˜

Recall that βp is de¬ned in Section 7.9. Set

˜

wp (A) := βp N2p (AI ).

Let xp (A, „ ) be the unique positive root of the equation

p’1 m 2p

wp (A) 1 wp (A)

„ (A) exp [ + ] = 1, (9.2)

z m+1 2z 2p

2

m=0

where „ (A) is de¬ned by (4.1).

Theorem 9.9.1 Let conditions (1.1), (1.2) and (9.1) hold. Then the rela-

tions (4.7) and

svA (D) ¤ xp (A, „ )

are valid.

Proof: The required result is due to Theorems 7.9.1 and 9.5.1, and Corol-

lary 9.4.3. 2

Substitute z = wp (A)x in (9.2) and apply Lemma 8.3.2. Then

wp (A)

xp (A, „ ) ¤ mp (A) := wp (A)δp ( ), (9.3)

„ (A)

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

if wp (A) ¤ pe„ (A)

e„ (A)

mp (A) := . (9.4)

wp (A)[ln (wp (A)/p„ (A))]’1/2p if wp (A) > pe„ (A)

Thus Theorem 9.9.1 and Corollary 9.4.3 imply

Corollary 9.9.2 Let relations (1.1), (1.2) and (9.1) hold. Then

svD (A) ¤ φp (A) and svA (D) ¤ mp (A).

In particular,

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

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

and ±(D) ’ mp (A) ¤ ±(A) ¤ ±(D) + φp (A).

9.10. Notes 149

9.10 Notes

The present chapter is based on the papers (Gil™, 2002) and (Gil™, 2003). The-

orem 9.1.1 supplements the well-known results on the invertibility of linear

operators, cf. (Harte, 1988).

As it was above mentioned, a lot of papers and books have been devoted

to the spectrum of linear operators. Mainly, the asymptotic distributions of

the eigenvalues are considered, cf. the books (Pietsch, 1987), (K¨nig, 1986),

o

and references therein. But the bounds and invertibility conditions have been

investigated considerably less than the asymptotic distributions. At the same

time, in particular, Theorems 9.6.1, 9.7.1 and 9.8.1 and their corollaries give

us explicit bounds for the spectrum of the considered operators.

References

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

joint operators, Adv. Appl. Mathematics, 28, 40-58.

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

ings of the American Mathematical Society (to appear).

[3] Harte R. (1988). Invertibility and Singularity for Bounded Linear Oper-

ators. Marcel Dekker, Inc. New York.

[4] K¨nig, H. (1986). Eigenvalue Distribution of Compact Operators,

o

Birkh¨user Verlag, Basel- Boston-Stuttgart.

a

[5] Pietsch, A. (1987). Eigenvalues and s-Numbers, Cambridge University

Press, Cambridge.

10. Multiplicative

Representations of

Resolvents

In the present chapter we introduce the notion of the multiplicative opera-

tor integral in a separable Hilbert space H. By virtue of the multiplicative

operator integral, we derive spectral representations for resolvents of various

classes of P -triangular operators. These representations are generalizations

of the classical spectral representation for the resolvent of a normal operator.

If the maximal resolution of the identity is discrete, then the multiplicative

integral is an operator product.

10.1 Operators with Finite Chains

of Invariant Projectors

Recall that I is the unit operator in H.

Lemma 10.1.1 Let P be a projector onto an invariant subspace of a bounded

linear operator A in H, P = 0 and P = I. Then

»R» (A) = ’(I ’ AP R» (A)P )(I ’ A(I ’ P )R» (A)(I ’ P )) (» ∈ σ(A)).

Denote E = I ’ P . Since

Proof:

A = (E + P )A(E + P ) and EAP = 0,

we have

A = P AE + P AP + EAE. (1.1)

M.I. Gil™: LNM 1830, pp. 151“161, 2003.

c Springer-Verlag Berlin Heidelberg 2003

152 10. Multiplicative Representations of Resolvents

Let us check the equality

R» (A) = P R» (A)P ’ P R» (A)P AER» (A)E + ER» (A)E. (1.2)

In fact, multiplying this equality from the left by A ’ I» and taking into

account the equalities (1.1), AP = P AP and P E = 0, we obtain the relation

((A ’ I»)P + (A ’ I»)E + P AE)(P R» (A)P ’

P R» (A)P AER» (A)E + ER» (A)E) =

P ’ P AER» (A)E + E + P AER» (A)E = I.

Similarly, multiplying (1.2) by A ’ I» from the right and taking into account

(1.1), we obtain I. Therefore, (1.2) is correct. Due to (1.2)

I ’ AR» (A) = (I ’ AR» (A)P )(I ’ AER» (A)E). (1.3)

But

I ’ AR» (A) = ’»R» (A). (1.4)

We thus arrow at the result. 2

Let Pk (k = 1, . . . , n) be a chain of projectors onto the invariant subspaces

of a bounded linear operator A. That is,

Pk APk = APk (k = 1, ..., n) (1.5)

and

0 = P0 H ‚ P1 H ‚ ... ‚ Pn’1 H ‚ Pn H = H. (1.6)

For bounded linear operators X1 , X2 , ..., Xn again put

’

Xk := X1 X2 ...Xn .

1¤k¤n

I.e. the arrow over the symbol of the product means that the indexes of the

co-factors increase from left to right.

Lemma 10.1.2 Let a bounded linear operator A have properties (1.5) and

(1.6). Then

’

»R» (A) = ’ (I ’ A∆Pk R» (A)∆Pk ) (» ∈ σ(A)),

1¤k¤n

where ∆Pk = Pk ’ Pk’1 (1 ¤ k ¤ n).

10.1. Operators with Finite Chains 153

Proof: Due to the previous lemma

»R» (A) = ’(I ’ APn’1 R» (A)Pn’1 )(I ’ A(I ’ Pn’1 )R» (A)(I ’ Pn’1 )).

But I ’ Pn’1 = ∆Pn . So equality (1.4) implies.

I ’ AR» (A) = (I ’ APn’1 R» (A)Pn’1 )(I ’ A∆Pn R» (A)∆Pn ).

Applying this relation to APn’1 , we get

I ’ APn’1 R» (A)Pn’1 =

(I ’ APn’2 R» (A)Pn’2 )(I ’ A∆Pn’1 R» (A)∆Pn’1 ).

Consequently,

I ’ AR» (A) = (I ’ APn’2 R» (A)Pn’2 )(I ’ A∆Pn’1 R» (A)∆Pn’1 )(I’

A∆Pn R» (A)∆Pn ).

Continuing this process, we arrive at the required result. 2.

Let us consider an operator of the form

n

A= ak ∆Pk + V (1.7)

k=1

where ak are some numbers, {Pk } is a chain of projectors de¬ned by (1.6)

and V is a nilpotent operator with the property

Pk’1 V Pk = V Pk (k = 1, ..., n). (1.8)

Lemma 10.1.3 Under conditions (1.7) and (1.8), the relation

’

A∆Pk

»R» (A) = ’ (I + )

» ’ ak

1¤k¤n

is valid for any » = ak (k = 1, ..., n).

Proof: It is not hard to check that

∆Pk

∆Pk R» (A)∆Pk = .

ak ’ »

Now the required result is due to the previous lemma. 2

154 10. Multiplicative Representations of Resolvents

10.2 Complete Compact Operators

Let A be a compact operator in H whose system of all the root vectors is

complete in H. Then there is an orthogonal normed basis (Schur™s basis)

{ek }, such that

k

Aek = ajk ej , (2.1)

j=1

cf. (Gohberg and Krein, 1969, Chapter 5). Moreover akk = »k (A) are the

eigenvalues of A with their multiplicities. Introduce the orthogonal projectors

k

Pk = (., ej )ej (k = 1, 2, ...).

j=1

If there exists a limit in the operator norm of the products

’

(I + Xk ) ≡ (I + X1 )(I + X2 )...(I + Xn ).

1¤k¤n

as n ’ ∞, then we denote this limit by

’

(I + Xk ).

1¤k¤∞

That is,

’

A∆Pk

(I + )

» ’ »k (A)

1¤k¤∞

is a limit in the operator norm of the sequence the operators

’

A∆Pk

Πn (») := (I + ) :=

» ’ »k (A)

1¤k¤n

A∆P1 A∆P2 A∆Pn

(I + )(I + )...(I + )

» ’ »1 (A) » ’ »2 (A) » ’ »n (A)

for » = »k (A). Here ∆Pk = Pk ’ Pk’1 , k = 1, 2, ... ; P0 = 0, again.

Lemma 10.2.1 Suppose that the system of all the root vectors of a compact

linear operator A is complete in H. Then

’

A∆Pk

»R» (A) = ’ (I + ) (» ∈ σ(A)). (2.2)

» ’ »k (A)

1¤k¤∞

10.2. Complete Compact Operators 155

Proof: Let An = APn . Lemma 10.1.3 implies the equality

»R» (An ) = ’Πn (»). (2.3)

Since A is compact, An tends to A in the operator norm as n tends to ∞.

Besides,

(An ’ »I)’1 Pn ’ (A ’ »I)’1

in the operator norm for any regular » . We arrive at the result. 2

Let A be a normal compact operator. Then

∞

A= »k (A)∆Pk .

k=1

Hence, A∆Pk = »k (A)∆Pk . Since ∆Pk ∆Pj = 0 for j = k, Lemma 10.2.1

gives us the equality

∞

A∆Pk

’»R» (A) = I + (I + ).

» ’ »k (A)

k=1

But

∞

I= ∆Pk .

k=1

Thus,

∞

[1 + (» ’ »k (A))’1 »k (A)]∆Pk =

»R» (A) = ’

k=1

∞

»∆Pk (» ’ »k (A))’1 .

’

k=1

Or

∞

∆Pk

R» (A) = .

»k (A) ’ »

k=1

Thus, Lemma 10.2.1 generalizes the well-known spectral representation for

the resolvent of a normal completely continuous operator.

Furthermore, according to (2.1), the nilpotent part V of A can be de¬ned

as

k’1

V ek = ajk ej . (2.4)

j=1

Therefore, Pk’1 V ∆Pk = Pk’1 A∆Pk = V ∆Pk and

A∆Pk = Pk A∆Pk = ∆Pk A∆Pk + Pk’1 A∆Pk = »k (A)∆Pk + V ∆Pk .

Now Lemma 10.2.1 implies the relation

’

(»k (A) + V )∆Pk

»R» (A) = ’ (I + ) (» ∈ σ(A)). (2.5)

» ’ »k (A)

1¤k¤∞

156 10. Multiplicative Representations of Resolvents

10.3 The Second Representation

for Resolvents of Complete

Compact Operators

Let V be a Volterra operator, de¬ned by (2.4). Then due to (2.5)

’

’1

(I ’ V ) = (I + V ∆Pk ). (3.1)

2¤k¤∞

Furthermore, according to (2.1) A = D + V , where D is de¬ned by Dek =

»k (A)ek . Clearly,

(A ’ »I)’1 = (D + V ’ I»)’1 = (D ’ I»)’1 (I + B» )’1 , (3.2)

where B» = V (D ’ I»)’1 . Due to Lemma 7.3.4 B» is a Volterra operator.

Moreover, Pk’1 B» Pk = B» Pk . Thus relation (3.1) implies

’

’1

(I ’ B» ∆Pk ).

(I + B» ) =

2¤k¤∞

But

V ∆Pk

B» ∆Pk = .

»k ’ »

Therefore,

’

V ∆Pk

’1

(I + B» ) = (I + ).

» ’ »k (A)

2¤k¤∞

Now (3.2) yields

Theorem 10.3.1 Suppose that the system of all the root vectors of a compact

linear operator A is complete in H. Then

’

V ∆Pk

) (» ∈ σ(A)),

R» (A) = R» (D) (I +

» ’ »k (A)

2¤k¤∞

where V is the nilpotent part of A and

∞

∆Pk

R» (D) = .

»k (A) ’ »

k=1

10.4. Operators with Compact Inverse Ones 157

10.4 Operators with Compact Inverse Ones

Let a linear operator A in H have a compact inverse one A’1 . Let the system

of the root vectors of A’1 (and therefore of A) is complete in H. Then due

to (2.1), there is an orthogonal normed basis (Schur™s basis) {ek }, such that

k

’1

A ek = bjk ej (4.1)

j=1

with entries bjk . The nilpotent part V0 and diagonal one D0 of A’1 are

de¬ned by

k’1

V0 ek = bjk ej . (4.2)

j=1

and D0 ek = bkk ek = »k (A’1 )ek . As above, put

k

Pk = (., ej )ej (k = 1, 2, ...).

j=1

Theorem 10.4.1 Let operator A have the compact inverse one A’1 . Let the

system of the root vectors of A’1 is complete in H. Then

’

»(1 + »k (A)V0 )∆Pk

) ’ I (» ∈ σ(A)).

» R» (A) = (I +

»k (A) ’ »

1¤k¤∞

The product converges in the operator norm.

Proof: Thanks to Lemma 10.2.1,

(A ’ »I)’1 = A’1 (I ’ »A’1 )’1 =

’

»A’1 ∆Pk

’1

A (I + )

1 ’ »k (A’1 )»

1¤k¤∞

for any regular » of A. But D0 ∆Pk = »k (A’1 )∆Pk . Hence,

’

»(»k (A’1 ) + V0 )∆Pk

’1 ’1

(A ’ »I) =A (I + ).

1 ’ »k (A’1 )»

1¤k¤∞

Thus, we have derived the relation

’

»(1 + »k (A)V0 )∆Pk

’1 ’1

(A ’ »I) =A (I + ).

»k (A) ’ »

1¤k¤∞

Taking into account that A(A ’ »I)’1 = I + »(A ’ »I)’1 , we arrive at the

required result. 2

158 10. Multiplicative Representations of Resolvents

10.5 Multiplicative Integrals

Let F be a function de¬ned on a ¬nite real segment [a, b] whose values are

bounded linear operators in H. We de¬ne the right multiplicative integral as

the limit in the uniform operator topology of the sequence of the products

’

(n) (n) (n)

(1 + δF (tk )) := (1 + δF (t1 ))(I + δF (t2 ))...(I + δF (t(n) ))

n

1¤k¤n

(n) (n)

as maxk |tk ’ tk’1 | tends to zero. Here

(n) (n) (n)

δF (tk ) = F (tk ) ’ F (tk’1 ) for k = 1, ..., n

(n) (n) (n)

and a = t0 < t1 < ... < tn = b. The right multiplicative integral we

denote by

’

(1 + dF (t)).

[a,b]

In particular, let P be an orthogonal resolution of the identity de¬ned on

[a, b], φ be a function integrable in the Riemann-Stieljes with respect to P ,

and A be a compact linear operator. Then the right multiplicative integral

’

(I + φ(t)AdP (t))

[a,b]

is the limit in the uniform operator topology of the sequence of the products

’

(n) (n) (n) (n) (n)

(I + φ(tk )A∆P (tk )) (∆P (tk ) = P (tk ) ’ P (tk’1 ))

1¤k¤n

(n) (n)

as maxk |tk ’ tk’1 | tends to zero.

10.6. Volterra Operators 159

10.6 Resolvents of Volterra Operators

Lemma 10.6.1 Let V be a Volterra operator with a m.r.i. P (t) de¬ned on

a ¬nite real segment [a, b]. Then the sequence of the operators

n

(n) (n)

Vn = P (tk’1 )V ∆P (tk ) (6.1)

k=1

(n) (n)

tends to V in the uniform operator topology as maxk |tk ’ tk’1 | tends to

zero.

Proof: We have

n

(n) (n)

V ’ Vn = ∆P (tk )V ∆P (tk ).

k=1

But thanks to the well known Lemma I.3.1 (Gohberg and Krein, 1970), the

sequence { V ’ Vn } tends to zero as n tends to in¬nity. This proves the

required result. 2

Lemma 10.6.2 Let V be a Volterra operator with a maximal resolution of

the identity P (t) de¬ned on a segment [a,b]. Then

’

’1

(I ’ V ) = (I + V dP (t)).

[a,b]

Proof: Due to Lemma 10.6.1, V is the limit in the operator norm of the

sequence of operators Vn , de¬ned by (6.1) . Due to Lemma 10.1.2,

’

(n)

’1

(I ’ Vn ) = (I + Vn ∆P (tk )).

1¤k¤n

Hence the required result follows. 2

Resolvents of P -Triangular Operators

10.7

In this section [a, b] is a ¬nite real segment, again.

Theorem 10.7.1 Let A be a P -triangular operators with a m.r.i. P (.) de-

¬ned on [a, b], a (compact) nilpotent part V and the diagonal part

b

D= φ(t)dP (t), (7.1)

a

where φ is a scalar function integrable in the Riemann-Stieljes sense with

respect to P (.). Then

’

V dP (t)

(I ’ ) (» ∈ σ(A)).

R» (A) = R» (D) (7.2)

φ(t) ’ »

[a,b]

160 10. Multiplicative Representations of Resolvents

Proof: By Lemma 7.3.4 V R» (D) is a Volterra operator. We invoke Lemma

10.6.3. It asserts that

’

’1

(I ’ V R» (D)dP (t)).

(I + V R» (D)) = (7.3)

[a,b]

But according to (7.1)

1

R» (D)dP (t) = dP (t).

φ(t) ’ »

Thus,

’

V dP (t)

’1

(I ’

(I + V R» (D)) = ).

φ(t) ’ »

[a,b]

Hence relation (3.2) yields the required result. 2

Furthermore, from (7.2) it follows that

b ’

dP (s) V dP (t)

(I ’

R» (A) = )

φ(s) ’ » φ(t) ’ »

a [a,b]

for all regular ». But dP (s)V dP (t) = 0 for t ¤ s. We thus get

Corollary 10.7.2 Let the hypothesis of Theorem 10.7.1 hold. Then

b ’

dP (s) V dP (t)

(I ’ ) (» ∈ σ(A)).

R» (A) =

φ(s) ’ » φ(t) ’ »

a [s,b]

Let us suppose that A is a normal operator. Then V = 0 and Theorem 10.7.1

yields

b

dP (s)

R» (A) = .

a φ(s) ’ »

Thus, Theorem 10.7.1 generalizes the classical representation for the resolvent

of a normal operator.

Corollary 10.7.3 Let the hypothesis of Theorem 10.7.1 hold. Then

’

2i(P (t)AI ’ Im φ(t))dP (t)

(I ’ ) (» ∈ σ(A)).

R» (A) = R» (D)

φ(t) ’ »

[a,b]

Indeed, since A = D + V , we have AI = VI + DI with

DI = (D ’ D— )/2i and VI = (V ’ V — )/2i.

But

P (t)V dP (t) = V dP (t), dP (t)V dP (t) = 0 and P (t)V — dP (t) = 0.

10.8. Notes 161

Thus, V dP (t) = 2iP (t)VI dP (t). Moreover, since DI dP (t) = Im φ(t)dP (t),

we get

V dP (t) = 2i[P (t)AI ’ Im φ(t)]dP (t).

Thus, applying Theorem 10.7.1, we get Corollary 10.7.3. In particular, let A

have a purely real spectrum. Then Corollary 10.7.3 implies the representation

b ’

dP (s) 2iAI dP (t)

(I ’

R» (A) = )

φ(s) ’ » φ(t) ’ »

a [s,b]

for all regular ». Let AR , VR and DR are the real components of A, V and D,

respectively. Repeating the above arguments, by Theorem 10.7.1, we easily

obtain the following result.

Corollary 10.7.4 Let the hypothesis of Theorem 10.7.1 hold. Then

’

2(P (t)AR ’ Re φ(t))dP (t)

(I ’ ) (» ∈ σ(A)).

R» (A) = R» (D)

φ(t) ’ »

[a,b]

10.8 Notes

The contents of Sections 10.1-10.3 and 10.5-10.8 is based on the papers (Gil™,

1973) and (Gil™, 1980). The results presented in Sections 10.4 and 10.8 are

probably new.

For more details about the multiplicative integral see (Gohberg and Krein,

1970), (Brodskii, 1971), (Feintuch and Saeks, 1982).

References

[1] Brodskii, M. S. (1971). Triangular and Jordan Representations of Linear

Operators, Transl. Math. Monogr., v. 32, Amer. Math. Soc. Providence,

R. I.

[2] Feintuch, A., Saeks, R. (1982). System Theory. A Hilbert Space Ap-

proach. Ac. Press, New York.

[3] Gil™, M. I. (1973). On the representation of the resolvent of a nonselfad-

joint operator by the integral with respect to a spectral function, Soviet

Math. Dokl., 14 : 1214-1217.

[4] Gil™, M. I. (1980). On spectral representation for the resolvent of linear

operators. Sibirskij Math. Journal, 21: 231.

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

Volterra Operators in Hilbert Space, Trans. Mathem. Monogr., v. 24,

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

11. Relatively

P -Triangular Operators

This chapter is devoted to operators of the type A = D + W , where D

is a normal boundedly invertible operator in a separable Hilbert space H,

and W has the following property: V := D’1 W is a Volterra operator in

H. If, in addition, A has a maximal resolutions of the identity, then it

is called a relatively P -triangular operator. Below we derive estimates for

the resolvents of various relatively P -triangular operators and investigate

spectrum perturbations of such operators.

11.1 De¬nitions and Preliminaries

Let P (.) be a maximal resolution of the identity de¬ned on a real segment

[a, b] (see Section 7.2). In the present chapter paper we consider a linear

operator A in H of the type

A=D+W (1.1)

where D is a normal boundedly invertible, generally unbounded operator and

W is linear operator with the properties

H ⊇ Dom (W ) ⊃ Dom (D) = Dom (A).

In addition,

P (t)W P (t)h = W P (t)h (t ∈ [a, b], h ∈ Dom (W )) (1.2)

and

DP (t)h = P (t)Dh (t ∈ [a, b], h ∈ Dom (A)). (1.3)

M.I. Gil™: LNM 1830, pp. 163“172, 2003.

c Springer-Verlag Berlin Heidelberg 2003

11. Relatively P -Triangular Operators

164

Moreover,

V := D’1 W is a Volterra operator . (1.4)

We have

P (t)D’1 W P (t)h = D’1 P (t)W P (t)h = D’1 W P (t)h

(t ∈ [a, b], h ∈ Dom (A)).

So

P (t)V P (t) = V P (t) (t ∈ [a, b]).

De¬nition 11.1.1 Let relations (1.1)-(1.4) hold. Then A is said to be a

relatively P -triangular operator (RPTO), D is the diagonal part of A and V

is the relatively nilpotent part of A.

Everywhere in the present chapter A denotes a relatively P -triangular op-

erator, D denotes its diagonal part and V ∈ Y denotes the relatively nilpotent

part of A.

Recall that Y is an ideal of linear compact operators in H with a norm

|.|Y and the following property: any Volterra operator V ∈ Y satis¬es the

inequalities

V k ¤ θk |V |k (k = 1, 2, ...), (1.5)

Y

√

where constants θk are independent of V and k θk ’ 0 (k ’ ∞). Under

(1.5) put

ni(V )’1

θk |V |k (θ0 = 1),

JY (V ) = Y

k=0

where ni (V ) is the ”nilpotency” index (see De¬nition 1.4.2). In the sequel

one can replace ni (V ) by ∞.

Let rl (D) = inf |σ(D)|, again. Since D is invertible, rl (D) > 0.

Lemma 11.1.2 Under conditions (1.1)-(1.5), operator A is boundedly in-

’1

vertible. Moreover, A’1 ¤ rl (D)JY (V ) and

A’1 D ¤ JY (V ).

Proof: Clearly, A = (D + W ) = D(I + V ). So A’1 = (I + V )’1 D’1 . Due

to (1.5)

(I + V )’1 ¤ JY (V ).

’1

Since D is normal, D’1 = rl (D). This proves the required results. 2

11.2. Resolvents of Relatively P -Triangular Operators 165

11.2 Resolvents of Relatively

P -Triangular Operators

Denote

»

w(», D) ≡ inf | ’ 1|

t

t∈σ(D)

and

ν(») := ni ((D ’ »I)’1 W ) = ni ((I ’ D’1 »)’1 V ) (» ∈ σ(D)).

Under (1.5) put

m’1

z ’1’k θk |V |k (z > 0).

JY (V, m, z) = Y

k=0

Lemma 11.2.1 Under conditions (1.1)-(1.5), let » ∈ σ(D). Then » is a

regular point of operator A. Moreover,

’1

R» (A) ¤ rl (D)JY (V, ν(»), w(», D)) (2.1)

and

R» (A)D ¤ JY (V, ν(»), w(», D)). (2.2)

Proof: According to (1.1),

A ’ »I = (D ’ »I)(I + (D ’ »I)’1 W ).

Consequently

R» (A) = (I + R» (D)W )’1 R» (D).

Taking into account that (D ’ »I)’1 = (I ’ D’1 »)’1 D’1 , we have

R» (A) = (I + (I ’ D’1 »)’1 D’1 W )’1 (I ’ D’1 »)’1 D’1 . (2.3)

But D is normal. Therefore,

1

(I ’ D’1 »)’1 ¤ .

w(», D)

Moreover, due to Lemma 7.3.3, (I ’ D’1 »)’1 V is a Volterra operator. So

ν(»)’1

’1 ’1 ’1

((I ’ D’1 »)’1 V )k ¤

(I + (I ’ D ¤

») V)

k=0

ν(»)’1

θk |((I ’ D’1 »)’1 V )k |Y ¤

k=0

11. Relatively P -Triangular Operators

166

ν(»)’1

θk w’k (», D)|V |k = w(», D)JY (V, ν(»), w(», D)).

Y

k=0

Thus (2.3) yields

R» (A) ¤ (I + (I ’ D’1 »)’1 V )’1 D’1 (I ’ D’1 »)’1 ¤

’1

JY (V, ν(»), w(», D))rl (D).

Therefore (2.1) is proved. Moreover, thanks to (2.3)

R» (A)D ¤ (I + (I ’ D’1 »)’1 V )’1 (I ’ D’1 »)’1 ¤

JY (V, ν(»), w(», D)).

So inequality (2.2) is also proved. 2

11.3 Invertibility of Perturbed RPTO

In the sequel Z is a linear operator in H satisfying the condition

m(Z) := D’1 Z < ∞. (3.1)

Lemma 11.3.1 Under conditions (1.1)-(1.5) and (3.1), let

JY (V )m(Z) < 1. (3.2)

Then the operator A + Z is boundedly invertible. Moreover, the inverse op-

erator satis¬es the inequality

JY (V )

(A + Z)’1 ¤ .

ρl (D)(1 ’ JY (V )m(Z))

Proof: Due to Lemma 11.1.2 we have

A’1 Z = A’1 DD’1 Z ¤ JY (V )m(Z).

But

(A + Z)’1 ’ A’1 = ’A’1 Z(A + Z)’1 = ’A’1 DD’1 Z(A + Z)’1 .

Hence,

(A + Z)’1 ¤ A’1 + A’1 DD’1 Z(A + Z)’1 ¤

A’1 + JY (V )m(Z) (A + Z)’1

and

(A + Z)’1 ¤ A’1 (1 ’ JY (V )m(Z))’1 .

Lemma 11.1.2 yields now the required result. 2

11.4. Resolvents of Perturbed RPTO 167

11.4 Resolvents of Perturbed RPTO

Recall that JY (V, ν(»), w(», D)) is de¬ned in Section 11.2.

Theorem 11.4.1 Under conditions (1.1)-(1.5) and (3.1), for a » ∈ σ(D),

let

m(Z)JY (V, ν(»), w(», D)) < 1. (4.1)

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

JY (V, ν(»), w(», D))

R» (A + Z) ¤ .

ρl (D)(1 ’ m(Z)JY (V, ν(»), w(», D)))

Proof: Due to Lemma 11.2.1,

R» (A)Z = R» (A)DD’1 Z ¤ m(Z)JY (V, ν(»), w(», D)).

But

R» (A + Z) ’ R» (A) = ’R» (A)ZR» (A + Z).

Hence,

R» (A + Z) ¤ R» (A) + R» (A)Z R» (A + Z) ¤

R» (A) + m(Z)JY (V, ν(»), w(», D)) R» (A + Z) .

Due to (4.1)

R» (A + Z) ¤ R» (A) (1 ’ m(Z)JY (V, ν(»), w(», D))’1 .

Now Lemma 11.2.1 yields the required result. 2

Theorem 11.4.1 implies

Corollary 11.4.2 Under conditions (1.1)-(1.5) and (3.1), for any µ ∈ σ(A+

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

m(Z)JY (V, ν(µ), |1 ’ µµ’1 |) ≥ 1. (4.2)

0

11.5 Relative Spectral Variations

De¬nition 11.5.1 Let A and B be linear operators in H. Then the quantity

µ

inf |1 ’ |

rsvA (B) := sup

»

µ∈σ(B) »∈σ(A)

will be called the relative spectral variation of B with respect to A. In addition,

rhd(A, B) := max{rsvA (B), rsvB (A)}

is said to be the relative Hausdor¬ distance between the spectra of A and B.

11. Relatively P -Triangular Operators

168

Put

ni ((D ’ »I)’1 W ).

ν0 = sup

˜ ν(») = sup

»∈σ(D) »∈σ(D)

In the sequel one can replace ν0 by ∞.

˜

Theorem 11.5.2 Under conditions (1.1)-(1.5) and (3.1), the equation

m(Z)JY (V, ν0 , z) = 1

˜ (5.1)

has a unique positive root z0 (Y, V, Z). Moreover,

rsvD (A + Z) ¤ z0 (Y, V, Z). (5.2)

Proof: Comparing equation (5.2) with inequality (4.2), we arrive at the

required result. 2

Lemma 8.3.1 and Theorem 11.5.2 imply

Corollary 11.5.3 Under conditions (1.1)-(1.5) and (3.1), we have the in-

equality

rsvD (A + Z) ¤ δY (A, Z), (5.3)

where

θj’1 |V |j’1 Z .

j

δY (A, Z) ≡ 2 max Y

j=1,2,...

Due to (5.3), for any µ ∈ σ(A + Z), there is a µ0 ∈ σ(D), such that

1 ¤ |µµ’1 | + δY (A, Z).

0

Thus, |µ| ≥ |µ0 |(1 ’ δY (A, Z)). Let µ = rl (A + Z). Then we get

Corollary 11.5.4 Under conditions (1.1)-(1.5) and (3.1), we have the in-

equality

rl (A + Z) ≥ rl (D) max{0, 1 ’ δY (A, Z)}.

11.6 Operators with von Neumann-Schatten

Relatively Nilpotent Parts

11.6.1 Invertibility conditions

Let A be a relatively P -triangular operator (RPTO). Throughout this section

it is assumed that its relatively nilpotent part

V := D’1 W ∈ C2p (6.1)

11.6. Neumann-Schatten Relatively Nilpotent Parts 169

for a natural p ≥ 1. Again put,

ni(V )’1

(p)

k

Jp (V ) = θk N2p (V ),

k=0

where

1

(p)

θk =

[k/p]!

and [.] means the integer part. Due to Corollary 6.9.4,

(p)

j

V j ¤ θj N2p (V ) (j = 1, 2, ...). (6.2)

Then Lemma 11.1.2 implies that under (1.1)-(1.4) and (6.1),

’1

A’1 ¤ rl (D)Jp (V ) and A’1 D ¤ Jp (V ). (6.3)

Moreover, due to Lemma 11.3.1, we get

Lemma 11.6.1 Under conditions (1.1)-(1.4), (3.1) and (6.1), let

m(Z)Jp (V ) < 1.

Then operator A + Z is boundedly invertible. Moreover,

Jp (V )

(A + Z)’1 ¤ .

(1 ’ m(Z)Jp (V ))rl (D)

Under (6.1) for a z > 0, put

p’1 j 2p

N2p (V ) b0 N2p (V )

ψp (V, z) = a0 exp [ ], (6.4)

z j+1 z 2p

j=0

where constants a0 , b0 can be taken from the relations

c

a0 = , b0 = c/2 for a c > 1, (6.5)

c’1

or

a0 = e1/2 , b0 = 1/2. (6.6)

Lemma 11.6.2 Under conditions (1.1)-(1.4), (3.1) and (6.1), let

m(Z)ψp (V, 1) < 1.

Then operator A + Z is boundedly invertible. Moreover,

ψp (V, 1)

(A + Z)’1 ¤ .

rl (D) (1 ’ m(Z)ψp (V, 1))

11. Relatively P -Triangular Operators

170

Proof: Due to Theorem 6.7.3,

(I ’ V )’1 ¤ ψp (V, 1). (6.7)

But A’1 = (I + V )’1 D’1 . So

A’1 ¤ ρ’1 (D)ψp (V, 1) and A’1 D ¤ ψp (V, 1).

l

Now, using the arguments of the proof of Lemma 11.3.1, we arrive at the

required result. 2

11.6.2 The norm of the resolvent

Recall that ν(») is de¬ned in Section 11.2. Relation (6.2) and Lemma 11.2.1,

under (6.1) imply

’1 ˜

R» (A) ¤ rl (D)Jp (V, ν(»), w(», D)) and

˜

R» (A)D ¤ Jp (V, ν(»), w(», D)) (» ∈ σ(A)),

where

(p)

m’1 k

θk N2p (V )

˜

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

z k+1

k=0

Now thanks to Theorem 11.4.1, we get

Theorem 11.6.3 Under conditions (1.1)-(1.4), (3.1) and (6.1), let

˜

m(Z)Jp (V, ν(»), w(», D)) < 1.

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

˜

Jp (V, ν(»), w(», D))

R» (A + Z) ¤ .

˜

rl (D) (1 ’ m(Z)Jp (V, ν(»), w(», D))

Put

V (») = (I ’ D’1 »)’1 V.

˜

Thanks to (2.3)

R» (A)D = (I + V (»))’1 (I ’ D’1 »)’1 .

˜

Taking into account (6.7), we get

˜

R» (A) ¤ ψp (V (»), 1)w(», D).

Hence, we arrive at the inequalities

’1

R» (A) ¤ rl (D)ψp (V, w(», D)) and R» (A)D ¤ ψp (V, w(», D)) (6.8)

for all » ∈ σ(A). Now, taking into account (6.8) and repeating the arguments

of the proof of Theorem 11.4.1, we arrive at the following result

11.6. Neumann-Schatten Relatively Nilpotent Parts 171

Corollary 11.6.4 Under conditions (1.1)-(1.4), (3.1) and (6.1), let

m(Z)ψp (V, w(», D)) < 1.

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

ψp (V, w(», D))

R» (A + Z) ¤ .

rl (D) (1 ’ m(Z)ψp (V, w(», D))

11.6.3 Localization of the spectrum

Corollary 11.6.4 immediately yield.

Corollary 11.6.5 Under conditions (1.1)-(1.4), (3.1) and (6.1), for any µ ∈

σ(A + Z), there is a µ0 ∈ σ(D), such that, either µ = µ0 , or

µ

m(Z)ψp (V, |1 ’ |) ≥ 1. (6.9)

µ0

Theorem 11.6.6 Under conditions (1.1)-(1.4), (3.1) and (6.1), the equa-

tion

p’1 2p

k N2p (V )

N2p (V )

m(Z) exp [(1 + )/2] = 1 (6.10)

z k+1 z 2p

k=0

has a unique positive root zp (V, Z). Moreover, rsvD (A + Z) ¤ zp (V, Z). In

particular, the lower spectral radius of A + Z satis¬es the inequality

rl (A + Z) ≥ rl (D) max{0, 1 ’ zp (V, Z)}.

Comparing (6.9) and (6.10), we have w(D, µ) ¤ zp (V, Z), as claimed.

Proof:

2

Substituting the equality z = N2p (V )x, in (6.10) and applying Lemma

8.3.2, we get

zp (V, Z) ¤ ∆p (V, Z),

where

if N2p (V ) ¤ m(Z)pe

m(Z)pe

∆p (V, Z) := .

N2p (V )[ln (N2p (V )/m(Z)p)]’1/2p if N2p (V ) > m(Z)pe

Thanks to Theorem 11.6.5 we arrive at the following

Corollary 11.6.7 Under conditions (1.1)-(1.5), (3.1) and (6.1), rsvD (A +

Z) ¤ ∆p (V, Z). In particular,

rl (A + Z) ≥ rl (D) max{0, 1 ’ ∆p (V, Z)}.

11. Relatively P -Triangular Operators

172

11.7 Notes

This 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).

12. Relatively Compact

Perturbations of Normal

Operators

The present chapter is devoted to linear operators of the type A = D + W+ +

W’ , where D is a normal invertible operator and D’1 W± are Volterra (com-

pact quasinilpotent) operators. Numerous di¬erential and integro-di¬erential

operators are examples of such operators. We derive estimates for the resol-

vents and bounds for the spectra.

12.1 Invertibility Conditions

Let H be a separable Hilbert space. In the present chapter we consider a

linear operator A in H of the type

A = D + W+ + W’ , (1.1)

where D is a normal bounded invertible, generally unbounded operator and

W± are linear operators, such that

H ⊇ Dom (W± ) ⊃ Dom (D) = Dom (A).

Let P (t) (’∞ ¤ t ¤ ∞) be a maximal resolution of the identity (m.r.i.) (see

Section 7.2). It is assumed that

P (t)W+ P (t)h = W+ P (t)h (h ∈ Dom (W+ ); t ∈ R), (1.2a)

P (t)W’ P (t)h = P (t)W’ h (h ∈ Dom (W’ ); t ∈ R) (1.2b)

M.I. Gil™: LNM 1830, pp. 173“180, 2003.

c Springer-Verlag Berlin Heidelberg 2003

174 12. Relatively Compact Perturbations

and

P (t)Dh = DP (t)h (h ∈ Dom (D); t ∈ R). (1.2c)

Again we use a normed ideal Y of linear compact operators in H with a

norm |.|Y and having the property: any Volterra operator V ∈ Y satis¬es the

inequalities V k ¤ θk |V |k (k = 1, 2, ...) where constants θk are independent

√ Y

of V and k θk ’ 0 (k ’ ∞). It is assumed that

V± ≡ D’1 W±

are Volterra operators from ideal Y . That is,

V± ≡ (D’1 W± )k ¤ θk |V± |k (k = 1, 2, ...).

k

(1.3)

Y

Recall that ni (V ) means the nilpotency index of a quasinilpotent operator

V (see De¬nition 7.5.2). Put

ni(V± )’1

θk |V± |k (θ0 = 1).

JY (V± ) = Y

k=0

Lemma 12.1.1 Under condition (1.2), (1.3), let

1 1

ζ0 (A) ≡ max { ’ V+ , ’ V’ } > 0. (1.4)

JY (V’ ) JY (V+ )

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

inverse operator satis¬es the inequality

1

A’1 ¤ . (1.5)

rl (D)ζ0 (A)

Recall that rl (.) denotes the lower spectral radius.

Proof: Condition (1.4) implies that either

V+ JY (V’ ) < 1 (1.6)

or

V’ JY (V+ ) < 1.

If (1.6) holds, then replacing in Lemma 11.3.1 A by D + W’ and Z by W+

we get the invertibility and estimate

JY (V’ )

A’1 ¤ =

rl (D)(1 ’ V+ JY (V’ ))

1

. (1.7)

’1

’ V+ )

rl (D)(JY (V’ )

Exchanging V’ and V+ , we have the estimate

1

A’1 ¤ .

’1

rl (D)(JY (V+ ) ’ V’ )

This and (1.7) imply the required result. 2

12.2. Estimates for Resolvents 175

12.2 Estimates for Resolvents

Under condition (1.3) denote

m’1

z ’1’k θk |V± |k (z > 0).

JY (V± , m, z) = (2.1)

Y

k=0

Lemma 12.2.1 Under conditions (1.2), the operator (D ’ »I)’1 W± is

quasinilpotent for any » ∈ σ(D).

Proof: Condition (1.2a) implies

P (t)V+ P (t) = P (t)D’1 W+ P (t) = D’1 P (t)W+ P (t) =

D’1 W+ P (t) = V+ P (t)

and

(I ’ D’1 »)’1 P (t) = P (t)(I ’ D’1 »)’1 (t ∈ (’∞, ∞)).

Thus, due to Lemma 7.3.3, the operator (I ’ D’1 »)’1 V+ is a Volterra one.

But

(I ’ D’1 »)’1 V+ = (D ’ »I)’1 W+ .

Similarly, we can prove that (D ’ »I)’1 W’ is a Volterra operator. 2

Put

ν± (») ≡ ni((D ’ »I)’1 W± ) = ni((I ’ »D’1 )’1 V± )

and

»

w(», D) ≡ inf | ’ 1|.

t

t∈σ(D)

Now we are in a position to formulate the main result of the chapter

Theorem 12.2.2 Under conditions (1.2) and (1.3), for a » ∈ σ(D), let

1

ζ(A, ») ≡ max { ’ V+ ,

JY (V’ , ν’ (»), w(», D))

1

’ V’ } > 0. (2.2)

JY (V+ , ν+ (»), w(», D))

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

1

R» (A) ¤ . (2.3)

ζ(A, »)ρl (D)

Proof: Condition (1.5) means that at least one of the following inequalities

holds:

JY (V’ , ν’ (»), w(», D)) V+ < 1

or

JY (V+ , ν+ (»), w(», D)) V’ < 1. (2.4)

176 12. Relatively Compact Perturbations

If (2.4) holds, then replacing in Lemma 11.4.1 operator A by D + W’ and

operator Z by W+ we get the regularity of » and estimate

JY (V+ , ν+ (»), w(», D))

R» (A) ¤ .

ρl (D)(1 ’ V’ JY (V+ , ν+ (»), w(», D))

Or

1

R» (A) ¤ .

’1

ρl (D)(JY (V+ , ν+ (»), w(», D)) ’ V’ )

Exchanging V+ and V’ , we get

1

R» (A) ¤ .

’1

’ V+ )

ρl (D)(JY (V’ , ν’ (»), w(», D))

These inequalities yield the required result. 2

12.3 Bounds for the Spectrum

Recall that rsvD (A) denotes the relative spectral variation of operator A with

respect to D. Again put

„ (A) := min{ V’ , V+ }, (3.1)

if V+ ≥ V’ ,

V+

˜

V := (3.2)

V’ if V’ > V+

and

ni ((D ’ »I)’1 V ).

˜

ν0 = sup

˜

»∈σ(D)

In the sequel one can replace ν0 by ∞.

˜

˜

Theorem 12.3.1 Under conditions (1.1), (1.2), let V ∈ Y and

V k ¤ θk |V |k (k = 1, 2, ...).

˜ ˜Y (3.3)

Then the equation

˜˜

„ (A)JY (V , ν0 , z) = 1 (3.4)

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

rsvD (A) ¤ zY (A). (3.5)

Proof: Due to Theorem 12.2.2,

˜˜

„ (A)JY (V , ν0 , w(D, µ)) ≥ 1

for any µ ∈ σ(A). Comparing this inequality with (3.4), we have w(D, µ) ¤

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

Theorem 12.3.1 and Lemma 8.3.1 give us the following result

12.4. Relatively Neumann-Schatten Parts 177

Corollary 12.3.2 Under conditions (1.1), (1.2) and (3.3), we have

rsvD (A) ¤ ∆Y (A),

where

θj’1 |V |j’1 „ (A).

j

˜

∆Y (A) := 2 max Y

j=1,2,...

Due to (3.3), for all µ ∈ σ(A), there is µ0 ∈ σ(D), such that

1 ¤ |µµ’1 | + ∆Y (A).

0

Thus |µ| ≥ |µ0 |(1 ’ ∆Y (A)). Now the previous corollary yields

Corollary 12.3.3 Let conditions (1.1), (1.2) and (3.3) hold. Then

rl (A) ≥ rl (D) max{0, 1 ’ ∆Y (A)}.

Remark 12.3.4 Everywhere below V± can be replaced by upper bounds.

12.4 Operators with Relatively

von Neumann - Schatten

O¬-diagonal Parts

12.4.1 Invertibility conditions

In this section it is assumed that V± = D’1 W± are quasinilpotent operators

belonging to the Neumann-Schatten ideal C2p with some integer p ≥ 1. That

is,

N2p (V± ) ≡ [T race (V± V± )p ]1/2p < ∞.

—

(4.1)

Put

ni(V )’1

(p)k

Jp (V± ) = θk N2p (V± ),

k=0

where

1

(p)

θk =

[k/p]!

and [.] means the integer part.

Lemma 12.4.1 Let relations (1.2) and (4.1) hold. In addition, let

’1 ’1

ζ2p (A) := max{Jp (V’ ) ’ V+ , Jp (V+ ) ’ V’ } > 0. (4.2)

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

1

A’1 ¤ . (4.3)

ζ2p (A)rl (D)

178 12. Relatively Compact Perturbations

Proof: Due to Corollary 6.9.4,

(p)

V k ¤ θk N2p (V ) (k = 1, 2, ...).

k

(4.4)

for a quasinilpotent operator V ∈ C2p . Now the required result is due to

Lemma 12.1.1. 2

Clearly,

p’1 ∞

j+pk

N2p (V± )

√

Jp (V± ) ¤ Ip (V± ) := .

k!

j=0 k=1

According to (4.4) one can replace Jp (V± ) in (4.2) by Ip (V± ).

Furthermore, for a z > 0, put

p’1 2p

j N2p (V± )

Np (V± )

ψp (V± , z) = exp [(1 + )/2]. (4.5)

z j+1 z 2p

j=0