ρ (D, µ)

j=0

9.5. Inner Bounds for Spectra 143

for any µ ∈ σ(A). Comparing this inequality with (4.6), we get ρ(D, µ) ¤

z2p (A). This inequality proves the required result. 2

˜

To estimate z2p (A), substitute z = xN2p (V ) in (4.6) and use Lemma

˜ ˜

8.3.2. Then z2p (A) ¤ φp (A) = N2p (V )δp (a), where a = N2p (V )/„ (A) and

if a ¤ pe

pe/a

δp (a) := . (4.8)

[ln (a/p)]’1/2p if a > pe

That is,

˜

if N2p (V ) ¤ „ (A)pe

pe„ (A)

φp (A) := .

N2p (V )[ln (N2p (V )/p„ (A))]’1/2p

˜ ˜ ˜

if N2p (V ) > „ (A)pe

(4.9)

Now the previous lemma yields

Corollary 9.4.3 Under conditions (1.1), (1.2) and (4.5), svD (A) ¤ φp (A).

In particular,

rs (A) ¤ rs (D) + φp (A),

provided D is bounded.

˜

In the case V ∈ C2 we have

if a ¤ e

e/a

δ1 (a) := (4.10)

[ln a]’1/2 if a > e

and

˜

if N2 (V ) ¤ „ (A)e

e„ (A)

φ1 (A) := . (4.11)

N2 (V ) [ln (N2 (V )/„ (A))]’1/2

˜ ˜ ˜

if N2 (V ) > „ (A)e

Now (4.7) implies

svD (A) ¤ z2 (A) ¤ φ1 (A). (4.12)

Remark 9.4.4 Everywhere below V± can be replaced by their upper bounds,

since the right roots of equations (4.4) and (4.6) increase, when the coe¬-

cients of these equations increase.

9.5 Inner Bounds for Spectra

Again, let there be a monotonically increasing continuous scalar-valued func-

tion F (z) (z ≥ 0) with the properties

F (0) = 0, F (∞) = ∞ (5.1)

such that the inequality

(»I ’ A)’1 ¤ F (ρ’1 (A, »)) (5.2)

144 9. Spectrum Localization

holds, where ρ(A, ») is the distance between σ(A) and a regular point » ∈ C

of A. Recall that „ (A) := min{ V’ , V+ } and denote by y(„, F ) the

unique positive root of the equation

„ (A)F (1/z) = 1 (z > 0). (5.3)

Now we are in a position to formulate the main result of the present section.

Theorem 9.5.1 Let A be de¬ned by (1.1) and conditions (1.2) and (5.2)

hold. Then

svA (D) ¤ y(„, F ). (5.4)

The proof of this theorem is presented below. Recall that

rl (A) := inf |σ(A)|, ±(A) := sup Re σ(A).

Corollary 9.5.2 Under the hypothesis of Theorem 9.5.1, the following in-

equalities are true:

rs (A) ≥ max{0, rs (D) ’ y(„, F )} if D is bounded, (5.5)

rl (A) ¤ rl (D) + y(„, F ) and (5.6)

±(A) ≥ ±(D) ’ y(„, F ) if ±(D) < ∞. (5.7)

Indeed, take µ in such a way that |µ| = rs (D). Then due to (5.4), there is

µ0 ∈ σ(A), such that |µ0 | ≥ rs (D) ’ y(„, F ). Hence, (5.5) follows. Similarly,

inequality (5.6) can be proved. Furthermore, take µ in such a way that

Re µ = ±(D). Due to (1.6) for some µ0 ∈ σ(A), |Re µ0 ’ ±(D)| ¤ y(„, F ).

So, either Re µ0 ≥ ±(D), or Re µ0 ≥ ±(D) ’ y(„, F ). Thus, inequality (5.7)

is also proved.

Proof of Theorem 9.5.1: Take the operator B+ = D + V+ . Then due

to (2.4) σ(B+ ) = σ(D). Due to to Lemma 8.4.2, for any µ ∈ σ(D), there is

µ0 ∈ σ(A), such that

|µ0 ’ µ| ¤ z’ , (5.8)

where z’ is the unique positive root of the equation

V’ F (1/z) = 1 (z ≥ 0).

Now, take B’ = D + V’ . Then due to relation (2.4), we get σ(B’ ) = σ(D).

Similarly to (5.8), we have that for any µ ∈ σ(D), there is µ0 ∈ σ(A), such

that

|µ0 ’ µ| ¤ z+ , (5.9)

where z+ is the unique positive root of the equation V+ F (1/z) = 1, since

A ’ B’ = V+ . Relations (5.8) and (5.9) prove the required result. 2

9.7. Hilbert-Schmidt Operators 145

9.6 Bounds for Spectra of Hilbert-Schmidt

Operators

Assume that

A ∈ C2 . (6.1)

Recall that g(A) is de¬ned in Section √ According to Lemma 6.5.1 every-

6.4.

where below one can replace g(A) by 2N2 (AI ).

Under (6.1), denote by y2 (A, „ ) the unique non-negative root of the equa-

˜

tion

1 g 2 (A)

„ (A)

exp [ + ] = 1, (6.2)

2z 2

z 2

where „ (A) is de¬ned by (4.1).

Theorem 9.6.1 Let conditions (1.1), (1.2) and (6.1) hold. Then the rela-

tions (4.12) and

svA (D) ¤ y2 (A, „ )

˜

are valid.

Proof: The required result is due to Theorems 6.4.2 and 9.5.1, and Corol-

lary 9.4.3. 2

Substitute z = g(A)x in (6.2) and use Lemma 8.3.2. Then we get

g(A)

˜

y2 (A, „ ) ¤ ∆2 (A) := g(A)δ1 (

˜ ),

„ (A)

where δ1 (a) is de¬ned by (4.10). That is,

if g(A) ¤ e„ (A)

e„ (A)

˜

∆2 (A) := . (6.3)

g(A)[ln (g(A)/„ (A))]’1/2 if g(A) > e„ (A)

Thus Theorem 9.6.1 and relations (4.12) imply

Corollary 9.6.2 Let relations (1.1), (1.2) and (6.1) hold. Then

˜

svD (A) ¤ φ1 (A) and svA (D) ¤ ∆2 (A).

In particular,

˜

max{0, rs (D) ’ ∆2 (A)} ¤ rs (A) ¤ rs (D) + φ1 (A),

˜

max{0, rl (D) ’ φ1 (A)} ¤ rl (A) ¤ rl (D) + ∆2 (A)

˜

and ±(D) ’ ∆2 (A) ¤ ±(A) ¤ ±(D) + φ1 (A).

146 9. Spectrum Localization

9.7 Von Neumann-Schatten Operators

Assume that

A ∈ C2p for an integer p > 1 (7.1)

and denote by yp (A, „ ) the unique non-negative root of the equation

p’1

(2N2p (A))m 1 (2N2p (A))2p

„ (A) exp [ + ] = 1, (7.2)

z m+1 2z 2p

2

m=0

where „ (A) is de¬ned by (4.1).

Theorem 9.7.1 Let conditions (1.1), (1.2) and (7.1) hold. Then the in-

equalities (4.7) and svA (D) ¤ yp (A, „ ) are valid.

Proof: The required result is due to Theorems 6.7.4 and 9.5.1, and Corol-

lary 9.4.4. 2

Substitute z = 2N2p (A)x in (7.2) and use Lemma 8.3.2. Then we arrive

at the inequality

2N2p (A)

yp (A, „ ) ¤ ∆p (A) := N2p (A)δp ( ),

„ (A)

where δp (a) is de¬ned by (4.8). That is,

if 2N2p(A) ¤ p e „ (A)

p e „ (A)

∆p (A) := .

2N2p (A)[ln (2N2p (A)/„ (A))]’1/2p if 2N2p (A) > p e „ (A)

(7.3)

Thus Theorem 9.7.1 and Corollary 9.4.3 imply

Corollary 9.7.2 Let relations (1.1), (1.2) and (7.1) hold. Then

svD (A) ¤ φp (A) and svA (D) ¤ ∆p (A).

In particular,

max{0, rs (D) ’ ∆p (A)} ¤ rs (A) ¤ rs (D) + φp (A),

max{0, rl (D) ’ φp (A)} ¤ rl (A) ¤ rl (D) + ∆p (A)

and ±(D) ’ ∆p (A) ¤ ±(A) ¤ ±(D) + φp (A).

9.8. Operators with Hilbert-Schmidt Components 147

9.8 Operators with Hilbert-Schmidt

Hermitian Components

In this section it is assumed that Dom (A) = Dom (A— ) and A has the

Hilbert-Schmidt imaginary component AI ≡ (A ’ A— )/2i:

N2 (AI ) = T race (AI )2 < ∞.

2

(8.1)

Recall that gI (A) is de¬ned in Section 7.7. Everywhere below on can replace

√

gI (A) by 2N2 (AI ).

Under (8.1), denote by yH (A, „ ) the unique non-negative root of the equa-

tion

2

„ (A) 1 gI (A)

exp [ + ] = 1. (8.2)

2z 2

z 2

Theorem 9.8.1 Let conditions (1.1), (1.2) and (8.1) hold. Then the rela-

tions (4.12) and

svA (D) ¤ yH (A, „ )

are valid.

Proof: The required result is due to Theorems 7.7.1 and 9.5.1, and Corol-

lary 9.4.3. 2

Substitute z = gI (A)x in (8.2) and apply Lemma 8.3.2. Then

gI (A)

yH (A, „ ) ¤ ∆H (A) := gI (A)δ1 ( ),

„ (A)

where δ1 (a) is de¬ned by (4.10). That is,

if gI (A) ¤ e„ (A)

e „ (A)

∆H (A) := .

gI (A)[ln (gI (A)/„ (A))]’1/2 if gI (A) > e „ (A)

Thanks to (4.12), Theorem 9.8.1 implies

Corollary 9.8.2 Let relations (1.1), (1.2) and (8.1) hold. Then

svD (A) ¤ φ1 (A) and svA (D) ¤ ∆H (A).

In particular,

max{0, rs (D) ’ ∆H (A)} ¤ rs (A) ¤ rs (D) + φ1 (A),

max{0, rl (D) ’ φ1 (A)} ¤ rl (A) ¤ rl (D) + ∆H (A)

and ±(D) ’ ∆H (A) ¤ ±(A) ¤ ±(D) + φ1 (A).

148 9. Spectrum Localization

9.9 Operators with Neumann-Schatten

Hermitian Components

In this section it is assumed that the Hermitian component AI = (A’A— )/2i

belongs to the Neumann-Schatten ideal C2p with some integer p > 1:

Np (AI ) = [T race A2p ]1/2p < ∞. (9.1)