Indeed, under the consideration, VA = 0, WA = 0. So

VA (I + VA )’1 = VA and WA (I + WA )’1 = WA .

Hence θ = VA WA . Now Lemma 5.4.2 yields the required result.

Furthermore, Lemmas 5.3.1 and 5.2.3 yield the following:

72 5. Block Matrices

Lemma 5.4.4 The inequalities

(I + VA )’1 ¤ M (VA ) := (1 + V D’1 ∆Pk )

2¤k¤m

and

(I + WA )’1 ¤ M (WA ) := (1 + W D’1 ∆Pk )

1¤k¤m’1

are valid.

5.5 Proof of Theorem 5.1.1

In the present section, D, V and W are the diagonal, upper diagonal and

lower diagonal parts of the matrix in (1.1), respectively, that is,

« «

A11 0 ... 0 0 A12 . . . A1m

¬0 0· ¬ . . . A2m ·

A22 . . . ·, V = ¬ 0 0

D=¬ · , (5.1)

. . . ... .

... . .

0 0 . . . Amm 0 0 ... 0

«

and

0 0 ... 0

¬ A21 ... 0 ·

0

W =¬ ·. (5.2)

. .

... .

Am1 Am2 ... 0

Recall that VA ≡ V D’1 , WA ≡ W D’1 .

Lemma 5.5.1 Let D be invertible. Then the inequalities

(I + VA )’1 ¤ Mup (5.3)

∞

and

(I + WA )’1 ¤ Mlow (5.4)

∞

are valid.

ˆ

Proof: Let π = {Pk , k = 1, .., m} be the chain of the projectors onto the

ˆ

standard basis. That is, for an h = (hk ) ∈ Cn ,

ˆ

Pk h = (h1 , ...., hνk , ..., 0),

ˆ

where νk = dim Pk . Then according to (5.1) D and V are π -diagonal and

ˆ

π -nilpotent operators, respectively. Moreover,

ˆ

k’1

∆Pk = Vk A’1 ∆Pk = Ajk A’1 ∆Pk .

’1 ˆ ˆ ˆ

VD kk kk

j=1

5.5. Proof of Theorem 5.1.1 73

But, clearly,

k’1

up

Ajk A’1 ∆Pk = max Ajk A’1

ˆ = vk .

∞ ∞

kk kk

j

j=1

Therefore inequality (5.3) is due to the previous lemma. Inequality (5.4) can

be similarly proved. 2

Lemma 5.5.2 Let D be invertible. Then the inequalities

VA (I + VA )’1 ¤ Mup ’ 1 (5.5)

∞

and

WA (I + WA )’1 ¤ Mlow ’ 1 (5.6)

∞

are valid.

Let B = (bjk )n be a positive matrix with the property

Proof: k=1

Bh ≥ h (5.7)

for any nonnegative h ∈ Cn . Then

n

B’I bjk ’ δjk ] = B ’ 1.

= max [ (5.8)

∞ ∞

j=1,...,n

k=1

Here δjk is the Kronecker symbol. Furthermore, since VA is nilpotent,

n’1

’1

|VA |k = (I ’ |VA |)’1 ’ I

¤

VA (I + VA ) ∞,

∞ ∞

k=1

where |VA | is the matrix whose entries are the absolute values of the entries

of VA . Moreover,

n’1

|VA |k h ≥ h

k=0

for any nonnegative h ∈ Cn . So according to (5.7) and (5.8)

VA (I + VA )’1 ¤ (I ’ |VA |)’1 ’ 1.

∞ ∞

Since

ˆ ˆ

|VA |∆Pk = VA ∆Pk ∞,

∞

inequality (5.3) with ’|VA | instead of VA yields inequality (5.5). Inequality

(5.6) can be proved similarly. 2

The assertion of Theorem 5.1.1 follows from Lemmas 5.4.2, 5.5.1 and

5.5.2.

74 5. Block Matrices

5.6 Notes

Many books and papers are devoted to the invertibility of block matrices, cf.

(Feingold and Varga, 1962), (Fiedler, 1960), (Gantmacher, 1967), (Ostrowski,

1961), etc. In these works, the Hadamard theorem is mainly generalized to

block matrices. Note that the generalized Hadamard theorem does not assert

that a block triangular matrix with nonsingular diagonal blocks is invertible.

But it is not hard to check that such a matrix is always invertible.

Theorem 5.1.1 gives us the invertibility conditions which improve well-

known results for matrices that are ”close” to block triangular matrices.

Moreover, we derive an estimate for the norm of the inverse matrices.

Chapter 5 is based on the paper (Gil™, 2002).

References

[1] Feingold D.G. and Varga, R.S. (1962). Block diagonally dominant ma-

trices and generalization of the Gershgorin circle theorem. Paci¬c J. of

Mathematics, 12, 1241-1250.

[2] Fiedler, M. (1960) Some estimates of spectra of matrices. Symposium

of the Numerical Treatment of Ordinary Di¬erential Equations, Integral

and Integro- Di¬erential Equations, Birkh´user Verlag, Rome, 33-36.

a

[3] Gantmacher, F. R. (1967). Theory of Matrices, Nauka, Moscow (in Rus-

sian).

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

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

[5] Horn, R. A. and Johnson, C. R. (1991). Topics in Matrix Analysis ,

Cambridge University Press, Cambridge.

[6] Ostrowski, A.M. (1961). On some metrical properties of operator matri-

ces and matrices partitioned into blocks. J. Math. Ann. Appl., 2, 161-

209.

6. Norm Estimates for

Functions of

Compact Operators in a

Hilbert Space

The present chapter contains the estimates for the norms of the resolvents

and analytic functions of Hilbert-Schmidt operators and resolvents of von

Neumann-Schatten operators.

6.1 Bounded Operators in a Hilbert Space

In this section we recall very brie¬‚y some basic notions of the theory of

operators in a Hilbert space. More details can be found in any textbook on

Hilbert spaces (e.g. (Ahiezer and Glazman, 1981), (Dunford and Schwartz,

1963) ).

In the sequel H denotes a separable Hilbert space with a scalar product

(., .) and the norm

(h, h) (h ∈ H).

h=

A sequence {hn } of elements of H converges strongly (in the norm) to h ∈ H

if hn ’ h ’ 0 as n ’ ∞. Any separable Hilbert space possesses an

orthonormal basis. This means that there is a sequence {ek ∈ H}, such that

(ek , ej ) = 0 if j = k and (ek , ek ) = 1 (j, k = 1, 2, ...)

M.I. Gil™: LNM 1830, pp. 75“96, 2003.

c Springer-Verlag Berlin Heidelberg 2003

76 6. Functions of Compact Operators

and any h ∈ H can be represented as

∞

h= ck ek

k=1

with ck = (h, ek ) (k = 1, 2, . . .). In addition, this series strongly converges.

If the closed linear span of vectors {vk ∈ H}∞ coincides with H, then the

k=1

set of these vectors is said to be complete in H.

A linear operator A acting in H is called a bounded one, if there is a

constant a such that

Ah ¤ a h for all h ∈ H.

The quantity

Ah

A = sup

h

h∈H

is called the norm of A. A sequence {An } of bounded linear operators con-

verges strongly to an operator A, if the sequence of elements {An h} strongly

converges to Ah for every h ∈ H. {An } converges in the uniform operator

topology (in the operator norm ) to an operator A, if An ’ A ’ 0 as

n ’ ∞. A bounded linear operator A— is called adjoint to A, if

(Af, g) = (f, A— g) for every h, g ∈ H.

The relation A = A— is true. A bounded operator A is a selfadjoint one,

if A = A— . A is a unitary operator, if AA— = A— A = I. Here and below

I ≡ IH is the identity operator in H. A selfadjoint operator A is positive

(negative) de¬nite, if

(Ah, h) ≥ 0 ((Ah, h) ¤ 0) for every h ∈ H.

A selfadjoint operator A is strongly positive (strongly negative) de¬nite, if

there is a constant c > 0, such that

(Ah, h) ≥ c (h, h) ((Ah, h) < ’c (h, h)) for every h ∈ H.

A bounded linear operator satisfying the relation AA— = A— A is called a

normal operator. It is clear that unitary and selfadjoint operators are ex-

amples of normal ones. The operator B ≡ A’1 is the inverse one to A, if

AB = BA = I. An operator P is called a projector if P 2 = P . If, in addition,

P — = P , then it is called an orthogonal projector (an orthoprojector).

A point » of the complex plane is said to be a regular point of an operator

A, if the operator R» (A) ≡ (A ’ I»)’1 (the resolvent) exists and is bounded.

The complement of all regular points of A in the complex plane is the spectrum

of A. The spectrum of A is denoted by σ(A). The spectrum of a selfadjoint

operator is real, the spectrum of a unitary operator lies on the unit circle.

6.2. Compact Operators in a Hilbert Space 77

The quantity

rs (A) = sups∈σ(A) |s|

is the spectral radius of A. An operator V is called a quasinilpotent one, if

its spectrum consists of zero, only. If there is a nontrivial solution e of the

equation

Ae = »(A)e,

where »(A) is a number, then this number is called an eigenvalue of operator

A, and e ∈ H is an eigenvector corresponding to »(A). Any eigenvalue is a

point of the spectrum. An eigenvalue »(A) has the (algebraic) multiplicity

r ¤ ∞ if

dim(∪∞ ker(A ’ »(A)I)k ) = r.

k=1

In the sequel »k (A), k = 1, 2, ... are the eigenvalues of A repeated according

to their multiplicities.

A vector v satisfying (A ’ »(A)I)n v = 0 for a natural n, is a root vector

of operator A corresponding to »(A).

6.2 Compact Operators in a Hilbert Space

All the results, presented in this section can be found, for instance, in (Go-

hberg and Krein, 1969, Chapters 2 and 3). The set of all linear completely

continuous (compact) operators in H is de¬ned by C∞ .

Recall that the spectrum of an operator from C∞ is either ¬nite, or the

sequence of the eigenvalues of A converges to zero, any nonzero eigenvalue

has the ¬nite multiplicity. Moreover, any normal operator A ∈ C∞ can be

represented in the form

∞

A= »k (A)Ek , (2.1)

k=1

where Ek are eigenprojectors of A, i.e. the projectors de¬ned by Ek h =

(h, dk )dk for all h ∈ H. Here dk are the normal eigenvectors of A. Recall

that eigenvectors of normal operators are mutually orthogonal. A completely

continuous positive de¬nite selfadjoint operator has non-negative eigenvalues,

only. Let A ∈ C∞ be positive de¬nite and represented by (2.1). Then we

write

∞

»β (A)Ek (β > 0).

β

A := k

k=1

A completely continuous quasinilpotent operator is called a Volterra operator.

Let {ek } be an orthogonal normal basis in H, and the series

∞

(Aek , ek ) (A ∈ C∞ )

k=1

78 6. Functions of Compact Operators

converges. Then the sum of this series is called the trace of A:

∞

T race A = T r A = (Aek , ek ).

k=1

An operator A satisfying the condition

T r (A— A)1/2 < ∞

is called a nuclear operator. An operator A, satisfying the relation

T r (A— A) < ∞

is said to be a Hilbert-Schmidt operator.

The eigenvalues »k ((A— A)1/2 ) (k = 1, 2, ...) of the operator (A— A)1/2

are called the singular numbers (s-numbers) of A and are denoted by sk (A).

That is,

sk (A) ≡ »k ((A— A)1/2 ) (k = 1, 2, ...).

Enumerate singular numbers of A taking into account their multiplicity and

in decreasing order. The set of completely continuous operators acting in a

Hilbert space and satisfying the condition

∞

sp (A) ]1/p < ∞,

Np (A) := [ k

k=1

for some p ≥ 1, is called the von Neumann - Schatten ideal and is denoted by

Cp . Np (.) is called the norm of the ideal Cp . It is not hard to show that

T r (AA— )p/2 .

p

Np (A) =

Thus, C1 is the ideal of nuclear operators (the Trace class) and C2 is the

ideal of Hilbert-Schmidt operators. N2 (A) is called the Hilbert-Schmidt norm.

Sometimes we will omit index 2 of the Hilbert-Schmidt norm, i.e.

T r (A— A).

N (A) := N2 (A) =

For any orthogonal normal basis {ek } we can write

∞

2 1/2

N2 (A) = ( Aek ) .

k=1

This equality is equivalent to the following one:

∞

|ajk |2 )1/2 ,

N2 (A) = ( (2.2)

j,k=1

6.3. Triangular Representations 79

where ajk = (Aek , ej ) (j, k = 1, 2, . . .) are entries of a Hilbert-Schmidt oper-

ator A in basis {ek }.

For all p ≥ 1, the following propositions are true (the proofs can be found

in the books (Gohberg and Krein, 1969, Section 3.7), and (Pietsch, 1988)):

If A ∈ Cp , then also A— ∈ Cp . If A ∈ Cp and B is a bounded linear

operator, then both AB and BA belong to Cp . Moreover,

Np (AB) ¤ Np (A) B and Np (BA) ¤ Np (A) B .

In addition, the inequality

n n

sp (A) (n = 1, 2, . . .)

p

|»j (A)| ¤ (2.3)

j

j=1 j=1

is valid, cf. (Gohberg and Krein, 1969, Theorem II.3.1).

Lemma 6.2.1 If A ∈ Cp and B ∈ Cq (1 < p, q < ∞), then AB ∈ Cs with

1 11

=+.

s pq

Moreover,

Ns (AB) ¤ Np (A)Nq (B). (2.4)

For the proof of this lemma see (Gohberg and Krein, 1969, Section III.7).

We need also the following result (Lidskij™s theorem).

Theorem 6.2.2 The trace of A ∈ C1 does not depend on a choice of an

orthogonal normal basis and

∞

Tr A = »k (A).

k=1

The proof of this theorem can be found in (Gohberg and Krein, 1969, Section

III.8).

6.3 Triangular Representations

of Compact Operators

Let R0 be a set in the complex plane and let > 0. By S(R0 , ) we denote

the -neighborhood of R0 . That is,

dist{R0 , S(R0 , )} ¤ .

80 6. Functions of Compact Operators

Lemma 6.3.1 Let A be a bounded operator and let > 0. Then there is a

δ > 0, such that, if a bounded operator B satis¬es the condition A ’ B ¤ δ,

then σ(B) lies in S(σ(A), ) and

R» (A) ’ R» (B) ¤

for any », which does not belong to S(σ(A), ).

For the proof of this lemma we refer the reader to the book (Dunford and

Schwartz, 1963, p. 585).

Lemma 6.3.2 Let V ∈ Cp , p > 1 be a Volterra operator. Then there is

a sequence of nilpotent operators, having ¬nite dimensional ranges and con-

verging to V in the norm Np (.).

Proof: Let T = V ’ V — . Due to the well-known Theorems 22.1 and 16.3

from the book (Brodskii, 1971), for an > 0, there is a ¬nite chain {Pk }n

k=0

of orthogonal projectors onto invariant subspaces of V :

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

such that with the notation

n

Pk’1 T ∆Pk (∆Pk = Pk ’ Pk’1 ),

Wn =

k=1

(k)

the inequality Np (Wn ’ V ) < is valid. Furthermore, let {em }∞ be an

m=1

orthonormal basis in ∆Pk H. Put

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 .

Since,

n k’1

Wn = ∆Pj T ∆Pk ,

k=1 j=1

the operators

n k’1

(j) (k)

Wnl = Ql T Ql

k=1 j=1

have ¬nite dimensional ranges and tend to Wn in the norm Np as l ’ ∞,

since T ∈ Cp . Thus, Wnl tend to V in the norm Np as l, n ’ ∞. Put

k

(l) (j)

Lk = Ql (k = 1, ..., n).

j=1

6.3. Triangular Representations 81

(l) (l) (l) n

Then Lk’1 Wnl Lk = Wnl Lk . Hence we easily have Wnl = 0. This proves

the lemma. 2

We recall the following well-known result, cf. (Gohberg and Krein, 1969,

Lemma I.4.2).

Lemma 6.3.3 Let M = H be the closed linear span of all the root vectors

of an operator A ∈ C∞ and let QA be the orthogonal projector of H onto

M ⊥ , where M ⊥ is the orthogonal complement of M in H. Then QA AQA is

a Volterra operator.

The previous lemma means that A can be represented by the matrix

BA A12

A= (3.1)

0 V1

acting in M • M ⊥ . Here BA = A(I ’ QA ), V1 = QA AQA is a Volterra

operator in QA H and A12 = (I ’ QA )AQA .

Theorem 6.3.4 Let A ∈ C∞ . Then there are a normal operator D and a

Volterra operator V , such that

A = D + V and σ(D) = σ(A). (3.2)

Moreover, A, D and V have the same invariant subspaces.

Proof: Let M be the linear closed span of all the root vectors of A, and

PA is the projector of H onto M . So the system of the root vectors of the

operator BA = APA is complete in M . Thanks to the well-known Lemma

I.3.1 from (Gohberg and Krein, 1969), there is an orthonormal basis (Schur™s

basis) {ek } in M , such that

j’1

BA ej = Aej = »j (BA )ej + ajk ek (j = 1, 2, ...). (3.3)

k=1

We have BA = DB + VB , where DB ek = »k (BA )ek , k = 1, 2, ... and VB =

BA ’ DB is a quasinilpotent operator. But according to (3.1) »k (BA ) =

»k (A), since V1 is a quasinilpotent operator. Moreover DB and VB have

the same invariant subspaces. Take the following operator matrix acting in

M • M ⊥:

DB 0 VB A12

D= and V = .

0 0 0 V1

Since the diagonal of V contains VB and V1 only, σ(V ) = σ(VB )∪σ(V1 ) = {0}.

So V is quasinilpotent and (3.2) is proved. From (3.1) and (3.3) it follows

that A, D and V have the same invariant subspace, as claimed. 2

82 6. Functions of Compact Operators

De¬nition 6.3.5 Equality (3.2) is said to be the triangular representation

of A. Besides, D and V will be called the diagonal part and nilpotent part of

A, respectively.

Lemma 6.3.6 Let A ∈ Cp , p ≥ 1. Let V be the nilpotent part of A. Then

there exists a sequence {An } of operators, having n-dimensional ranges, such

that

σ(An ) ⊆ σ(A), (3.4)

and

n ∞

p

|»(A)|p as n ’ ∞.

|»(An )| ’ (3.5)

k=1 k=1

Moreover,

Np (An ’ A) ’ 0 and Np (Vn ’ V ) ’ 0 as n ’ ∞, (3.6)

where Vn are the nilpotent parts of An (n = 1, 2, ...).

Proof: Again, let M be the linear closed span of all the root vectors of A,

and PA the projector of H onto M . So the system of root vectors of the

operator BA = APA is complete in M . Let DB and VB be the nilpotent

parts of BA , respectively. According to (3.3), put

n

Pn = (., ek )ek .

k=1

Then

σ(BA Pn ) = σ(DB Pn ) = {»1 (A), ..., »n (A)}. (3.7)

In addition, DB Pn and VB Pn are the diagonal and nilpotent parts of BA Pn ,

respectively. Due to Lemma 6.3.2, there exists a sequence {Wn } of nilpotent

operators having n-dimensional ranges and converging in Np to the operator

V1 . Put

BA Pn Pn A12

An = .

0 Wn

Then the diagonal part of An is

DB Pn 0

Dn =

0 0

and the nilpotent part is

V B Pn Pn A12

Vn = .

0 Wn

So relations (3.6) are valid. According to (3.7), relation (3.5) holds. Moreover

Np (Dn ’DB ) ’ 0. So relation (3.5) is also proved. This ¬nishes the proof. 2

6.4. Resolvents of Hilbert-Schmidt Operators 83

6.4 Resolvents of Hilbert-Schmidt Operators

Let A be a Hilbert-Schmidt operator. The following quantity plays a key role

in this section:

∞

2

|»k (A)|2 ]1/2 ,

’

g(A) = [N2 (A) (4.1)

k=1

where N2 (A) is the Hilbert-Schmidt norm of A, again. Since

∞ ∞

2

»2 (A)| = |T race A2 |,

|»k (A)| ≥ | k

k=1 k=1

one can write

g 2 (A) ¤ N2 (A) ’ |T race A2 |.

2

(4.2)

If A is a normal Hilbert-Schmidt operator, then g(A) = 0, since

∞

2

|»k (A)|2

N2 (A) =

k=1

in this case. Let AI = (A ’ A— )/2i. We will also prove the inequality

N2 (A ’ A— )

2

2 2

g (A) ¤ = 2N2 (AI ) (4.3)

2

(see Lemma 6.5.2 below). Again put ρ(A, ») := inf t∈σ(A) |» ’ t|.

Theorem 6.4.1 Let A be a Hilbert-Schmidt operator. Then

∞

g k (A)

√

R» (A) ¤ (» ∈ σ(A)). (4.4)

ρk+1 (A, ») k!

k=0

Proof: Due to Lemma 6.3.6 there exists a sequence {An } of operators,

having n-dimension ranges, such that the relations (3.4),

N2 (An ) ’ N2 (A) and g(An ) ’ g(A) as n ’ ∞ (4.5)

are valid. But due to Corollary 2.1.2,

n’1

g k (An )

√

R» (An ) ¤ (» ∈ σ(An )).

ρk+1 (An , ») k!

k=0

According to (3.4) ρ(An , ») ≥ ρ(A, »). Now, letting n ’ ∞ in the latter

relation, we arrive at the stated result. 2

An additional proof of this theorem can be found in (Gil™, 1995, Chapter

2).

84 6. Functions of Compact Operators

Theorem 6.4.1 is precise. Inequality (4.4) becomes the equality

R» (A) = ρ’1 (A, »),

if A is a normal operator, since g(A) = 0 in this case.

Note that for an arbitrary constant c > 1, Schwarz™s inequality implies

the relations

∞ √k k

∞ ∞ k 2k ∞

xk cx cx 1 1/2 c 2

ecx /2 (x ≥ 0).

√= √ ¤ =

k! j=0 cj c’1

k! k=0 ck k!

k=0 k=0

(4.6)

With c = 2, we have

∞

√

xk 2

√ ¤ 2 ex (x ≥ 0).

k!

k=0

Now Theorem 6.4.1 implies the inequality

b0 g 2 (A)

a0

R» (A) ¤ exp [ 2 ] for all regular », (4.7)

ρ(A, ») ρ (A, »)

where according to (4.6), one can take.

√

c c

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

c’1 2

(4.8)

Moreover, letting n ’ ∞ in Theorem 2.14.1, we get (4.7) with

a0 = e1/2 and b0 = 1/2. (4.9)

We thus have proved

Theorem 6.4.2 Let A ∈ C2 . Then there are nonnegative constants a0 , b0 ,

such that estimate (4.7) is valid. Moreover, a0 and b0 can be taken as in

(4.8) or in (4.9).

In particular, if V ∈ C2 is a quasinilpotent operator, then

2

a0 b0 N2 (V )

R» (V ) ¤ exp [ ] for all » = 0.

|»|2

|»|

6.5 Equalities for Eigenvalues

of a Hilbert-Schmidt Operator

Lemma 6.5.1 Let V be a Volterra operator and VI ≡ (V ’ V — )/2i ∈ C2 .

2 2

Then V ∈ C2 . Moreover, N2 (V ) = 2N2 (VI ).

6.5. Equalities for Eigenvalues 85

Proof: By Theorem 6.2.2 we have T race V 2 = T race (V — )2 = 0, because

V is a Volterra operator. Hence,

N2 (V ’ V — ) = T race (V ’ V — )2 = T race (V 2 + V V — + V — V + (V — )2 )

2

= T race (V V — + V — V ) = 2T race (V V — ).

We arrive at the result. 2

Lemma 6.5.2 Let A ∈ C2 . Then

∞ ∞

2 2 2

|Im »k (A)|2 = N2 (V ),

2

’ |»k (A)| = ’2

N2 (A) 2N2 (AI )

k=1 k=1

where V is the nilpotent part of A.

Proof: Let D be the diagonal part of A. By (3.3), it is simple to check that

V D— is a Volterra operator (see also Lemma 7.3.4). By Theorem 6.2.2,

T race V D— = T race V — D = 0. (5.1)

From the triangular representation (3.2) it follows that

T r AA— = T r (D + V )(D— + V — ) = T r AA— = T r (DD— ) + T r (V V — ).

Besides, due to (3.2) σ(A) = σ(D). Thus,

∞

2

|»k (A)|2 .

N2 (D) =

k=1

So the relation

∞

2 2

|»k (A)|2

’

N2 (V )= N2 (A)

k=1

is proved. Furthermore, from the triangular representation (3.2) it follows

that

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

I

Hence, thanks to (5.1), 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

2 2 2

2N2 (AI ) ’ 2N2 (DI ) = N2 (V ).

86 6. Functions of Compact Operators

Besides, due to (3.2) σ(A) = σ(D). Thus,

∞

2

|Im »k (A)|2 ,

N (DI ) =

k=1

and we arrive at the required result. 2

Replace in Lemma 6.5.2, operator A by Aeit and Aei„ with real numbers

t, „ . Then we get

Corollary 6.5.3 Let A ∈ C2 . Then

∞

2 it — ’it

|eit »k (A) ’ e’it »k (A)|2 =

N (Ae ’ A e )’

k=1

∞

2 i„ — ’i„

|ei„ »k (A) ’ e’i„ »k (A)|2 (t, „ ∈ R).

’A e )’

N (Ae

k=1

In particular, take t = 0 and „ = π/2. Then due to Corollary 6.5.3,

∞ ∞

2 2 2

|Re »k (A)|2

N (AI ) ’ |Im »k (A)| = N (AR ) ’ (5.2)

k=1 k=1

with AR = (A + A— )/2 .

6.6 Operators Having

Hilbert-Schmidt Powers

Assume that for some positive integer p > 1,

Ap is a Hilbert-Schmidt operator. (6.1)

Note that under (6.1) A can, in general, be a noncompact operator. Below

in this section we will give a relevant example.

Theorem 6.6.1 Let (6.1) hold for some integer p > 1. Then

∞

g k (Ap )

(»p ∈ σ(Ap )),

√

R» (A) ¤ T»,p (6.2)

k+1 (Ap , »p ) k!

ρ

k=0

where

p’1

Ak »p’k’1 ,

T»,p = (6.3)

k=0

and

ρ(Ap , »p ) = inf |tp ’ »p |

t∈σ(A)

is the distance between σ(Ap ) and the point »p .

6.7. Resolvents of Neumann-Schatten Operators 87

Proof: We use the identity

p’1

p p

Ak »p’k’1 = (A ’ I»)T»,p .

A ’ I» = (A ’ I»)

k=0

This implies

(A ’ I»)’1 = T»,p (Ap ’ I»p )’1 . (6.4)

Thus,

(A ’ I»)’1 ¤ T»,p (Ap ’ I»p )’1 . (6.5)

Applying Theorem 6.4.1 to the resolvent (Ap ’I»p )’1 = R»p (Ap ), we obtain:

∞

g k (Ap )

p

(»p ∈ σ(Ap )).

√

R (A ) ¤

»p

ρk+1 (Ap , »p ) k!

k=0

This and (6.4) complete the proof. 2

According to (6.5) Theorem 6.4.2 gives us

Corollary 6.6.2 Let condition (6.1) hold for some integer p > 1. Then

b0 g 2 (Ap )

a0 T»,p

exp [ 2 p p ] (»p ∈ σ(Ap )),

R» (A) ¤

ρ(Ap , »p ) ρ (A , » )

where constants a0 and b0 can be taken from (4.8) or from (4.9).

Example 6.6.3 Consider a noncompact operator satisfying condition (6.1).

Let H be an orthogonal sum of Hilbert spaces H1 and H2 : H = H1 • H2 ,

and let A be a linear operator de¬ned in H by the formula

B1 T

A= ,

0 B2

where B1 and B2 are bounded linear operators acting in H1 and H2 , respec-

tively, and a bounded linear operator T maps H2 into H1 . Evidently A2 is

de¬ned by the matrix

2

B1 B1 T + T B2

2

A= .

2

0 B2

If B1 , B2 ∈ C2 and T is a noncompact one, then A2 ∈ C2 , while A is a

noncompact operator.

88 6. Functions of Compact Operators

6.7 Resolvents of Neumann-Schatten

Operators

Let

A ∈ C2p for some integer p > 1. (7.1)

Then due to (2.4), condition (6.1) holds. So we can directly apply Theorem

6.6.1, but in appropriate situations the following result is more convenient.

Theorem 6.7.1 Let A ∈ C2p (p = 2, 3, ...). Then

p’1 ∞

(2N2p (A))pk+m

√

R» (A) ¤ (» ∈ σ(A)). (7.2)

ρpk+m+1 (A, ») k!

m=0 k=0

The proofs of this theorem and the next one are presented in the next section.

An additional proof of Theorem 6.7.1 can be found in (Gil™, 1995, Section

2.6). Put

1

(p)

θj = , (7.3)

[j/p]!

where [x] means the integer part of a real number x. Now the previous

theorem implies

Corollary 6.7.2 Let A ∈ C2p (p = 2, 3, ...). Then

(p)

∞

θj (2N2p (A))j

R» (A) ¤ (» ∈ σ(A)). (7.4)

ρj+1 (A, »)

j=0

Theorem 6.7.3 Under condition (7.1) there are constants a0 , b0 > 0, such

that the estimate

p’1

(2N2p (A))m b0 (2N2p (A))2p

R» (A) ¤ a0 ] (» ∈ σ(A))

exp [ (7.5)

ρm+1 (A, ») ρ2p (A, »)

m=0

holds. These constants can be taken as in (4.8) or in (4.9).

Since, condition (7.1) implies AI ≡ (A ’ A— )/2i ∈ C2p , additional estimates

for the resolvent under condition (7.1) are derived in Section 7.9 below.

6.8 Proofs of Theorems 6.7.1 and 6.7.3

We need the following result.

6.8. Proof of Theorems 6.7.1 and 6.7.3 89

Lemma 6.8.1 Let A be a linear operator acting in a Euclidean space Cn

with n = jp and integers p ≥ 1, j > 1. Then

p’1 j kp+m

N2p (V )

√

R» (A) ¤ (» ∈ σ(A)) (8.1)

ρpk+m+1 (A, ») k!

m=0 k=0

where V is the nilpotent part of A, . is the Euclidean norm.

Proof: Since A = D + V , where D is the diagonal part of A,

(A ’ »I)’1 = (D + V ’ »I)’1 = (D ’ »I)’1 (I + B» )’1 (8.2)

where B» = ’V R» (D). By the identity

p’1 p

(I ’ B» )(I + B» + . . . + B» ) = I ’ B»

we have

(A ’ »I)’1 = (D + V ’ »I)’1 =

p’1 p

(D ’ »I)’1 (I + B» + . . . + B» )(I ’ B» )’1 . (8.3)

pj

n

Clearly, B» is a nilpotent operator. So B» = B» = 0 and

j

p kp

B» )’1

(I ’ = B» .

k=0

Thus,

p’1 j

p’1 p kp+m

’1

B» )’1

(I ’ B» ) ’

= (I + B» + . . . + B» )(I = B» .

m=0 k=0

Hence,

p’1 j

kp+m

R» (A) = R» (D) B» . (8.4)

m=0 k=0

Taking into account that σ(A) = σ(D), we can assert that R» (D) is a

bounded operator for all regular points » of A. But

p p

N2 (B» ) ¤ N2p (B» ) (8.5)

(see relation (2.4)). Now let us use Theorem 2.5.1. It gives

p

k

N2 (B» )

pk p

k

¤√

¤

B» γn,k N2 (B» ) .

k!

Thus, (8.5) ensures the estimate

kp

N2p (B» )

pk

√

¤

B» .

k!

90 6. Functions of Compact Operators

Furthermore, it is clear that

N2p (B» ) = N2p (V R» (D)) ¤ N2p (V ) R» (D) .

Since D is normal and σ(A) = σ(D),

R» (D) = ρ’1 (D, ») = ρ’1 (A, »). (8.6)

Hence,

N2p (V )

N2p (B» ) ¤ . (8.7)

ρ(A, »)

Thus,

kp

N2p (V )

pk

√.

¤

B»

ρkp (A, ») k!

m m

Evidently, B» ¤ N2p (B» ). Now relation (8.7) implies

m

N2p (V )

m

¤m

B» .

ρ (A, »)

Consequently,

kp+m

N2p (V )

pk+m

√.

¤

B»

ρkp+m (A, ») k!

Taking into account (8.4), we have

p’1 j

kp+m

N2p (V )

√.

R» (A) ¤ R» (D)

ρkp+m (A, ») k!

m=0 k=0

Now (8.6) yields the required result. 2

Letting j ’ ∞ in the last lemma, we easily get

Corollary 6.8.2 Let A ∈ C2p (p = 1, 2, ...). Then

p’1 ∞ kp+m

N2p (V )

√

R» (A) ¤ (» ∈ σ(A)),

ρpk+m+1 (A, ») k!

m=0 k=0

where V is the nilpotent part of A.

Proof of Theorem 6.7.1: Due to inequality (2.3) N2p (D) ¤ N2p (A).

Thus, the triangular representation implies

N2p (V ) ¤ N2p (A) + N2p (D) ¤ 2N2p (A). (8.8)

Now, the required result follows from the previous lemma. 2

To prove Theorem 6.7.3, we need the gollowing

6.9. Functions of Hilbert-Schmidt Operators 91

Lemma 6.8.3 Under condition (7.1), let V be the nilpotent part of A. Then

there are constants a0 , b0 > 0, such that the inequality

p’1

(N2p (V ))m b0 (N2p (V ))2p

R» (A) ¤ a0 ] (» ∈ σ(A))

exp [

ρm+1 (A, ») ρ2p (A, »)

m=0

holds. These constants can be taken as in (4.8) or in (4.9).

Proof: According to (8.3),

p’1

p

’1 ’1 k

(I ’ B» )’1 .

(A ’ »I) ¤ (D ’ »I) B» (8.9)

k=0

p

It is not hard to check that B» is a Volterra operator (see also Lemma 7.3.4)

p

below. Moreover, B» ∈ C2 . Due to Theorem 6.4.2,

p

2

p

(I ’ B» )’1 ¤ e1/2 eN2 (B» )/2 . (8.10)

But

p p p

N2 (B» ) ¤ N2p (B» ) ¤ N2p (V ) R» (D) p .

p p

In addition, (8.6) implies that N2 (B» ) ¤ N2p (A)ρ’p (A, »). Now relations

(8.7), (8.9) and (8.10) yield the required result. 2

The assertion of Theorem 6.7.3 follows from the previous lemma and

(8.8).

6.9 Regular Functions of

Hilbert-Schmidt Operators

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

and f be a scalar-valued function, which is analytic on a neighborhood of

σ(A). Let a contour C consist of a ¬nite number of recti¬able Jordan curves,

oriented in the positive sense customary in the theory of complex variables.

Suppose that C is the boundary of an open set M ⊃ σ(A) and M ∪ C is

contained in the domain of analycity of f . We de¬ne f (A) by the equality

1

f (A) = ’ f (»)R» (A)d» (9.1)

2πi C

(see the book by Dunford and Schwartz (1966, p. 568)).

Theorem 6.9.1 Let A be a Hilbert-Schmidt operator and let f be a holo-

morphic function on a neighborhood of the closed convex hull co(A) of the

spectrum of A. Then

∞

g k (A)

(k)

f (A) ¤ sup |f (»)| . (9.2)

(k!)3/2

k=0 »∈co(A)

92 6. Functions of Compact Operators

Proof: Thanks to Corollary 6.3.6, there is a sequence {An } of operators

having n-dimensional ranges, such that relations (4.5) hold. Corollary 2.7.2

implies

n’1

g k (An )

sup |f (k) (»)|

f (An ) ¤ . (9.3)

(k!)3/2

»∈co(An )

k=0

Due to the well-known Lemma VII.6.5 from (Dunford and Schwartz, 1966)

we have

f (An ) ’ f (A) ’ 0 as n ’ ∞.

Letting n ’ ∞ in (9.3), due to Lemma 6.3.6, we arrive at the stated result.

2

Theorem 6.9.1 is precise: inequality (9.2) becomes the equality

f (A) = sup |f (µ)|,

µ∈σ(A)

if A is a normal operator and

sup |f (»)| = sup |f (»)|,

»∈co(A) »∈σ(A)

because g(A) = 0 in this case.

Corollary 6.9.2 Let A be a Hilbert-Schmidt operator. Then

∞

tk g k (A)

At ±(A)t

¤e for all t ≥ 0,

e

(k!)3/2

k=0

where ±(A) = sup Re σ(A). In addition,

m

m!rs (A)g k (A)

m’k

m

¤

A (m = 1, 2, ...).

(m ’ k)!(k!)3/2

k=0

Recall that rs (A) is the spectral radius of operator A. In particular, if V ∈ C2

is a Volterra operator, then

m

N2 (V )

m

¤√

V (m = 1, 2, ...). (9.4)

m!

Note that an independent proof of inequality (9.4) can be found in Section

2.3 of the book (Gil™, 1995). In addition, that inequality allows us to estimate

a power of a Volterra von Neumann - Schatten operator.

Lemma 6.9.3 For some integer p ≥ 1, let V ∈ C2p be a Volterra operator.

Then

pk+m

N2p (V )

kp+m

√

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

V (9.5)

k!

6.10. A Relation for Determinants 93

p

Relation (2.4) implies N2 (V p ) ¤ N2p (V ). Thus V p ∈ C2 . Due to

Proof:

(9.4)

kp

N2 (V p )

k N2p (V )

pk

¤√ ¤√

V .

k! k!

Since V m ¤ N2p (V ),

m

kp kp+m

N2p (V ) N2p (V )

pk+m m

√ √

¤V ¤

V ,

k! k!

as claimed. 2

Inequality (9.5) can be rewritten in the following way:

Corollary 6.9.4 For some integer p ≥ 1, let V ∈ C2p be a Volterra operator.

Then

(p) j

V j ¤ θj N2p (V ) (j = 1, 2, ...).

(p)

Recall that θj is given in Section 6.7.

6.10 A Relation between

Determinants and Resolvents

Let A ∈ C2 . Then the generalized determinant

∞

(1 ’ »k )e»k (»k ≡ »k (A))

det2 (I ’ A) :=

k=1

is ¬nite (Dunford and Schwartz, 1963, p. 1038). The following theorem is due

to Carleman (see the next section), but we suggest a new proof and correct

a misprint in Theorem XI.6.27 of the book (Dunford and Schwartz, 1963).

Theorem 6.10.1 Let A ∈ C2 . Then

(I ’ A)’1 det2 (I ’ A) ¤ exp [(N2 (A) + 1)/2].

2

(10.1)

Proof: Thanks to Theorem 2.11.1,

1

(I ’An )’1 det (I ’An ) ¤ [1+ (N2 (An )’2Re T race (An )+1)](n’1)/2

2

n’1

for any n-dimensional operator An . Hence,

(I ’ An )’1 det (I ’ An ) ¤ exp [(N2 (An ) ’ 2Re T race (An ) + 1)/2].

2

Rewrite this relation as

(I ’ An )’1 det (I ’ An )exp [Re T race (An )] ¤ exp [(N2 (An ) + 1)/2].

2

94 6. Functions of Compact Operators

Or

n

’1

|(1 ’ »k (An ))e»k (An ) | ¤ exp [(N2 (An ) + 1)/2].

2

(I ’ An )

k=1

Hence,

(I ’ An )’1 det2 (I ’ An ) ¤ exp [(N2 (An ) + 1)/2].

2

(10.2)

Let An , n = 1, 2, ... converge to A in the norm N2 (.) and satisfy conditions

(3.4). Then

det2 (I ’ An ) ’ det2 (I ’ A).

This ¬nishes the proof. 2

Replacing in (10.1) A by »’1 A, we get

Corollary 6.10.2 Let A ∈ C2 . Then

2

1 1 N2 (A)

’1 ’1

(»I ’ A) det2 (I ’ » A) ¤ ] (» ∈ σ(A)). (10.3)

exp [ +

2|»|2

|»| 2

In particular, if V ∈ C2 is quasinilpotent, then

2

1 1 N2 (V )

’1

(»I ’ V ) ¤ exp [ + ] (» = 0). (10.4)

2|»|2

|»| 2

Moreover, relation (10.4) implies.

Corollary 6.10.3 Let A ∈ C2 . Then

g 2 (A)

1 1

R» (A) ¤ ] (» ∈ σ(A)).

exp [ + 2 (10.5)

ρ(A, ») 2 2ρ (A, »)

Indeed, due to (3.2), R» (A) = R» (D)(I + V R» (D))’1 . Lemma 7.3.4 below

yields that V R» (D) is a Volterra operator. So according to (10.4),

2

1 N2 (V R» (D))

’1 ’1

¤ exp [ + ]¤

(I + V R» (D)) (I + V R» (D))

2 2

2 2

1 N2 (V ) R» (D)

exp [ + ].

2 2

But due to Lemma 6.5.2, N2 (V ) = g(A). Thus,

g 2 (A)

1 1

R» (A) ¤ R» (D)(I + V R» (D))’1 ¤ exp [ + 2 ],

ρ(A, ») 2 2ρ (A, »)

as claimed.

Note that Corollary 6.10.3 gives us an additional proof of Theorem 6.4.2.

6.11. Notes 95

6.11 Notes

Theorems 6.4.1, 6.7.1 and 6.9.1 were derived in the papers (Gil™, 1979a), (Gil™,

1992) and (Gil™, 1979b), respectively (see also (Gil™, 1995, Chapter 2)), but

in the present chapter we suggest the new proofs. Theorems 6.4.2 and 6.7.3

are probably new.

In the book (Dunford and Schwartz, 1963, p. 1038), instead of (10.3), it

is erroneously stated that

2

1 N2 (A)

’1 ’1

(»I ’ A) det2 (I ’ » A) ¤ |»| exp [ + ].

2|»|2

2

Note that the very interesting estimates for the resolvents of operators from

Cp are established in the papers (Dechevski and Persson, 1994 and 1996).

References

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

in a Hilbert Space. Pitman Advanced Publishing Program, Boston.

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

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

R. I.

[3] Dechevski, L. T. and Persson, L. E. (1994). Sharp generalized Carle-

man inequalities with minimal information about the spectrum, Math.

Nachr., 168, 61-77.

[4] Dechevski, L. T. and Persson, L. E. (1996). On sharpness, applications

and generalizations of some Carleman type inequalities, Tˆhuku Math.

o

J., 48, 1-22.

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

eral Theory. Interscience publishers, New York.