<< стр. 6(всего 11)СОДЕРЖАНИЕ >>
 Gil , M. I. (1992). One estimate for the norm of a function of a
quasihermitian operator, Studia Mathematica, 103(1), 17-24.
 GilвЂ™, M. I. (1993). Estimates for Norm of matrix-valued and
operator-value functions, Acta Applicandae Mathematicae 32, 59-
87.
 GilвЂ™, M. I. (1995). Norm Estimations for Operator-Valued Func-
tions and Applications. Marcel Dekker, Inc, New York.

 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.
 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
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
Deп¬Ѓnition 7.2.2 and Lemma 7.5.1 this means that

A = D + V and Пѓ(D) = Пѓ(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 |Пѓ(D)| > 0. (1.3)

M.I. GilвЂ™: LNM 1830, pp. 123вЂ“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 := Dв€’1 V

is a Volterra one. Let V belong to a norm ideal Y with a norm |В·|Y , introduced
in Section 7.4. Namely, there are positive numbers Оёk (k в€€ N), with
1/k
в†’ 0 as k в†’ в€ћ,
Оёk

such that
V k в‰¤ Оёk |V |k . (1.4)
Y

Put
ni(W )в€’1
Оёk |W |k .
JY (W ) = Y
k=0

Recall that ni(W ) is the nilpotency index of W . Clearly,
в€ћ
Оёk |V |k
Y
JY (W ) в‰¤ .
k+1
rl (D)
k=0

Theorem 8.1.1 Under conditions (1.1)-(1.4), let
в€’1
rl (D) B JY (W ) < 1.

Then the operator A + B is invertible. Moreover,

JY (W )
(A + B)в€’1 в‰¤ .
rl (D) в€’ 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 Aв€’1 < 1,
1

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

Aв€’1
Aв€’1 1
в‰¤ .
2
1 в€’ B1 Aв€’1
1

Clearly A2 = A1 + B1 = (I + B1 Aв€’1 )A1 . Hence
Proof: 1

в€ћ
Aв€’1 Aв€’1 (в€’Aв€’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 )в€’1
в€’1 в€’1 в€’1 в€’1
W k Dв€’1
A = (I + D V) D =
k=0

в€’1
So Aв€’1 в‰¤ 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 в€— V )p ]1/2p < в€ћ (1.5)
for some integer p в‰Ґ 1. Put
ni(W )в€’1
(p)
Оёk |W |k ,
Jp (W ) = Y
k=0

(p)
where Оёk are deп¬Ѓned 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)в€’1 в‰¤ .
rl (D) в€’ B Jp (W )

Note that Theorem 7.5.5 implies under (1.5) the inequality

Aв€’1 в‰¤ П€p (V, rl (D)), (1.6)

where
pв€’1 j 2p
N2p (V ) b0 N2p (V )
П€p (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
в€љ
c c
a0 = and b0 = for any c > 1, in particular, a0 = 2 and b0 = 1,
cв€’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 П€p (V, rl (D)) < 1.

Then operator A + B is invertible. Moreover,

П€p (V, rl (D))
(A + B)в€’1 в‰¤ .
1 в€’ B П€p (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 О» be a
regular point of A1 and

B1 (A1 в€’ О»I)в€’1 < 1 (B1 = A1 в€’ A2 ).

Then О» is regular also for A2 , and

(A1 в€’ О»I)в€’1
в€’1
(A2 в€’ О»I) в‰¤ .
1 в€’ B1 (A1 в€’ О»I)в€’1

Again put ОЅ(О») = ni (V RО» (D)). Recall that one can replace ОЅ(О») by в€ћ.
Under (1.4), let
mв€’1
Оёk |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, ОЅ(О»), ПЃ(D, О»)) < 1.

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

JY (V, ОЅ(О»), ПЃ(D, О»))
(A + B в€’ О»I)в€’1 в‰¤ .
1 в€’ B JY (V, ОЅ(О»), ПЃ(D, О»))

Under (1.5) denote
mв€’1 k
N2p (V )
Лњ
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
Лњ
B Jp (V, ОЅ(О»), ПЃ(D, О»)) < 1.

Then О» is a regular point of A + B. Moreover,
Лњ
Jp (V, ОЅ(О»), ПЃ(D, О»))
в€’1
(A + B в€’ О»I) в‰¤ .
Лњ
1 в€’ B Jp (V, ОЅ(О»), ПЃ(D, О»))

Furthermore, put
pв€’1 2p
k b0 N2p (V )
N2p (V )
] (z > 0, V в€€ C2p ),
П€p (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 П€p (V, ПЃ(D, О»)) < 1.

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

П€p (V, ПЃ(D, О»))
(A + B в€’ О»I)в€’1 в‰¤ .
1 в€’ B П€p (V, ПЃ(D, О»))

8.3 Roots of Scalar Equations
Consider the scalar equation
в€ћ
ak z k = 1 (3.1)
k=1

where the coeп¬ѓcients ak , k = 1, 2, ... have the property

Оі0 в‰Ў 2 max |ak | < в€ћ.
k

k

We will need the following
Lemma 8.3.1 Any root z0 of equation (3.1) satisп¬Ѓes the estimate |z0 | в‰Ґ
1/Оі0 .
в€’1
Proof: Set in (3.1) z0 = xОі0 . We have
в€ћ в€ћ
в€’k в€’k
ak Оі0 xk |ak |Оі0 |x|k .
в‰¤
1= (3.2)
k=1 k=1
8. Bounded Perturbations of P -Triangular Operators
128

But
в€ћ в€ћ
в€’k
2в€’k = 1
|ak |Оі0 в‰¤
k=1 k=1
в€’1 в€’1
and therefore, |x| в‰Ґ 1. Hence, |z0 | = Оі0 |x| в‰Ґ Оі0 . 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
pв€’1
1 1 1
exp [ (1 + 2p )] = a (a в‰Ў const > 0) (3.3)
y j+1 2 y
j=0

satisп¬Ѓes the inequality za в‰¤ Оґp (a), where

if a в‰¤ pe,
pe/a
Оґp (a) := . (3.4)
[ln (a/p)]в€’1/2p if a > pe

Proof: Assume that
pe в‰Ґ a. (3.5)
Since the function
pв€’1
1 1 1
f (y) в‰Ў exp [ (1 + 2p )]
y j+1 2 y
j=0

is nonincreasing and f (1) = pe, we have za в‰Ґ 1. But due to (3.3),
pв€’1
в€’j в€’2p
za exp [(1 + za )/2] в‰¤ 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 в‰¤ 1. But
pв€’1
xj+1 в‰¤ pxp в‰¤ p exp [xp в€’ 1] в‰¤ p exp [(x2p + 1)/2 в€’ 1]
j=0

= p exp [x2p /2 в€’ 1/2] (x в‰Ґ 1).
So
pв€’1
1 1 1 1
exp [ (1 + 2p )] в‰¤ p exp [ 2p ] (y в‰¤ 1).
f (y) =
y j+1 2 y y
j=0
8.4. Spectral Variations 129

But za в‰¤ 1 under (3.6). We thus have
1
a = f (za ) в‰¤ p exp [ 2p ].
za
Or
2p
za в‰¤ 1/ln (a/p).
This п¬Ѓnishes the proof. 2

8.4 Spectral Variations
Deп¬Ѓnition 8.4.1 Let A and B be linear operators in H. Then the quantity

inf |Вµ в€’ О»|
svA (B) := sup
Вµв€€Пѓ(B) О»в€€Пѓ(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 Hausdorп¬Ђ 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 в‰Ў A1 в€’ A2 < в€ћ.

RО» (A1 ) в‰¤ F (ПЃв€’1 (A1 , О»)) (О» в€€ Пѓ(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 (в€ћ) = в€ћ. Then

svA1 (A2 ) в‰¤ 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 в‰¤ qF (ПЃв€’1 (A, Вµ)) for all Вµ в€€ Пѓ(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 ПЃ(A, Вµ) в‰¤ 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) в‰¤ zY (V, B)

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

ОЅ0 в€’1
Оёk |V |k
Y
1 = B JY (ОЅ0 , V, z) в‰Ў B k+1
z
k=0

and
ОЅ0 = sup ni (V RО» (D)).
О»в€€Пѓ(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)
в‰¤ zp (V, B), where zp (V, B) is the extreme right-hand (positive) root of the
equation
pв€’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) в‰¤ Оґp (V, B), where

N2p (V )
Оґp (V, B) := N2p (V )Оґp ( )
B

and Оґp (.) is deп¬Ѓned by (3.4). We thus have derived

Corollary 8.4.5 Under the hypothesis of Theorem 8.4.4, svD (A + B) в‰¤
Оґp (V, B).

8.5 Perturbations of Compact Operators
First assume that
A в€€ 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) в‰¤ z1 (A, B),
Лњ
where z1 (A, B) is the extreme right-hand (positive) root of the equation
Лњ

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
Лњ
Applying Lemma 8.3.2 to this equation, we get z1 (A, B) в‰¤ в€†1 (A, B), where
Лњ
Лњ
в€†1 (A, B) := g(A)Оґ1 (g(A)/ B )

where Оґ1 is deп¬Ѓned by (3.4) with p = 1. So

if g(A) в‰¤ e B ,
eB
Лњ
в€†1 (A, B) := . (5.3)
g(A) [ln (g(A)/ B )]в€’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 Вµ в€€ Пѓ(A +
Лњ
B), there is a Вµ0 в€€ Пѓ(A), such that |Вµ в€’ Вµ0 | в‰¤ Оґ1 (A, B). In particular,
Лњ
rs (A + B) в‰¤ rs (A) + в€†1 (A, B), (5.4)
Лњ Лњ
rl (A + B) в‰Ґ max{rl (A) в€’ в€†1 (A, B), 0} and О±(A + B) в‰¤ О±(A) + в€†1 (A, B).
(5.5)

Remark 8.5.3 According to Lemma 6.5.2, in Theorem 8.5.1 and its corol-
в€љ
lary, one can replace g(A) by 2N2 (AI ).

Now let
A в€€ 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) в‰¤ yp (A, B), where yp (A, B) is the extreme right-hand
Лњ Лњ
(positive) root of the equation
pв€’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) в‰¤ в€†p (A, B), where
Лњ

в€†p (A, B) := 2N2p (A)Оґp (2N2p (A)/ B ).

Recall that Оґp (.) is deп¬Ѓned by (3.4). So

if 2N2p (A) в‰¤ ep B ,
pe B
в€†p (A, B) := .
2N2p (A) [ln (2N2p (A)/p B )]в€’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 Вµ в€€ Пѓ(A +
B), there is a Вµ0 в€€ Пѓ(A), such that |Вµ в€’ Вµ0 | в‰¤ в€†p (A, B). In particular,
Лњ
relations (5.4) and (5.5) hold with в€†p (A, B) instead of в€†1 (A, B).

Remark 8.5.6 According to Theorem 7.9.1, in Theorem 8.5.4 and its corol-
Лњ
lary one can replace 2N2p (A) by ОІp N2p (AI ).

8.6 Perturbations of Operators with Compact
Hermitian Components
First, let Dom (A) = Dom (Aв€— ) and

A в€’ Aв€— в€€ 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) в‰¤ x1 (A, B), where x1 (A, B) is the extreme right-hand
(positive) root of the equation
в€ћk
gI (A)
в€љ
1= B . (6.2)
k!z k+1
k=0
в€љ
Recall that gI (A) в‰¤ 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 П„1 (A, B), where

П„1 (A, B) = gI (A)Оґ1 (gI (A)/ B ).
8.6. Operators with Compact Components 133

So, according to (3.4),
if gI (A) в‰¤ e B ,
eB
П„1 (A, B) = .
gI (A) [ln (gI (A)/ B )]в€’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 Вµ в€€ Пѓ(A + B), there is a Вµ0 в€€ Пѓ(A), such that |Вµ в€’ Вµ0 |
в‰¤ П„1 (A, B). In particular,
rs (A + B) в‰¤ rs (A) + П„1 (A, B),
rl (A + B) в‰Ґ max{rl (A) в€’ П„1 (A, B), 0} and О±(A + B) в‰¤ О±(A) + П„1 (A, B).
Now let
A в€’ Aв€— в€€ 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) в‰¤ xp (A, B), where xp (A, B) is the extreme right-hand
Лњ Лњ
(positive) root of the equation
pв€’1 в€ћ pk+m
wp (A)
в€љ
1= B . (6.5)
k!z pk+m+1
m=0 k=0

where
Лњ
wp (A) = ОІp N2p (AI ).
According to Theorem 7.9.1, one can replace (6.5) by the equation
pв€’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)Оґp (wp (A)/ B ),
where Оґp (a) is deп¬Ѓned by (3.4). That is,
if wp (A) в‰¤ ep B
pe B
mp (A, B) = .
wp (A) [ln (wp (A)/p B )]в€’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 Вµ в€€
Пѓ(A + B), there is a Вµ0 в€€ Пѓ(A), such that |Вµ в€’ Вµ0 | в‰¤ mp (A, B). In particular,
rs (A + B) в‰¤ rs (A) + mp (A, B),
rl (A + B) в‰Ґ max{rl (A) в€’ П„p (A, B), 0} and О±(A + B) в‰¤ О±(A) + П„p (A, B).
Лњ Лњ
8. Bounded Perturbations of P -Triangular Operators
134

8.7 Notes
The material of this chapter is based on the papers (GilвЂ™, 2002) and (GilвЂ™,
2003). About the well-known perturbations results see (Kato, 1966),
(BaumgВґrtel, 1985) and references therein.
a

References
 BaumgВґrtel, H. (1985). Analytic Perturbation Theory for Matri-
a
ces and Operators. Operator Theory, Advances and Appl., 52.
BirkhВґuser Verlag, Basel, Boston, Stuttgart.
a
 GilвЂ™, M. I. (2002). Invertibility and spectrum localization of non-
 GilвЂ™, M. I. (2003). Inner bounds for spectra of linear operators,
Proceedings of the American Mathematical Society , (to appear)

 Kato, T. (1966). Perturbation Theory for Linear Operators,
Springer-Verlag. New York.

 Ostrowski, A. M. (1973). Solution of Equations in Euclidean and
Banach spaces. Academic Press, New York - London.
9. Spectrum Localization
Operators

In the present chapter we consider operators of the form A = D + V+ + Vв€’ ,
where D is a normal operator and VВ± are Volterra (quasinilpotent compact)
operators. Numerous integral, integro-diп¬Ђerential operators and inп¬Ѓnite 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 в‰¤ t в‰¤ b; в€’в€ћ в‰¤ a < b в‰¤ в€ћ) be a maximal resolution of the
identity in a separable Hilbert space H (see Section 7.2). Let

A = D + V + + Vв€’ , (1.1)

where D is a normal operator and VВ± are Volterra operators satisfying the
conditions
P (t)V+ P (t) = V+ P (t); P (t)Vв€’ P (t) = P (t)Vв€’ ,
P (t)Dh = DP (t)h (a в‰¤ t в‰¤ b; h в€€ Dom (D)). (1.2)
As above, Y is a norm ideal of compact operators in H, which is complete in
an auxiliary norm | В· |Y . In addition, there are positive numbers Оёk (k в€€ N)
1/k
with Оёk в†’ 0 (k в†’ в€ћ), for which, for arbitrary Volterra operators V в€€ Y ,
V k в‰¤ Оёk |V |k (k = 1, 2, ...). It is assumed that VВ± в€€ Y . That is,
Y

VВ± в‰¤ Оёk |VВ± |k (k = 1, 2, ...).
k
(1.3)
Y

M.I. GilвЂ™: LNM 1830, pp. 135вЂ“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 |Пѓ(D)| > 0. (1.4)

Since D is normal and VВ± are Volterra operators, due to Lemma 7.3.3 con-
ditions (1.2) are enough to guarantee that

WВ± := Dв€’1 VВ±

are also Volterra operators. Under (1.2)-(1.4), put
ni(WВ± )в€’1
Оёk |WВ± |k ,
JY (WВ± ) в‰Ў (1.5)
Y
k=0

where ni(V ) again denotes the вЂњnilpotency indexвЂќ 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
О¶Y (A) := max { в€’ W+ , в€’ Wв€’ } > 0. (1.6)
JY (Wв€’ ) JY (W+ )
Then A is boundedly invertible, and
1
Aв€’1 в‰¤ . (1.7)
rl (D)О¶Y (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 (WВ± ) by
в€ћ
Оёk |WВ± |k .
IY (WВ± ) := Y
k=0

Consider operators, whose oп¬Ђ-diagonal parts belong to the Neumann-Schatten
ideal C2p with some integer p в‰Ґ 1:

VВ± в€€ C2p . (1.8)

Again put
1
(p)
Оёj = ,
[j/p]!
where [x] means the integer part of a real number x. For a Volterra operator
V в€€ C2p denote
ni(V )в€’1
(p)k
Jp (V ) = Оёk 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
О¶2p (A) в‰Ў max{ в€’ W+ , в€’ Wв€’ } > 0. (1.9)
Jp (Wв€’ ) Jp (W+ )
Then operator A represented by (1.1) is boundedly invertible. Moreover,
1
Aв€’1 в‰¤ .
О¶2p (A)rl (D)
It is simple to see that, in this corollary, one can replace Jp (WВ± ) by I2p (WВ± ),
where
pв€’1 в€ћ j+pk
N2p (WВ± )
в€љ
I2p (WВ± ) := .
k!
j=0 k=0
Moreover, put
pв€’1
j 2p
П€p (WВ± ) := a0 N2p (WВ± ) exp [b0 N2p (WВ± )].
j=0

Besides, the constants a0 , b0 can be taken as in the relations
в€љ
c c
a0 = and b0 = for any c > 1, in particular, a0 = 2 and b0 = 1,
cв€’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
Лњ в€’ W+ , в€’ Wв€’ } > 0.
О¶p (A) := max{
П€p (Wв€’ ) П€p (W+ )
Then operator A represented by (1.1) is boundedly invertible. Moreover,
1
Aв€’1 в‰¤ .
Лњ
О¶p (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
Оё0 := (D + Vв€’ )в€’1 V+ < 1. (2.1)
Then operator A represented by (1.1) is boundedly invertible. Moreover,
(D + Vв€’ )в€’1
Aв€’1 в‰¤ . (2.2)
1 в€’ Оё0
138 9. Spectrum Localization

Proof: According to (1.1) we have

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

Thanks to (1.2) and Lemma 7.5.1,

Пѓ(D + VВ± ) = Пѓ(D). (2.4)

So according to (1.4), D + VВ± is invertible. Moreover, under condition (2.1),
the operator
I + (D + Vв€’ )в€’1 V+
is invertible and
в€ћ
в€’1 в€’1
((D + Vв€’ )в€’1 V+ )k в‰¤
в‰¤
(I + (D + Vв€’ ) V+ )
k=0

в€ћ
Оё0 = (1 в€’ Оё0 )в€’1 .
k

k=0

So due to (2.3)

Aв€’1 в‰¤ (I + (D + Vв€’ )в€’1 V+ )в€’1 (D + Vв€’ )в€’1 .

This proves the required result. 2

Proof of Theorem 9.1.1: Since VВ± в€€ Y ,

(D + Vв€’ )в€’1 V+ = (I + Dв€’1 Vв€’ )в€’1 Dв€’1 V+ = (I + Wв€’ )в€’1 W+ в‰¤

ni(Wв€’ )в€’1
в€ћ
k k
Wв€’ в‰¤
W+ Wв€’ = W+
k=0 k=0

ni(Wв€’ )в€’1
Оёk |Wв€’ |k = W+ JY (Wв€’ ).
W+ Y
k=0

Hence
Оё0 в‰¤ W+ JY (Wв€’ ).
But condition (1.6) implies that at least one of the following inequalities

W+ JY (Wв€’ ) < 1 (2.5)

or
Wв€’ 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, Dв€’1 = rl (D)в€’1 . Thus,
в€ћ
в€’1 в€’1 в€’1 в€’1 k
в‰¤D
(D + Vв€’ ) = (I + Wв€’ ) D Wв€’ =
k=0

ni(Wв€’ )в€’1 ni(Wв€’ )в€’1
в€’1 k в€’1
Оёk |Wв€’ |k = rl (D)в€’1 JY (Wв€’ ).
в‰¤ rl (D)
rl (D) Wв€’ Y
k=0 k=0

Thus, under (2.5), Lemma 9.2.1 yields the inequality
JY (Wв€’ ) 1
Aв€’1 в‰¤ = . (2.7)
0 в€’1
rl (D)(1 в€’ W+ JY (Wв€’ )) rl (D)(JY (Wв€’ ) в€’ W+ )
Interchanging Wв€’ and W+ , under condition (2.6), we get
1
Aв€’1 в‰¤ .
0 в€’1
rl (D)(JY (W+ ) в€’ Wв€’ )
This relation and (2.7) yield the required result. 2

Proof of Theorem 9.1.3: Due to Theorem 6.7.3,

(D + VВ± )в€’1 = (I + WВ± )в€’1 Dв€’1 в‰¤ П€p (WВ± )ПЃв€’1 (D).
l

(D + Vв€’ )в€’1 V+ = (I + Wв€’ )в€’1 W+ в‰¤ П€p (Wв€’ ) W+

and
(D + V+ )в€’1 Vв€’ в‰¤ П€p (W+ ) Wв€’ .
Now the required result is due to Lemma 9.2.1. 2

9.3 Resolvents of Quasinormal Operators
For a V в€€ Y , denote
mв€’1
z в€’1в€’k Оёk |V |k (z > 0).
JY (V, m, z) := Y
k=0

Due to Lemma 7.3.4, (D в€’ О»I)в€’1 VВ± is a quasinilpotent operator for any
О» в€€ Пѓ(D). Put
ОЅВ± (О») в‰Ў ni((D в€’ О»I)в€’1 VВ± ).
Everywhere below we can replace ОЅВ± (О») by в€ћ.
Again, RО» (A) is the resolvent and ПЃ(D, О») = inf zв€€Пѓ(D) |s в€’ z|.
140 9. Spectrum Localization

Lemma 9.3.1 Under conditions (1.2), (1.3), for a О» в€€ Пѓ(D), let

1
О¶(A, О») в‰Ў max{ в€’ V+ ,
JY (Vв€’ , ОЅв€’ (О»), ПЃ(D, О»))

1
в€’ Vв€’ } > 0. (3.1)
JY (V+ , ОЅ+ (О»), ПЃ(D, О»))
Then О» is a regular point of operator A represented by (1.1). Moreover,

1
RО» (A) в‰¤ . (3.2)
О¶(A, О»)ПЃ(D, О»)

Since, D is a normal operator, (D в€’ О»I)в€’1 = ПЃв€’1 (О», D). Thus,
Proof:

|(D в€’ О»I)в€’1 VВ± |Y в‰¤ (D в€’ О»I)в€’1 |VВ± |Y =

ПЃв€’1 (О», D)|VВ± |Y .
Hence,
ОЅв€’ (О»)в€’1
(D в€’ О»I)в€’1 V+ Оёk |(D в€’ О»I)в€’1 Vв€’ |k в‰¤
Y
k=0

ОЅв€’ (О»)в€’1
Оёk ПЃв€’1в€’k (О», D)|Vв€’ |k = V+ JY (Vв€’ , ОЅв€’ (О»), ПЃ(D, О»)).
V+ Y
k=0

Similarly,
ОЅ+ (О»)в€’1
в€’1
Оёk |(D в€’ О»I)в€’1 V+ |k в‰¤ Vв€’ JY (V+ , ОЅ+ (О»), ПЃ(D, О»)).
(D в€’ О»I) Vв€’ Y
k=0

Now Theorem 9.1.1 with

A в€’ О»I = D + V+ + Vв€’ в€’ О»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 Вµ в€€ Пѓ(A),
there is a Вµ0 в€€ Пѓ(D), such that, either Вµ = Вµ0 , or both the inequalities

V+ JY (Vв€’ , ОЅв€’ (Вµ), |Вµ в€’ Вµ0 |) в‰Ґ 1 and

Vв€’ JY (V+ , ОЅ+ (Вµ), |Вµ в€’ Вµ0 |) в‰Ґ 1 (3.3)
are true.
9.4. Upper Bounds for Spectra 141

This result is exact in the following sense: if either Vв€’ = 0, or (and) V+ = 0,
then due to the latter corollary,

Пѓ(A) = Пѓ(D). (3.4)

Now let condition (1.8) hold. Put
(p)
mв€’1 k
Оёk N2p (VВ± )
Лњ
Jp (VВ± , 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 О» в€€ Пѓ(D), let

1
О¶2p (О», A) в‰Ў max{ в€’ V+ ,
Лњ
Jp (Vв€’ , ОЅв€’ (О»), ПЃ(D, О»))

1
в€’ Vв€’ } > 0.
Лњ
Jp (V+ , ОЅ+ (О»), ПЃ(D, О»))
Then О» is a regular point of operator A, represented by (1.1). Moreover,

1
RО» (A) в‰¤ .
ПЃ(D, О»)О¶2p (О», A)

Furthermore, thanks to Theorem 9.1.3 with Aв€’О» instead of A, we can replace
Лњ
Jp by the function

pв€’1 j 2p
N2p (VВ± ) b0 N2p (VВ± )
П€p (VВ± , 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 О» в€€ Пѓ(D), let

1
Лњ
О¶2p (О», A) в‰Ў max{ в€’ V+ ,
П€p (Vв€’ , ПЃ(D, О»))

1
в€’ Vв€’ } > 0.
П€p (V+ , ПЃ(D, О»))
Then О» is a regular point of operator A, represented by (1.1). Moreover,

1
RО» (A) в‰¤ .
ПЃ(D, О»)О¶2p (О», 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

П„ (A) := min{ Vв€’ , V+ }, (4.1)

if V+ в‰Ґ Vв€’
V+
Лњ
V := (4.2)
Vв€’ if Vв€’ > V+
and
ni ((D в€’ О»I)в€’1 V ).
Лњ
ОЅ0 = sup
Лњ
О»в€€Пѓ(D)

In the sequel one can replace ОЅ0 by в€ћ.
Лњ
Лњ
Theorem 9.4.1 Under conditions (1.1), (1.2), let V в€€ Y and

V k в‰¤ Оёk |V |k (k = 1, 2, ...).
Лњ ЛњY (4.3)

Then the equation
ЛњЛњ
П„ (A)JY (V , ОЅ0 , z) = 1 (4.4)
has a unique positive root zY (A). Moreover, svD (A) в‰¤ zY (A).

Proof: Due to Corollary 9.3.2,
ЛњЛњ
П„ (A)JY (V , ОЅ0 , ПЃ(D, Вµ)) в‰Ґ 1

for any Вµ в€€ Пѓ(A). Comparing this inequality with (4.4), we have ПЃ(D, Вµ) в‰¤
zY (A). This inequality proves the theorem. 2

Lemma 9.4.2 Under the conditions (1.1), (1.2) and
Лњ
V в€€ C2p (p = 1, 2, ...), (4.5)

the equation
pв€’1 jЛњ 2p Лњ
N2p (V ) N2p (V )
1
П„ (A) exp [ (1 + )] = 1. (4.6)
z j+1 z 2p
2
j=0

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

svD (A) в‰¤ z2p (A). (4.7)

Proof: Due to Corollary 9.3.4,
pв€’1 jЛњ 2p Лњ
N2p (V ) N2p (V )
1
)] в‰Ґ 1
П„ (A) exp [ (1 + 2p
ПЃj+1 (D, Вµ) 2 ПЃ (D, Вµ)
j=0
9.5. Inner Bounds for Spectra 143

for any Вµ в€€ Пѓ(A). Comparing this inequality with (4.6), we get ПЃ(D, Вµ) в‰¤
z2p (A). This inequality proves the required result. 2

Лњ
To estimate z2p (A), substitute z = xN2p (V ) in (4.6) and use Lemma
Лњ Лњ
8.3.2. Then z2p (A) в‰¤ П†p (A) = N2p (V )Оґp (a), where a = N2p (V )/П„ (A) and

if a в‰¤ pe
pe/a
Оґp (a) := . (4.8)
[ln (a/p)]в€’1/2p if a > pe
That is,
Лњ
if N2p (V ) в‰¤ П„ (A)pe
peП„ (A)
П†p (A) := .
N2p (V )[ln (N2p (V )/pП„ (A))]в€’1/2p
Лњ Лњ Лњ
if N2p (V ) > П„ (A)pe
(4.9)
Now the previous lemma yields
Corollary 9.4.3 Under conditions (1.1), (1.2) and (4.5), svD (A) в‰¤ П†p (A).
In particular,
rs (A) в‰¤ rs (D) + П†p (A),
provided D is bounded.
Лњ
In the case V в€€ C2 we have
if a в‰¤ e
e/a
Оґ1 (a) := (4.10)
[ln a]в€’1/2 if a > e
and
Лњ
if N2 (V ) в‰¤ П„ (A)e
eП„ (A)
П†1 (A) := . (4.11)
N2 (V ) [ln (N2 (V )/П„ (A))]в€’1/2
Лњ Лњ Лњ
if N2 (V ) > П„ (A)e

Now (4.7) implies
svD (A) в‰¤ z2 (A) в‰¤ П†1 (A). (4.12)
Remark 9.4.4 Everywhere below VВ± can be replaced by their upper bounds,
since the right roots of equations (4.4) and (4.6) increase, when the coeп¬ѓ-
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 в‰Ґ 0) with the properties

F (0) = 0, F (в€ћ) = в€ћ (5.1)

such that the inequality

(О»I в€’ A)в€’1 в‰¤ F (ПЃв€’1 (A, О»)) (5.2)
144 9. Spectrum Localization

holds, where ПЃ(A, О») is the distance between Пѓ(A) and a regular point О» в€€ C
of A. Recall that П„ (A) := min{ Vв€’ , V+ } and denote by y(П„, F ) the
unique positive root of the equation

П„ (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 deп¬Ѓned by (1.1) and conditions (1.2) and (5.2)
hold. Then
svA (D) в‰¤ y(П„, F ). (5.4)

The proof of this theorem is presented below. Recall that

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

Corollary 9.5.2 Under the hypothesis of Theorem 9.5.1, the following in-
equalities are true:

rs (A) в‰Ґ max{0, rs (D) в€’ y(П„, F )} if D is bounded, (5.5)

rl (A) в‰¤ rl (D) + y(П„, F ) and (5.6)
О±(A) в‰Ґ О±(D) в€’ y(П„, F ) if О±(D) < в€ћ. (5.7)

Indeed, take Вµ in such a way that |Вµ| = rs (D). Then due to (5.4), there is
Вµ0 в€€ Пѓ(A), such that |Вµ0 | в‰Ґ rs (D) в€’ y(П„, F ). Hence, (5.5) follows. Similarly,
inequality (5.6) can be proved. Furthermore, take Вµ in such a way that
Re Вµ = О±(D). Due to (1.6) for some Вµ0 в€€ Пѓ(A), |Re Вµ0 в€’ О±(D)| в‰¤ y(П„, F ).
So, either Re Вµ0 в‰Ґ О±(D), or Re Вµ0 в‰Ґ О±(D) в€’ y(П„, 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) Пѓ(B+ ) = Пѓ(D). Due to to Lemma 8.4.2, for any Вµ в€€ Пѓ(D), there is
Вµ0 в€€ Пѓ(A), such that
|Вµ0 в€’ Вµ| в‰¤ zв€’ , (5.8)
where zв€’ is the unique positive root of the equation

Vв€’ F (1/z) = 1 (z в‰Ґ 0).

Now, take Bв€’ = D + Vв€’ . Then due to relation (2.4), we get Пѓ(Bв€’ ) = Пѓ(D).
Similarly to (5.8), we have that for any Вµ в€€ Пѓ(D), there is Вµ0 в€€ Пѓ(A), such
that
|Вµ0 в€’ Вµ| в‰¤ z+ , (5.9)
where z+ is the unique positive root of the equation V+ F (1/z) = 1, since
A в€’ Bв€’ = 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 в€€ C2 . (6.1)
Recall that g(A) is deп¬Ѓned in Section в€љ 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, П„ ) the unique non-negative root of the equa-
Лњ
tion
1 g 2 (A)
П„ (A)
exp [ + ] = 1, (6.2)
2z 2
z 2
where П„ (A) is deп¬Ѓned 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) в‰¤ y2 (A, П„ )
Лњ
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)
Лњ
y2 (A, П„ ) в‰¤ в€†2 (A) := g(A)Оґ1 (
Лњ ),
П„ (A)

where Оґ1 (a) is deп¬Ѓned by (4.10). That is,

if g(A) в‰¤ eП„ (A)
eП„ (A)
Лњ
в€†2 (A) := . (6.3)
g(A)[ln (g(A)/П„ (A))]в€’1/2 if g(A) > eП„ (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

Лњ
svD (A) в‰¤ П†1 (A) and svA (D) в‰¤ в€†2 (A).

In particular,

Лњ
max{0, rs (D) в€’ в€†2 (A)} в‰¤ rs (A) в‰¤ rs (D) + П†1 (A),

Лњ
max{0, rl (D) в€’ П†1 (A)} в‰¤ rl (A) в‰¤ rl (D) + в€†2 (A)
Лњ
and О±(D) в€’ в€†2 (A) в‰¤ О±(A) в‰¤ О±(D) + П†1 (A).
146 9. Spectrum Localization

9.7 Von Neumann-Schatten Operators
Assume that
A в€€ C2p for an integer p > 1 (7.1)
and denote by yp (A, П„ ) the unique non-negative root of the equation
pв€’1
(2N2p (A))m 1 (2N2p (A))2p
П„ (A) exp [ + ] = 1, (7.2)
z m+1 2z 2p
2
m=0

where П„ (A) is deп¬Ѓned 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) в‰¤ yp (A, П„ ) 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, П„ ) в‰¤ в€†p (A) := N2p (A)Оґp ( ),
П„ (A)

where Оґp (a) is deп¬Ѓned by (4.8). That is,

if 2N2p(A) в‰¤ p e П„ (A)
p e П„ (A)
в€†p (A) := .
2N2p (A)[ln (2N2p (A)/П„ (A))]в€’1/2p if 2N2p (A) > p e П„ (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) в‰¤ П†p (A) and svA (D) в‰¤ в€†p (A).

In particular,

max{0, rs (D) в€’ в€†p (A)} в‰¤ rs (A) в‰¤ rs (D) + П†p (A),

max{0, rl (D) в€’ П†p (A)} в‰¤ rl (A) в‰¤ rl (D) + в€†p (A)
and О±(D) в€’ в€†p (A) в‰¤ О±(A) в‰¤ О±(D) + П†p (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 (Aв€— ) and A has the
Hilbert-Schmidt imaginary component AI в‰Ў (A в€’ Aв€— )/2i:

N2 (AI ) = T race (AI )2 < в€ћ.
2
(8.1)

Recall that gI (A) is deп¬Ѓned in Section 7.7. Everywhere below on can replace
в€љ
gI (A) by 2N2 (AI ).
Under (8.1), denote by yH (A, П„ ) the unique non-negative root of the equa-
tion
2
П„ (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) в‰¤ yH (A, П„ )
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, П„ ) в‰¤ в€†H (A) := gI (A)Оґ1 ( ),
П„ (A)

where Оґ1 (a) is deп¬Ѓned by (4.10). That is,

if gI (A) в‰¤ eП„ (A)
e П„ (A)
в€†H (A) := .
gI (A)[ln (gI (A)/П„ (A))]в€’1/2 if gI (A) > e П„ (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) в‰¤ П†1 (A) and svA (D) в‰¤ в€†H (A).

In particular,

max{0, rs (D) в€’ в€†H (A)} в‰¤ rs (A) в‰¤ rs (D) + П†1 (A),

max{0, rl (D) в€’ П†1 (A)} в‰¤ rl (A) в‰¤ rl (D) + в€†H (A)
and О±(D) в€’ в€†H (A) в‰¤ О±(A) в‰¤ О±(D) + П†1 (A).
148 9. Spectrum Localization

9.9 Operators with Neumann-Schatten
Hermitian Components
In this section it is assumed that the Hermitian component AI = (Aв€’Aв€— )/2i
belongs to the Neumann-Schatten ideal C2p with some integer p > 1:

Np (AI ) = [T race A2p ]1/2p < в€ћ. (9.1)
 << стр. 6(всего 11)СОДЕРЖАНИЕ >>