Multiplicative Representations of Resolvents

10. Multiplicative Representations of Resolvents

In the present chapter we introduce the notion of the multiplicative operator 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 λRλ (A) = −(I − AP Rλ (A)P )(I − A(I − P )Rλ (A)(I − P )) (λ ∈ σ(A)). Proof:

Denote E = I − P . Since A = (E + P )A(E + P ) and EAP = 0,

we have A = P AE + P AP + EAE. M.I. Gil’: LNM 1830, pp. 151–161, 2003. c Springer-Verlag Berlin Heidelberg 2003


(1.1)

152

10. Multiplicative Representations of Resolvents

Let us check the equality Rλ (A) = P Rλ (A)P − P Rλ (A)P AERλ (A)E + ERλ (A)E.

(1.2)

In fact, multiplying this equality from the left by A − Iλ and taking into account the equalities (1.1), AP = P AP and P E = 0, we obtain the relation ((A − Iλ)P + (A − Iλ)E + P AE)(P Rλ (A)P − P Rλ (A)P AERλ (A)E + ERλ (A)E) = P − P AERλ (A)E + E + P AERλ (A)E = I. Similarly, multiplying (1.2) by A − Iλ from the right and taking into account (1.1), we obtain I. Therefore, (1.2) is correct. Due to (1.2) I − ARλ (A) = (I − ARλ (A)P )(I − AERλ (A)E).

(1.3)

I − ARλ (A) = −λRλ (A).

(1.4)

But 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)

0 = P0 H ⊂ P1 H ⊂ ... ⊂ Pn−1 H ⊂ Pn H = H.

(1.6)

and

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 λRλ (A) = −

→ 

(I − A∆Pk Rλ (A)∆Pk ) (λ ∈ σ(A)),

1≤k≤n

where ∆Pk = Pk − Pk−1 (1 ≤ k ≤ n).

10.1. Operators with Finite Chains

Proof:

153

Due to the previous lemma

λRλ (A) = −(I − APn−1 Rλ (A)Pn−1 )(I − A(I − Pn−1 )Rλ (A)(I − Pn−1 )). But I − Pn−1 = ∆Pn . So equality (1.4) implies. I − ARλ (A) = (I − APn−1 Rλ (A)Pn−1 )(I − A∆Pn Rλ (A)∆Pn ). Applying this relation to APn−1 , we get I − APn−1 Rλ (A)Pn−1 = (I − APn−2 Rλ (A)Pn−2 )(I − A∆Pn−1 Rλ (A)∆Pn−1 ). Consequently, I − ARλ (A) = (I − APn−2 Rλ (A)Pn−2 )(I − A∆Pn−1 Rλ (A)∆Pn−1 )(I− A∆Pn Rλ (A)∆Pn ). Continuing this process, we arrive at the required result. 2. Let us consider an operator of the form A=

n


ak ∆Pk + V

(1.7)

k=1

where ak are some numbers, {Pk } is a chain of projectors defined by (1.6) and V is a nilpotent operator with the property Pk−1 V Pk = V Pk (k = 1, ..., n). Lemma 10.1.3 Under conditions (1.7) and (1.8), the relation λRλ (A) = −

→ 

(I +

1≤k≤n

A∆Pk ) λ − ak

is valid for any λ = ak (k = 1, ..., n). Proof:

It is not hard to check that ∆Pk Rλ (A)∆Pk =

∆Pk . ak − λ

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

(1.8)

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 = λk (A) are the eigenvalues of A with their multiplicities. Introduce the orthogonal projectors Pk =

k


(., 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,

→ 

(I +

1≤k≤∞

A∆Pk ) λ − λk (A)

is a limit in the operator norm of the sequence the operators Πn (λ) :=

→ 

(I +

1≤k≤n

(I +

A∆Pk ) := λ − λk (A)

A∆P1 A∆P2 A∆Pn )(I + )...(I + ) λ − λ1 (A) λ − λ2 (A) λ − λn (A)

for λ = λk (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

λRλ (A) = −

→  1≤k≤∞

(I +

A∆Pk ) (λ ∈ σ(A)). λ − λk (A)

(2.2)

10.2. Complete Compact Operators

Proof:

155

Let An = APn . Lemma 10.1.3 implies the equality λRλ (An ) = −Πn (λ).

(2.3)

Since A is compact, An tends to A in the operator norm as n tends to ∞. Besides, (An − λI)−1 Pn → (A − λI)−1 in the operator norm for any regular λ . We arrive at the result. 2 Let A be a normal compact operator. Then A=

∞


λk (A)∆Pk .

k=1

Hence, A∆Pk = λk (A)∆Pk . Since ∆Pk ∆Pj = 0 for j = k, Lemma 10.2.1 gives us the equality −λRλ (A) = I +

∞


(I +

k=1

But I=

∞


A∆Pk ). λ − λk (A)

∆Pk .

k=1

Thus, λRλ (A) = − −

∞


[1 + (λ − λk (A))−1 λk (A)]∆Pk =

k=1 ∞


λ∆Pk (λ − λk (A))−1 .

k=1

Or Rλ (A) =

∞
k=1

∆Pk . λk (A) − λ

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 defined 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 = λk (A)∆Pk + V ∆Pk . Now Lemma 10.2.1 implies the relation

λRλ (A) = −

→  1≤k≤∞

(I +

(λk (A) + V )∆Pk ) (λ ∈ σ(A)). λ − λk (A)

(2.5)

156

10. Multiplicative Representations of Resolvents

10.3

The Second Representation for Resolvents of Complete Compact Operators

Let V be a Volterra operator, defined by (2.4). Then due to (2.5) (I − V )−1 =

→ 

(I + V ∆Pk ).

(3.1)

2≤k≤∞

Furthermore, according to (2.1) A = D + V , where D is defined by Dek = λk (A)ek . Clearly, (A − λI)−1 = (D + V − Iλ)−1 = (D − Iλ)−1 (I + Bλ )−1 ,

(3.2)

where Bλ = V (D − Iλ)−1 . Due to Lemma 7.3.4 Bλ is a Volterra operator. Moreover, Pk−1 Bλ Pk = Bλ Pk . Thus relation (3.1) implies (I + Bλ )−1 =

→ 

(I − Bλ ∆Pk ).

2≤k≤∞

But

V ∆Pk . λk − λ

Bλ ∆Pk = Therefore,

→ 

(I + Bλ )−1 =

(I +

2≤k≤∞

V ∆Pk ). λ − λk (A)

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 Rλ (A) = Rλ (D)

→ 

(I +

2≤k≤∞

V ∆Pk ) (λ ∈ σ(A)), λ − λk (A)

where V is the nilpotent part of A and Rλ (D) =

∞
k=1

∆Pk . λk (A) − λ

10.4. Operators with Compact Inverse Ones

10.4

157

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 A−1 ek =

k


bjk ej

(4.1)

j=1

with entries bjk . The nilpotent part V0 and diagonal one D0 of A−1 are defined by k−1
bjk ej . (4.2) V0 ek = j=1 −1

and D0 ek = bkk ek = λk (A

Pk =

)ek . As above, put k


(., 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 λ Rλ (A) =

→ 

(I +

1≤k≤∞

λ(1 + λk (A)V0 )∆Pk ) − I (λ ∈ σ(A)). λk (A) − λ

The product converges in the operator norm. Proof:

Thanks to Lemma 10.2.1, (A − λI)−1 = A−1 (I − λA−1 )−1 = A−1

→ 

(I +

1≤k≤∞

λA−1 ∆Pk ) 1 − λk (A−1 )λ

for any regular λ of A. But D0 ∆Pk = λk (A−1 )∆Pk . Hence, → 

(A − λI)−1 = A−1

(I +

λ(λk (A−1 ) + V0 )∆Pk ). 1 − λk (A−1 )λ

(I +

λ(1 + λk (A)V0 )∆Pk ). λk (A) − λ

1≤k≤∞

Thus, we have derived the relation (A − λI)−1 = A−1

→  1≤k≤∞

Taking into account that A(A − λI)−1 = I + λ(A − λI)−1 , we arrive at the required result. 2

158

10. Multiplicative Representations of Resolvents

10.5

Multiplicative Integrals

Let F be a function defined on a finite real segment [a, b] whose values are bounded linear operators in H. We define the right multiplicative integral as the limit in the uniform operator topology of the sequence of the products → 

(n)

1≤k≤n

(n)

(n)

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

and a = t0 denote by

< t1

(n)

< ... < tn 

= b. The right multiplicative integral we

→

(1 + dF (t)). [a,b]

In particular, let P be an orthogonal resolution of the identity defined 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)

1≤k≤n (n)

(n)

(n)

(n)

(I + φ(tk )A∆P (tk )) (∆P (tk ) = P (tk ) − P (tk−1 )) (n)

as maxk |tk − tk−1 | tends to zero.

10.6. Volterra Operators

10.6

159

Resolvents of Volterra Operators

Lemma 10.6.1 Let V be a Volterra operator with a m.r.i. P (t) defined on a finite real segment [a, b]. Then the sequence of the operators Vn =

n
k=1

(n)

(n)

P (tk−1 )V ∆P (tk )

(6.1) (n)

tends to V in the uniform operator topology as maxk |tk zero. Proof:

(n)

− tk−1 | tends to

We have V − Vn =

n
k=1

(n)

(n)

∆P (tk )V ∆P (tk ).

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 infinity. 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) defined on a segment [a,b]. Then  → −1 (I + V dP (t)). (I − V ) = [a,b]

Proof: Due to Lemma 10.6.1, V is the limit in the operator norm of the sequence of operators Vn , defined by (6.1) . Due to Lemma 10.1.2, (I − Vn )−1 =

→  1≤k≤n

(n)

(I + Vn ∆P (tk )).

Hence the required result follows. 2

10.7

Resolvents of P -Triangular Operators

In this section [a, b] is a finite real segment, again. Theorem 10.7.1 Let A be a P -triangular operators with a m.r.i. P (.) defined on [a, b], a (compact) nilpotent part V and the diagonal part  b φ(t)dP (t), (7.1) D= a

where φ is a scalar function integrable in the Riemann-Stieljes sense with respect to P (.). Then  → V dP (t) ) (λ ∈ σ(A)). (7.2) (I − Rλ (A) = Rλ (D) φ(t) −λ [a,b]

160

10. Multiplicative Representations of Resolvents

Proof: By Lemma 7.3.4 V Rλ (D) is a Volterra operator. We invoke Lemma 10.6.3. It asserts that  → (I − V Rλ (D)dP (t)). (7.3) (I + V Rλ (D))−1 = [a,b]

But according to (7.1) Rλ (D)dP (t) = Thus, −1

(I + V Rλ (D))

1 dP (t). φ(t) − λ 

→

= [a,b]

(I −

V dP (t) ). φ(t) − λ

Hence relation (3.2) yields the required result. 2 Furthermore, from (7.2) it follows that  Rλ (A) =

a

b

dP (s) φ(s) − λ



→

[a,b]

(I −

V dP (t) ) φ(t) − λ

for all regular λ. 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  Rλ (A) =

a

b

dP (s) φ(s) − λ



→

[s,b]

(I −

V dP (t) ) (λ ∈ σ(A)). φ(t) − λ

Let us suppose that A is a normal operator. Then V = 0 and Theorem 10.7.1 yields  b dP (s) Rλ (A) = . φ(s) −λ a 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) Rλ (A) = Rλ (D) ) (λ ∈ σ(A)). (I − φ(t) − λ [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) ) Rλ (A) = (I − φ(t) − λ a φ(s) − λ [s,b] for all regular λ. 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) ) (λ ∈ σ(A)). Rλ (A) = Rλ (D) (I − φ(t) − λ [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 Approach. Ac. Press, New York. [3] Gil’, M. I. (1973). On the representation of the resolvent of a nonselfadjoint 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.