[6] Dunford, N. and Schwartz, J. T. (1963). Linear Operators, part II. Spec-

tral Theory. Interscience publishers, New York, London.

[7] Gil™, M. I. (1979a). An estimate for norms of resolvent of completely

continuous operators, Mathematical Notes, 26 , 849-851.

[8] Gil™, M. I. (1979b). Estimates for norms of functions of a Hilbert-Schmidt

operator (in Russian), Izvestiya VUZ, Matematika, 23, 14-19. English

translation in Soviet Math., 23, 13-19.

[9] Gil™ , M. I. (1992). On estimate for the norm of a function of a quasi-

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

96 6. Functions of Compact Operators

[10] Gil™, M. I. (1995). Norm Estimations for Operator-valued Functions and

Applications. Marcel Dekker, Inc., New York.

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

Linear Nonselfadjoint Operators, Trans. Mathem. Monographs, v. 18,

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

[12] 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.

[13] Pietsch, A. (1988). Eigenvalues and s-numbers, Cambridge University

Press, Cambridge.

7. Functions of

Non-compact Operators

The present chapter is concerned with the estimates for the norms of resol-

vents and analytic functions of so called P -triangular operators. Roughly

speaking, a P -triangular operator is a sum of a normal operator and a com-

pact quasinilpotent one, having a su¬ciently rich set of invariant subspaces.

In particular, we consider the following classes of P -triangular operators: op-

erators whose Hermitian components are compact operators, and operators,

which are represented as sums of unitary operators and compact ones.

7.1 Terminology

Let A be a linear operator acting in a separable Hilbert space H. Let there

be a linear manifold Dom (A), such that the relation f ∈ Dom (A) implies

Af ∈ H. Then the set Dom (A) is called the domain of A. Let Dom (A) be

dense in H. Then the set of vectors g satisfying

|(Af, g)| ¤ c f for all f ∈ Dom (A)

with a constant c is the domain of the adjoint operator A— and, besides,

(Af, g) = (f, A— g) for all f ∈ Dom (A) and g ∈ Dom (A— ).

An unbounded operator A is selfadjoint, if

Dom (A) = Dom (A— ) and Ah = A— h (h ∈ Dom (A)).

An unbounded selfadjoint operator possesses an unbounded real spectrum.

An unbounded operator A is normal, if

Dom (AA— ) = Dom (A— A) and AA— h = A— Ah (h ∈ Dom (A— A)).

M.I. Gil™: LNM 1830, pp. 97“121, 2003.

c Springer-Verlag Berlin Heidelberg 2003

98 7. Functions of Non-compact Operators

An unbounded normal operator has an unbounded spectrum.

De¬nition 7.1.1 Let A be a linear operator in H. Then A is said to be a

quasi-normal operator, if it is a sum of a normal operator and a compact

one.

Operator A is said to be a quasi-Hermitian operator, if it is a sum of a

selfadjoint operator and a compact one.

Let A be bounded and AI ≡ (A ’ A— )/2i ∈ Cp (p ≥ 1). Then, as it is

well-known (see e.g. (Gohberg and Krein, 1969, Section II.6)), the nonreal

spectrum consists of the eigenvalues having ¬nite multiplicities, and

∞ ∞

p

|»j (AI )|p (n = 1, 2, . . .)

|Im »j (A)| ¤ (1.1)

k=1 k=1

where »j (A) and »j (AI ) are the eigenvalues with their multiplicities of A and

AI , respectively.

P -Triangular Operators

7.2

A family of orthogonal projectors P (t) in H (i.e. P 2 (t) = P (t) and P — (t) =

P (t)) de¬ned on a (¬nite or in¬nite) segment [a, b] of the real axis is a an

orthogonal resolution of the identity if for all t, s ∈ [a, b],

P (a) = 0, P (b) = I ≡ the unit operator and P (t)P (s) = P (min(t, s))

An orthogonal resolution of the identity P (t) is left-continuous, if P (t ’ 0) =

P (t) for all t ∈ (a, b] in the sense of the strong topology.

De¬nition 7.2.1 Let P (t) be a left-continuous orthogonal resolution of the

identity in H de¬ned on a (¬nite or in¬nite) real segment [a, b]. Then P (.)

will be called a maximal resolution of the identity (m.r.i.), if its every gap

P (t0 + 0) ’ P (t0 ) (if it exists) is one-dimensional.

Moreover, we will say that an m.r.i. P (.) belongs to a linear operator A

(or A has an m.r.i. P (.)), if

P (t)AP (t)h = AP (t)h for all t ∈ [a, b] and h ∈ Dom (A). (2.1)

Recall that a linear operator V is called a Volterra one, if it is compact and

quasinilpotent.

De¬nition 7.2.2 Let a linear generally unbounded operator A have an m.r.i.

P (.) de¬ned on [a, b]. In addition, let

A = D + V, (2.2)

7.3. Properties of Volterra Operators 99

where D is a normal operator and V is a Volterra one, having the following

properties:

P (t)V P (t) = V P (t) (t ∈ [a, b]) (2.3)

and

DP (t)h = P (t)Dh (t ∈ [a, b], h ∈ Dom (A)). (2.4)

Then A will be called a P -triangular operator. In addition, equality (2.2),

and operators D and V will be called the triangular representation, diagonal

part and nilpotent part of A, respectively.

Clearly, this de¬nition is in accordance with the de¬nition of the triangular

representation of compact operators (see Section 6.3).

7.3 Some Properties of Volterra Operators

Lemma 7.3.1 Let a compact operator V in H, have a maximal orthogonal

resolution of the identity P (t) (a ¤ t ¤ b) (that is, condition (2.3) hold). If,

in addition,

(P (t0 + 0) ’ P (t0 ))V (P (t0 + 0) ’ P (t0 )) = 0 (3.1)

for every gap P (t0 + 0) ’ P (t0 ) of P (t) (if it exists), then V is a Volterra

operator.

Proof: Since the set of the values of P (.) is a maximal chain of projectors,

the required result is due to Corollary 1 to Theorem 17.1 of the book by

Brodskii (1971). 2

In particular, if P (t) is continuous in t in the strong topology and (2.3)

holds, then V is a Volterra operator. We also need the following

Lemma 7.3.2 Let V be a Volterra operator in H, and P (t) a maximal or-

thogonal resolution of the identity satisfying equality (2.3). Then equality

(3.1) holds for every gap P (t0 + 0) ’ P (t0 ) of P (t) (if it exists).

Proof: Since the set of the values of P (.) is a maximal chain of projectors,

the required result is due to the well-known equality (I.3.1) from the book

by Gohberg and Krein (1970). 2

Lemma 7.3.3 Let V and B be bounded linear operators in H having the

same m.r.i. P (.). In addition, let V be a Volterra operator. Then V B and

BV are Volterra operators, and P (.) is their m.r.i.

Proof: It is obvious that

P (t)V BP (t) = V P (t)BP (t) = V BP (t). (3.2)

100 7. Functions of Non-compact Operators

Now let Q = P (t0 + 0) ’ P (t0 ) be a gap of P (t). Then according to Lemma

7.3.2 equality (3.1) holds. Further, we have

QV BQ = QV B(P (t0 + 0) ’ P (t0 )) =

QV [P (t0 + 0)BP (t0 + 0) ’ P (t0 )BP (t0 )] =

QV [(P (t0 ) + Q)B(P (t0 ) + Q) ’ P (t0 )BP (t0 )] =

QV [P (t0 )BQ + QBP (t0 )].

Since QP (t0 ) = 0 and P (t) projects onto the invariant subspaces, we obtain

QV BQ = 0. Due to Lemma 7.3.1 this relation and equality (3.2) imply that

V B is a Volterra operator. Similarly we can prove that BV is a Volterra one.

2

Lemma 7.3.4 Let A be a (generally unbounded) P -triangular operator. Let

V and D be the nilpotent and diagonal parts of A, respectively. Then for any

regular point » of D, the operators V R» (D) and R» (D)V are Volterra ones.

Besides, A, V R» (D) and R» (D)V have the same m.r.i.

Proof: Due to (2.4)

P (t)R» (D) = R» (D)P (t) for all t ∈ [a, b].

Now Lemma 7.3.3 ensures the required result. 2

7.4 Powers of Volterra Operators

Let Y be a a norm ideal of compact linear operators in H. That is, Y is

algebraically a two- sided ideal, which is complete in an auxiliary norm | · |Y

for which |CB|Y and |BC|Y are both dominated by C |B|Y .

In the sequel we suppose that there are positive numbers θk (k ∈ N),

with

1/k

θk ’ 0 as k ’ ∞,

such that

V k ¤ θk |V |k (4.1)

Y

for an arbitrary Volterra operator

V ∈ Y. (4.2)

Recall that C2p (p = 1, 2, ...) is the von Neumann-Schatten ideal of compact

operators with the ¬nite ideal norm

N2p (K) ≡ [T race (K — K)p ]1/2p (K ∈ C2p ).

7.5. Resolvents of P -Triangular Operators 101

Let V ∈ C2p be a Volterra operator. Then due to Corollary 6.9.4,

(p)

j

V j ¤ θj N2p (V ) (j = 1, 2, ...) (4.3)

where

1

(p)

θj =

[j/p]!

and [x] means the integer part of a positive number x. Inequality (4.3) can

be written as

pk+m

N2p (V )

kp+m

√

¤ (k = 0, 1, 2, ...; m = 0, ..., p ’ 1).

V (4.4)

k!

Resolvents of P -Triangular Operators

7.5

Lemma 7.5.1 Let A be a P -triangular operator. Then σ(A) = σ(D), where

D is the diagonal part of A.

Proof: Let » be a regular point of the operator D. According to the trian-

gular representation (2.2) we obtain

R» (A) = (D + V ’ »I)’1 = R» (D)(I + V R» (D))’1 . (5.1)

Operator V R» (D) for a regular point » of the operator D is a Volterra one

due to Lemma 7.3.4. Therefore,

∞

’1

(V R» (D))k (’1)k

(I + V R» (D)) =

k=0

and the series converges in the operator norm. Thus,

∞

(V R» (D))k (’1)k .

R» (A) = R» (D) (5.2)

k=0

Hence, it follows that » is the regular point of A.

Conversely let » ∈ σ(A). According to the triangular representation (2.2)

we obtain

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

Operator V R» (A) for a regular point » of operator A is a Volterra one due

to Lemma 7.3.3. So

∞

’1

(V R» (A))k

(I ’ V R» (A)) =

k=0

102 7. Functions of Non-compact Operators

and the series converges in the operator norm. Thus,

∞

(V R» (A))k .

R» (D) = R» (A)

k=0

Hence, it follows that » is the regular point of D. This ¬nishes the proof. 2

Furthermore, for a natural number m and a z ∈ (0, ∞), under (4.2), put

m’1

θk |V |k

Y

JY (V, m, z) = . (5.3)

k+1

z

k=0

De¬nition 7.5.2 A number ni(V ) is called the nilpotency index of a nilpo-

tent operator V , if V ni(V ) = 0 = V ni(V )’1 . If V is quasinilpotent but not

nilpotent we write ni(V ) = ∞.

Everywhere below one can replace ni(V ) by ∞.

Theorem 7.5.3 Let A be a P -triangular operator and let its nilpotent part

V belong to a norm ideal Y with the property (4.1). Then

ν(»)’1

θk |V |k

Y

R» (A) ¤ JY (V, ν(»), ρ(A, »)) := (5.4)

k+1 (A, »)

ρ

k=0

for all regular » of A. Here ν(») = ni(V R» (D)), and D is the diagonal part

of A.

Proof: Due to Lemma 7.3.4, V R» (D) ∈ Y is a Volterra operator. So

according to (4.1),

(V R» (D))k ¤ θk |V R» (D)|k .

Y

But

|V R» (D)|Y ¤ |V |Y R» (D) ,

and thanks to Lemma 7.5.1,

1 1

R» (D) = = .

ρ(D, ») ρ(A, »)

So

θk |V |k

(V R» (D))k ¤ Y

.

k (A, »)

ρ

Relation (5.2) implies

ν(»)’1

(V R» (D))k ¤

R» (A) ¤ R» (D)

k=0

7.5. Resolvents of P -Triangular Operators 103

ν(»)’1

θk |V |k

1 Y

,

k (A, »)

ρ(A, ») ρ

k=0

as claimed. 2

For a natural number m, a Volterra operator V ∈ C2p and a z ∈ (0, ∞),

put

(p)

m’1 k

θk N2p (V )

˜

Jp (V, m, z) := . (5.5)

z k+1

k=0

Theorem 7.5.3 and inequality (4.3) yield

Corollary 7.5.4 Let A be a P -triangular operator and its nilpotent part V ∈

C2p for an integer p ≥ 1. Then

ν(»)’1 (p) k

θk N2p (V )

˜

R» (A) ¤ Jp (V, ν(»), ρ(A, »)) ≡ (» ∈ σ(A)). (5.6)

ρk+1 (A, »)

k=0

In particular, let A be a P -triangular operator, whose nilpotent part V is a

Hilbert-Schmidt operator. Then due to the previous corollary

ν(»)’1 k

N2 (V )

√

˜

R» (A) ¤ J1 (V, ν(»), ρ(A, »)) ≡ (» ∈ σ(A)). (5.7)

ρk+1 (A, ») k!

k=0

Furthermore, inequality (5.6) implies

p’1 ∞ pk+j

N2p (V )

√.

R» (A) ¤ (5.8)

ρpk+j+1 (A, ») k!

j=0 k=0

Theorem 7.5.5 Let A be a P -triangular operator and its nilpotent part V ∈

C2p for some integer p ≥ 1. Then there are constants a0 , b0 > 0, such that

p’1 j 2p

N2p (V ) b0 N2p (V )

R» (A) ¤ a0 ] (» ∈ σ(A)).

exp [ 2p (5.9)

ρj+1 (A, ») ρ (A, »)

j=0

One can take

√

c c

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

c’1 2

(5.10)

or

a0 = e1/2 and b0 = 1/2. (5.11)

104 7. Functions of Non-compact Operators

Due to Lemma 6.8.3, for any Volterra operator V ∈ C2p ,

Proof:

p’1

j 2p

’1

(I ’ V ) ¤ a0 N2p (V ) exp [ b0 N2p (V )].

j=0

But V R» (D) is a Volterra operator due to Lemma 7.3.4. Hence,

p’1

j 2p

’1

¤ a0 N2p (V R» (D)) exp [ b0 N2p (V R» (D))] ¤

(I + V R» (D))

j=0

p’1 j 2p

N2p (V ) N2p (V )

¤ a0 exp [ b0 2p ].

ρj (A, ») ρ (A, »)

j=0

Now (5.1) implies the required result. 2

7.6 Triangular Representations of

Quasi-Hermitian Operators

Theorem 7.6.1 Let a linear generally unbounded operator A satisfy the con-

ditions Dom (A— ) = Dom (A) and

AI = (A ’ A— )/2i ∈ Cp (1 < p < ∞). (6.1)

Then A admits the triangular representation (2.2).

Proof: First, let A be bounded. Then, as it is proved by L. de Branges

(1965a, p. 69), under condition (6.1), there are a maximal orthogonal reso-

lution of the identity P (t) de¬ned on a ¬nite real segment [a, b] and a real

nondecreasing function h(t), such that

b b

A= h(t)dP (t) + 2i P (t)AI dP (t). (6.2)

a a

The second integral in (6.2) is understood as the limit in the operator norm

of the operator Stieltjes sums

n

1

Ln = [P (tk’1 ) + P (tk )]AI ∆Pk ,

2

k=1

where

(n)

tk = tk ; ∆Pk = P (tk ) ’ P (tk’1 ); a = t0 < t1 . . . < tn = b.

7.6. Representations of Quasi-Hermitian Operators 105

We can write Ln = Wn + Tn with

n n

1

Wn = P (tk’1 )AI ∆Pk and Tn = ∆Pk AI ∆Pk . (6.3)

2

k=1 k=1

The sequence {Tn } converges in the operator norm due to the well-known

Lemma I.5.1 from the book (Gohberg and Krein, 1970). We denote its limit

by T . Clearly, T is selfadjoint and P (t)T = T P (t) for all t ∈ [a, b]. Put

b

D= h(t)dP (t) + 2iT. (6.4)

a

Then D is normal and satis¬es condition (2.4).

Furthermore, it can be directly checked that Wn is a nilpotent operator:

(Wn )n = 0. Besides, the sequence {Wn } converges in the operator norm,

because the second integral in (6.2) and {Tn } converge in this norm. We

denote the limit of the sequence {2iWn } by V . It is a Volterra operator,

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

Volterra one (see for instance Lemma 2.17.1 from the book (Brodskii, 1971)).

From this we easily obtain relations (2.1)-(2.4). So, for bounded operators

the theorem is proved.

Now let A be unbounded. Due to De Branges (1965a, p. 69), there is

a maximal orthogonal resolution of the identity P (t), ’∞ ¤ t ¤ ∞ and

nondecreasing functions h(t), such that under (6.1), the representation

∞ ∞

A= h(t)dP (t) + 2i P (t)AI dP (t) (6.5)

’∞ ’∞

holds. The integrals in (6.5) are understood as the limits of corresponding

integrals from (6.2) when a ’ ’∞ and b ’ ∞. Besides, the ¬rst integral

in (6.5) is the strong limit on Dom(A) of the ¬rst integral in (6.2), and the

second integral in (6.5) is the limit in the uniform operator topology of the

second integral in (6.2). Take

An = A(P (n) ’ P (’n)). (6.6)

Clearly, An is bounded and has the property (6.1). So as it was proved

above, according to (6.5) it has a triangular representation with the m.r.i.

P (.). Hence, letting n ’ ∞, we get the required result. 2

Corollary 7.6.2 Let A be an unbounded operator with the property (6.1).

Then it is P -triangular and there is a sequence of bounded P -triangular op-

erators An , such that An ’ A— ∈ Cp ,

n

σ(An ) ⊆ σ(A), Np (An ’ A— ) ’ 0 and Np (Vn ’ V ) ’ 0

as n ’ 0, where Vn and V are the nilpotent part of An and A, respectively.

Moreover, operators A and An (n = 1, 2, ...) have the same m.r.i.

Indeed, taking An as in (6.6), we arrive at the result due to (2.2) and the

previous theorem.

106 7. Functions of Non-compact Operators

7.7 Resolvents of Operators with

Hilbert-Schmidt Hermitian Components

In this section we obtain an estimate for the norm of the resolvent of a gener-

ally unbounded quasi-Hermitian operator under the conditions Dom (A) =

Dom (A— ) and

AI = (A ’ A— )/2i is a Hilbert-Schmidt operator. (7.1)

Let us introduce the quantity

∞

√ 2

(Im »k (A))2 ]1/2 .

gI (A) ≡ ’

2 [N2 (AI ) (7.2)

k=1

Theorem 7.7.1 Let condition (7.1) hold. Then

∞ k

gI (A)

√

R» (A) ¤ (» ∈ σ(A)). (7.3)

ρk+1 (A, ») k!

k=0

Moreover, there are constants a0 , b0 > 0, such that

b0 g 2 (A)

a0

exp [ 2 I

R» (A) ¤ ] (» ∈ σ(A)). (7.4)

ρ(A, ») ρ (A, »)

These constants can be taken from (5.10) or from (5.11).

Here ρ(A, ») is the distance between the spectrum σ(A) of A and a complex

point », again.

To prove this theorem we need the following

Lemma 7.7.2 Let an operator A satisfy the condition (7.1). Then it admits

the triangular representation (due to Theorem 7.6.1). Moreover, N2 (V ) =

gI (A), where V is the nilpotent part of A.

Proof: First, assume that A is a bounded operator. Let D be the diagonal

part of A. From the triangular representation (2.2) it follows that

’4T r A2 = T r (A ’ A— )2 = T r (D + V ’ D— ’ V — )2 .

I

By Theorem 6.2.2 and Lemma 7.3.3, T race V D— = T race V — D = 0. Hence,

omitting simple calculations, we obtain

’4T r A2 = T r (D ’ D— )2 + T r (V ’ V — )2 .

I

That is, N2 (AI ) = N2 (VI ) + N2 (DI ), where VI = (V ’ V — )/2i and DI =

2 2 2

(D ’ D— )/2i. Taking into account Lemma 6.5.1, we arrive at the equality

2N 2 (AI ) ’ 2N 2 (DI ) = N 2 (V ).

7.7. Operators with Property Ap ’ (A— )p ∈ C2 107

Recall that the nonreal spectrum of a quasi-Hermitian operator consists of

isolated eigenvalues. Besides, due to Lemma 7.5.1 σ(A) = σ(D). Thus,

∞

2

|Im »k (A)|2 ,

N (DI ) =

k=1

and we arrive at the result, if A is bounded. If A is unbounded, then the

required assertion is due to Corollary 7.6.2 and the just proved result for

bounded operators. 2

Proof of Theorem 7.7.1: Inequality (7.3) follows from Corollary 7.5.4

and Lemma 7.7.2. Inequality (7.4) follows from Theorem 7.5.5 and Lemma

7.7.2. 2

7.8 Operators with the Property

Ap ’ (A— )p ∈ C2

Theorem 7.8.1 Let a bounded linear operator A satisfy the condition

Ap ’ (A— )p ∈ C2 (p = 2, 3, ...). (8.1)

Then

∞

gI (Ap )

k

(»p ∈ σ(Ap )).

√

R» (A) ¤ T»,p

k+1 (Ap , »p ) k!

ρ

k=0

Here ρ(Ap , »p ) is the distance between the spectrum σ(Ap ) of Ap and »p , and

p’1

Ak »p’k’1 .

T»,p =

k=0

Indeed, this result follows from Theorem 7.7.1 and the obvious relation

R» (A) = T»,p R»p (Ap ). (8.2)

Moreover, relations (7.4) and (8.2) yield

Corollary 7.8.2 Let condition (8.1) hold. Then there are constants a0 , b0 >

0, such that

b0 gI (Ap )

2

a0 T»,p

exp [ 2 p p ] (»p ∈ σ(Ap )).

R» (A) ¤ p , »p )

ρ(A ρ (A , » )

These constants can be taken from (5.10) or from (5.11).

108 7. Functions of Non-compact Operators

7.9 Resolvents of Operators with

Neumann - Schatten Hermitian

Components

In this section we obtain estimates for the resolvent of an operator A, assum-

ing that Dom (A) = Dom (A— ) and

AI := (A ’ A— )/2i ∈ C2p for some integer p > 1. (9.1)

That is,

∞

»2p (AI ) < ∞.

2p

N2p (AI ) = j

j=1

Put

π

if p = 2m’1 , m = 1, 2, ...

2(1 + ctg ( 4p ) )

˜

βp := . (9.2)

2p

2(1 + exp(2/3)ln2 ) otherwise

Theorem 7.9.1 Let condition (9.1) hold. Then there is a constant βp , de-

pending on p, only, such that

p’1 ∞

(βp N2p (AI ))kp+m

√

R» (A) ¤ (» ∈ σ(A)). (9.3)

ρpk+m+1 (A, ») k!

m=0 k=0

Besides,

˜

βp ¤ βp . (9.4)

Moreover, there are constants a0 , b0 > 0, such that

p’1

(βp N2p (AI ))m b0 (βp N2p (AI ))2p

R» (A) ¤ a0 ] (» ∈ σ(A)).

exp [

ρm+1 (A, ») ρ2p (A, »)

m=0

(9.5)

Constants a0 , b0 can be taken from (5.10) or from (5.11).

In order to prove this theorem we need the following result.

Lemma 7.9.2 Let A satisfy condition (9.1). Then it admits the triangular

representation (due to Theorem 7.6.1), and the nilpotent part V of A satis¬es

the relation

N2p (V ) ¤ βp N2p (AI ).

Proof: Let V ∈ Cp (2 ¤ p < ∞) be a Volterra operator. Then due to the

well-known Theorem III.6.2 from the book by Gohberg and Krein (1970, p.

118), there is a constant γp , depending on p, only, such that

Np (VI ) ¤ γp Np (VR ) (VI = (V + V — )/2i, VR := (V + V — )/2). (9.6)

7.10. Regular Functions of Bounded Operators 109

Besides, as it is proved in (Gohberg and Krein, 1970, pages 123 and 124),

p p

¤ γp ¤ .

2π exp(2/3)ln2

Moreover,

π

if p = 2n (n = 1, 2, ...),

γp = ctg

2p

cf. (Gohberg and Krein, 1970, page 124). Take

βp = 2(1 + γ2p ).

Now let D be the diagonal part of A. Let VI , DI be the imaginary com-

ponents of V and D, respectively. According to (2.2) AI = VI + DI . First,

assume that A is a bounded operator. Due to (1.1), the condition AI ∈ C2p

entails the inequality N2p (DI ) ¤ N2p (AI ). Therefore,

N2p (VI ) ¤ N2p (AI ) + N2p (DI ) ¤ 2N2p (AI ).

Hence, due to (9.6),

N2p (V ) ¤ N2p (VR ) + N2p (VI ) ¤ N2p (VI )(1 + γ2p ) = βp N2p (AI ),

as claimed.

Now let A be unbounded. To obtain the stated result in this case it is

su¬cient to apply Corollary 7.6.2 and the just obtained result for bounded

operators. 2

The assertion of Theorem 7.9.1 follows from (5.8), Theorem 7.5.5 and

Lemma 7.9.2.

7.10 Regular Functions of Bounded

Quasi-Hermitian Operators

Let A be a bounded linear operator in a H and let f (z) be a scalar-valued

function which is analytic on some neighborhood of σ(A). Again, put

1

f (A) = ’ f (»)R» (A)d», (10.1)

2πi C

where C is a closed Jordan contour, surrounding σ(A) and having the pos-

itive orientation with respect to σ(A). Below we also consider unbounded

operators.

Theorem 7.10.1 Let a bounded linear operator A satisfy the condition

AI is a Hilbert-Schmidt operator. (10.2)

110 7. Functions of Non-compact Operators

In addition, let f be a holomorphic function on a neighborhood of the closed

convex hull co(A) of σ(A). Then

∞ k

gI (A)

sup |f (k) (»)|

f (A) ¤ . (10.3)

(k!)3/2

k=0 »∈co(A)

Recall that the quantity gI (A) is de¬ned by the equality (7.2).

Theorem 7.10.1 is precise: the inequality (10.3) becomes the equality

f (A) = sup |f (µ)|, (10.4)

µ∈σ(A)

if A is a normal operator and

sup |f (»)| = sup |f (»)|, (10.5)

»∈co(A) »∈σ(A)

because gI (A) = 0 for a normal operator A.

Example 7.10.2 Let a bounded operator A satisfy condition (10.2). Then

Theorem 7.10.1 gives us the inequality

m m’k k

m!rs (A)gI (A)

m

¤

A

(m ’ k)!(k!)3/2

k=0

for any integer m ≥ 1. Recall that rs (A) is the spectral radius of A.

Example 7.10.3 Let a bounded operator A satisfy the condition (10.2).

Then Theorem 7.10.1 gives us the inequality

∞

tk gI (A)

k

At ±(A)t

¤e for all t ≥ 0,

e (10.6)

(k!)3/2

k=0

where ±(A) = sup Re σ(A).

7.11 Proof of Theorem 7.10.1

Let Pk (k = 0, ..., n) be a ¬nite chain of orthogonal projectors:

0 = Range(P0 ) ‚ Range(P1 ) ‚ .... ‚ Range(Pn ) = H.

We need the following

Lemma 7.11.1 Let a bounded operator A in H have the representation

n

φk ∆Pk + V (∆Pk = Pk ’ Pk’1 ),

A=

k=1

7.11. Proof of Theorem 7.10.1 111

where φk (k = 1, ..., n) are numbers and V is a Hilbert-Schmidt operator

satisfying the relations

Pk’1 V Pk = V Pk (k = 1, ..., n).

In addition, let f be holomorphic on a neighborhood of the closed convex hull

co(A) of σ(A). Then

∞ k

N2 (V )

sup |f (k) (»)|

f (A) ¤ .

(k!)3/2

k=0 »∈co(A)

Proof: Put

n

D= φk ∆Pk .

k=1

Clearly, the spectrum of D consists of numbers φk (k = 1, ..., n). It is simple

to check that V n = 0. That is, V is a nilpotent operator. Due to Lemma

7.5.1, σ(D) = σ(A). Consequently, φk (k = 1, ..., n) are eigenvalues of A.

(k)

Furthermore, let {em }∞ be an orthogonal normal basis in ∆Pk H. Put

m=1

l

(k)

(., e(k) )e(k) (k = 1, ..., n; l = 1, 2, ....).

Ql = m m

m=1

(k)

strongly converge to ∆Pk as l ’ ∞. Moreover,

Clearly, Ql

(k) (k) (k)

Ql ∆Pk = ∆Pk Ql = Ql .

Then the operators

n

(k)

Dl = φk Ql

k=1

strongly tend to D as l ’ ∞. We can write out,

n k’1

V= ∆Pi V ∆Pk .

k=1 i=1

Introduce the operators

n k’1

(i) (k)

Wl = Ql V Ql .

k=1 i=1

(k)

Since projectors Ql strongly converge to ∆Pk as l ’ ∞, and V is compact,

operators Wl converge to V in the operator norm. So the ¬nite dimensional

operators

Tl := Dl + Wl

112 7. Functions of Non-compact Operators

strongly converge to A and f (Tl ) strongly converge to f (A). Thus,

f (A) ¤ sup f (Tl ) , (11.1)

l

thanks to the Banach-Steinhaus theorem (see e.g. (Dunford and Schwartz,

1966)). But Wl are nilpotent, and Wl and Dl have the same invariant sub-

spaces. Consequently, due to Lemma 7.5.1,

σ(Dl ) = σ(Tl ) ⊆ σ(A) = {φk }.

The dimension of Tl is nl. Due to Lemma 2.8.2 and Corollary 6.9.2, we have

the inequality

ln’1 k

N2 (Wl )

(k)

f (Tl ) ¤ sup |f (»)| .

(k!)3/2

k=0 »∈co(A)

Letting l ’ ∞ in this inequality, we prove the stated result. 2

Lemma 7.11.2 Let A be a bounded P -triangular operator, whose nilpotent

part V ∈ C2 . Let f be a function holomorphic on a neighborhood of co(A).

Then

∞

N k (V )

sup |f (k) (»)| 2 3/2 .

f (A) ¤

(k!)

k=0 »∈co(A)

Proof: Let D be the diagonal part of A. According to (2.4) and the von

Neumann Theorem (Ahiezer and Glazman, 1981, Section 92), there exists a

bounded measurable function φ, such that

b

D= φ(t)dP (t).

a

De¬ne the operators

n n

Vn = P (tk’1 )V ∆Pk and Dn = φ(tk )∆Pk

k=0 k=1

(n)

(tk = tk , a = t0 ¤ t1 ¤ ... ¤ tn = b; ∆Pk = P (tk ) ’ P (tk’1 ), k = 1, ..., n).

Besides, put Bn = Dn +Vn . Then the sequence {Bn } strongly converges to A

due to the triangular representation (2.2). According to (10.1) the sequence

{f (Bn )} strongly converges to f (A). The inequality

f (A) ¤ sup f (Bn ) (11.2)

n

is true thanks to the Banach-Steinhaus theorem. Since the spectral resolution

of Bn consists of n < ∞ projectors, Lemma 7.11.1 yields the inequality

∞ k

N2 (Vn )

|f (k) (»)|

f (Bn ) ¤ sup . (11.3)

(k!)3/2

k=0 »∈co(Bn )

7.12. Functions of Unbounded Operators 113

Thanks to Lemma 7.5.1 we have σ(Bn ) = σ(Dn ). Clearly, σ(Dn ) ⊆ σ(D).

Hence,

σ(Bn ) ⊆ σ(A). (11.4)

Due to the well-known Theorem III.7.1 from (Gohberg and Krein, 1970),

{N2 (Vn )} tends to N2 (V ) as n tends to in¬nity. Thus (11.2), (11.3) and

(11.4) imply the required result. 2

The assertion of Theorem 7.10.1 immediately follows from Lemmas 7.11.2

and 7.7.2. 2

7.12 Regular Functions of

Unbounded Operators

Let an operator A be unbounded with a dense domain Dom (A) = Dom (A— ).

In addition, let the conditions (10.2) and

β(A) := inf Re σ(A) > ’∞ (12.1)

hold. Let f be analytic on some neighborhood of σ(A) and M an open set

containing σ(A) whose boundary C consists of a ¬nite number of Jordan

arcs such that f is analytic on M ∪ C. Let C have positive orientation with

respect to M . Then we de¬ne the function of the operator A by the equality

1

f (A) = f (∞)I ’ f (»)R» (A)d», (12.2)

2πi C

cf. (Dunford and Schwartz, 1966, p. 601).

Without loss of generality we assume that f (∞) = 0. In the other case

we can consider the function f (») ’ f (∞).

Theorem 7.12.1 Under the conditions D(A) = D(A— ), (10.2) and (12.1),

let f be regular on a neighborhood of co(A) and f (∞) = 0. Then inequality

(10.3) is true.

Proof: Due to Theorem 7.6.1, under (10.2), A admits the triangular repre-

sentation. According to (2.4) and the von Neumann Theorem (Ahiezer and

Glazman, 1981, Section 92), there exists a P -measurable scalar function φ,

such that ∞

D= φ(t)dP (t).

’∞

In addition, as it was shown in the proof of Theorem 7.6.1, the function

Re φ(t) nondecreases as t increases. Thus due to Lemma 7.5.1,

inf Re φ(t) = β(D) = β(A) > ’∞ and sup |Im φ(t)| < ∞.

t t

114 7. Functions of Non-compact Operators

So the operators An = AP (n) are bounded. Moreover, relations σ(An ) ⊆

σ(A) and

(I» ’ A)’1 P (n) = (I» ’ An )’1 P (n) (12.3)

hold. Hence, due to (12.2)

1

f (»)(I» ’ An )’1 d»P (n) = f (An )P (n).

f (A)P (n) =

2πi C

Due to Theorem 7.10.1,

∞ k

gI (An )

|f (k) (»)|

f (An ) ¤ sup .

(k!)3/2

k=0 »∈co(An )

Letting in this relation n ’ ∞, we get inequality (10.3). Thus, the theorem

is proved. 2

Theorem 12.1 is exact. Indeed, under (12.1), let A be normal and (10.5)

hold. Then we have equality (10.4), provided f (∞) = 0.

Furthermore, under conditions (12.1) and (10.2), put

c0 +i∞

1

’At

et» (I» + A)’1 d» (c0 > ’β(A)).

e := (12.4)

2πi c0 ’i∞

Since the non-real spectrum of A is bounded, the integral in (12.4) converges

in the sense of the Laplace transformation. Clearly, function ezt is non-

analytic at in¬nity. Let An = AP (n). According to (12.4) and (12.3),

c0 +i∞

1

’At

et» (I» + An )’1 P (n)d» = e’An t P (n).

e P (n) =

2πi c0 ’i∞

Due to Theorem 7.10.1

∞

tk gI (An )

k

’tβ(An )

exp [’An t] ¤ e ,

(k!)3/2

k=0

since An is bounded. Letting in this relation n ’ ∞, we get

Theorem 7.12.2 Let conditions (10.2) and (12.1) hold. Then

∞

tk gI (A)

k

’tβ(A)

exp [’At] ¤ e (t ≥ 0).

(k!)3/2

k=0

This theorem is exact. Indeed, let A be normal. Then we have

exp [’At] = e’tβ(A) , t ≥ 0.

7.13. Triangular Representations of Functions 115

For a scalar-valued function h de¬ned on [0, ∞), let the integral

∞

e’At h(t)dt

¦(A) =

0

strongly converges. Denote,

∞

k

gI (A)

e’tβ(A) |h(t)|tk dt (k = 0, 1, ...).

θk (¦, A) :=

(k!)3/2 0

Then due to Theorem 7.12.2,

∞

¦(A) ¤ θk (¦, A),

k=0

provided the integrals and series converge. In particular, let

β(A) > 0. (12.5)

Then ∞

1

’m

e’At tm’1 dt (m = 1, 2, ...).

A =

(m ’ 1)! 0

In this case

∞

k

gI (A)

e’tβ(A) tm+k’1 dt.

θk (¦, A) =

(m ’ 1)!(k!)3/2 0

Therefore

∞ k

(k + m ’ 1)!gI (A)

’m

¤

A . (12.6)

(m ’ 1)!(k!)3/2 β(A)m+k

k=0

In addition, under (12.5)

∞

1

’ν

e’At tν’1 dt (0 < ν < 1),

A =

“(ν) 0

where “(.) is the Euler Gamma-function. Hence,

∞ k

“(ν + k)gI (A)

’ν

¤

A . (12.7)

“(ν)(k!)3/2 β(A)k+ν

k=0

7.13 Triangular Representations of

Regular Functions

Lemma 7.13.1 Let a bounded operator A admit a triangular representation

with some maximal resolution of the identity. In addition, let f be a function

analytic on a neighborhood of σ(A). Then the operator f (A) admits the

triangular representation with the same maximal resolution of the identity.

Moreover, the diagonal part Df of f (A) is de¬ned by Df = f (D), where D

is the diagonal part of A.

116 7. Functions of Non-compact Operators

Proof: Due to representations (10.1) and (2.2),

1 1

f (»)R» (D)(I + V R» (D))’1 d».

f (A) = ’ f (»)R» (A)d» = ’

2πi 2πi

C C

Consequently,

1

f (A) = ’ f (»)R» (D)d» + W = f (D) + W, (13.1)

2πi C

where

1

f (»)R» (D)[(I + V R» (D))’1 ’ I]d».

W =’

2πi C

But

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

We thus get

1

W= f (»)ψ(»)d»,

2πi C

where

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

Let P (.) be a maximal resolution of the identity of A. Due to Lemma 7.3.3,

for each » ∈ C, ψ(») is a Volterra operator with the same m.r.i. Since P (t)

is a bounded operator, we have by Lemma 7.3.2

(P (t0 + 0) ’ P (t0 ))W (P (t0 + 0) ’ P (t0 )) =

1

f (»)(P (t0 + 0) ’ P (t0 ))ψ(»)(P (t0 + 0) ’ P (t0 ))d» = 0

2πi C

for every gap P (t0 + 0) ’ P (t0 ) of P (t). Thus, W is a Volterra operator

thanks to Lemma 7.3.1. This and (13.1) prove the lemma. 2

7.14 Triangular Representations

of Quasiunitary Operators

A bounded linear operator A is called a quasiunitary operator, if A— A ’ I is

a completely continuous operator.

Lemma 7.14.1 Let A be a linear operator in H satisfying the condition

A— A ’ I ∈ Cp (1 ¤ p < ∞), (14.1)

and let the operator I ’ A be invertible. Then the operator

B = i(I ’ A)’1 (I + A) (14.2)

(Cayley™s transformation of A) is bounded and satis¬es the condition B ’

B — ∈ Cp .

7.15. Functions of Quasiunitary Operators 117

Proof: According to (14.1) we can write down

A = U (I + K0 ) = U + K,

where U is a unitary operator and both operators K and K0 belong to Cp

—

and K0 = K0 . Consequently,

B — = ’i(I + U — + K — )(I ’ U — ’ K — )’1 =

’i(I + U ’1 + K — )(I ’ U ’1 ’ K — )’1 .

That is, B — = ’i(I + U + K0 )(U ’ I ’ K0 )’1 , since K0 = K — U . But (14.2)

clearly forces

B = i(I + U + K)(I ’ U ’ K)’1 .

Thus, 2BI = (B ’ B — )/i = T1 + T2 , where

T1 = (I + U )[(I ’ U ’ K)’1 + (U ’ I ’ K0 )’1 ]

and

T2 = K(I ’ U ’ K)’1 + K0 (U ’ I ’ K0 )’1 .

Since K, K0 ∈ Cp , we conclude that T2 ∈ Cp . It remains to prove that

T1 ∈ Cp . Let us apply the identity

(I ’ U ’ K)’1 + (U ’ I ’ K0 )’1 = ’(I ’ U ’ K)’1 (K0 + K)(U ’ I ’ K0 )’1 .

Hence, T1 ∈ Cp . This completes the proof. 2

Lemma 7.14.2 Under condition (14.1), let A have a regular point on the

unit circle. Then A admits the triangular representation (2.2)-(2.4).

Proof: Without any loss of generality we assume that A has on the unit

circle a regular point »0 = 1. In the other case we can consider instead of A

the operator A»’1 .

0

Let us consider in H the operator B de¬ned by (14.2). By the previous

lemma it satis¬es (14.2) and therefore due to Theorem 7.6.1 has a triangular

representation. The transformation inverse to (14.2) must be de¬ned by the

formula

A = (B ’ iI)(B + iI)’1 . (14.3)

Now Lemma 7.13.1 ensures the result. 2

7.15 Resolvents and Analytic Functions

of Quasiunitary Operators

Assume that A has a regular point on the unit circle and

AA— ’ I is a nuclear operator. (15.1)

118 7. Functions of Non-compact Operators

Due to Lemma 7.14.1, from (1.1), (14.3) and (15.1) it follows that

∞

(|»k (A)|2 ’ 1) < ∞,

k=1

where »k (A), k = 1, 2, ... are the nonunitary eigenvalues with their multiplic-

ities, that is, the eigenvalues with the property |»k (A)| = 1. Under (15.1)

put

∞

—

(|»k (A)|2 ’ 1)]1/2 .

‘(A) = [T r (A A ’ I) ’

k=1

If A is a normal operator, then ‘(A) = 0. Let A have the unitary spectrum,

only. That is, σ(A) lies on the unit circle. Then

‘(A) = [T r (A— A ’ I)]1/2 .

Moreover, if the condition

∞

(|»k (A)|2 ’ 1) ≥ 0 (15.2)

k=1

holds, then

∞ ∞

—

(s2 (A) (|»k (A)|2 ’ 1) = T r (D— D ’ I) ≥ 0

T r (A A ’ I) = ’ 1) ≥

k

k=1 k=1

and therefore, under (15.2),

‘(A) ¤ [T r (A— A ’ I)]1/2 . (15.3)

Theorem 7.15.1 Under condition (15.1), let an operator A have a regular

point on the unit circle. Then

∞

‘k (A)

√

R» (A) ¤ (» ∈ σ(A)). (15.4)

k!ρk+1 (A, »)

k=0

Moreover, there are constants a0 , b0 > 0, independent of », such that

b0 ‘2 (A)

a0

R» (A) ¤ ] (» ∈ σ(A)).

exp [ 2 (15.5)

ρ(A, ») ρ (A, »)

These constants can be taken from (5.10) or from (5.11).

To prove this theorem we need the following

Lemma 7.15.2 Under condition (15.1), let an operator A have a regular

point on the unit circle. Then A admits the triangular representation (2.2)

(due to Lemma 7.14.2). Moreover, N2 (V ) = ‘(A), where V is the nilpotent

part of A.

7.15. Functions of Quasiunitary Operators 119

Proof: By Lemma 7.3.3 and Theorem 6.2.2 T r (D— V ) = T r (V — D) = 0.

Employing the triangular representation (2.2) we obtain

T r (A— A ’ I) = T r [(D + V )— (D + V ) ’ I)] =

T r (D— D ’ I) + T r (V — V ).

Since D is a normal operator and the spectra of D and A coincide due to

Lemma 7.5.1, then we can write

∞

—

(|»k (A)|2 ’ 1).

T r (D D ’ I) =

k=1

This equality implies the required result. 2

Proof of Theorem 7.15.1: Inequality (15.4) follows from relation (5.7)

and Lemma 7.15.2. Inequality (15.5) follows from Theorem 7.5.5 and Lemma

7.15.2. 2

Let us extend Theorem 7.15.1 to the case

Ap (A— )p ’ I ∈ C1 (15.6)

for some integer number p > 1.

Corollary 7.15.3 Under condition (15.6), let an operator A have a regular

point on the unit circle. Then

∞

‘k (Ap )

(»p ∈ σ(Ap )),

√

R» (A) ¤ T»,p

k+1 (Ap , »p ) k!

ρ

k=0

where

p’1

Ak »p’k’1 and ρ(Ap , »p ) = inf |tp ’ »p |.

T»,p =

t∈σ(A)

k=0

Moreover, there are constants a0 , b0 > 0, independent of », such that

b0 ‘2 (Ap )

a0 T»,p

exp [ 2 p p ] (»p ∈ σ(Ap )).

R» (A) ¤ p , »p )

ρ(A ρ (A , » )

These constants can be taken from (5.10) or from (5.11).

This result is due Theorem 7.15.1 and identity (8.3).

Theorem 7.15.4 Let a linear operator A satisfy condition (15.1) and have

a regular point on the unit circle. If, in addition, f is a holomorphic function

on a neighborhood of closed convex hull co(A) of σ(A), then

∞

‘k (A)

(k)

f (A) ¤ sup |f (»)| . (15.7)

(k!)3/2

k=0 »∈co(A)

120 7. Functions of Non-compact Operators

Proof: The result immediately follows from Lemmas 7.15.2 and 7.11.2. 2

Theorem 7.15.4 is precise: inequality (15.7) becomes equality (10.4) if A

is a unitary operator and (10.5) holds, because ‘(A) = 0 in this case.

Example 7.15.5 Let an operator A satisfy the condition (15.1) and have a

regular point on the unit circle. Then the inequality

m

m!rs (A)‘k (A)

m’k

m

¤

A

(m ’ k)!(k!)3/2

k=0

holds for any integer m ≥ 1. Recall that rs (A) is the spectral radius of A.

7.16 Notes

Notions similar to De¬nitions 7.1.1, 7.2.1 and 7.2.2 can be found in the books

(Gohberg and Krein, 1970) and (Brodskii, 1971), as well as in the papers

(Branges, 1963, 1965a and 1965b) and (Brodskii, Gohberg, and Krein, 1969).

Theorem 7.5.3 is probably new.

The results presented in Sections 7.6-7.8 are based on Chapter 3 of the

book (Gil™, 1995), but the proofs are considerably improved.

Theorem 7.9.1 was established in the paper (Gil™, 1993). Theorems 7.10.1

and 7.12.1 were derived in the paper (Gil™, 1992).

Triangular representations of quasi-Hermitian and quasiunitary operators

can be found, in particular, in the paper by V. Brodskii, I. Gohberg and M.

Krein (1969), and references given therein.

References

[1] Ahiezer, N. I. and Glazman, I. M. (1981). Theory of Linear Op-

erators in a Hilbert Space. Pitman Advanced Publishing Program,

Boston, London, Melburn.

[2] Branges, L. de. (1963). Some Hilbert spaces of analytic functions

I, Proc. Amer. Math. Soc. 106, 445-467.

[3] Branges, L. de. (1965a). Some Hilbert spaces of analytic functions

II, J. Math. Analysis and Appl., 11, 44-72.

[4] Branges, L. de. (1965b). Some Hilbert spaces of analytic functions

III, J. Math. Analysis and Appl., 12, 149-186.

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

Linear Operators, Transl. Math. Mongr., v. 32, Amer. Math. Soc.,

Providence, R.I.

7.13. Notes 121

[6] Brodskii, V.M., Gohberg, I.C. and Krein M.G. (1969). General the-

orems on triangular representations of linear operators and mul-

tiplicative representations of their characteristic functions, Funk.

Anal. i Pril., 3,1-27 (in Russian); English Transl., Func. Anal.

Appl. 3, 255-276.

[7] Dunford, N and Schwartz, J. T. (1966). Linear Operators, part I.

General Theory. Interscience publishers, New York, London.

[8] Dunford, N and Schwartz, J. T. (1963). Linear Operators, part II.

Spectral Theory. Interscience publishers, New York, London.