11. Relatively P -Triangular Operators
This chapter is devoted to operators of the type A = D + W , where D is a normal...
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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
Definitions and Preliminaries
Let P (.) be a maximal resolution of the identity defined 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)
DP (t)h = P (t)Dh (t ∈ [a, b], h ∈ Dom (A)).
(1.3)
and
M.I. Gil’: LNM 1830, pp. 163–172, 2003. c Springer-Verlag Berlin Heidelberg 2003
164
Moreover,
11. Relatively P -Triangular Operators
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]). Definition 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 operator, 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 satisfies the inequalities (1.5) V k ≤ θk |V |kY (k = 1, 2, ...), √ k where constants θk are independent of V and θk → 0 (k → ∞). Under (1.5) put ni(V )−1 JY (V ) = θk |V |kY (θ0 = 1), k=0
where ni (V ) is the ”nilpotency” index (see Definition 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 invertible. Moreover, A−1 ≤ rl−1 (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 ). Since D is normal, D−1 = rl−1 (D). This proves the required results. 2
11.2. Resolvents of Relatively P -Triangular Operators
11.2
165
Resolvents of Relatively P -Triangular Operators
Denote w(λ, D) ≡ inf | t∈σ(D)
λ − 1| t
and ν(λ) := ni ((D − λI)−1 W ) = ni ((I − D−1 λ)−1 V ) (λ ∈ σ(D)). Under (1.5) put JY (V, m, z) =
m−1
z −1−k θk |V |kY (z > 0).
k=0
Lemma 11.2.1 Under conditions (1.1)-(1.5), let λ ∈ σ(D). Then λ is a regular point of operator A. Moreover, Rλ (A) ≤ rl−1 (D)JY (V, ν(λ), w(λ, D))
(2.1)
Rλ (A)D ≤ JY (V, ν(λ), w(λ, D)).
(2.2)
and Proof:
According to (1.1), A − λI = (D − λI)(I + (D − λI)−1 W ).
Consequently
Rλ (A) = (I + Rλ (D)W )−1 Rλ (D).
Taking into account that (D − λI)−1 = (I − D−1 λ)−1 D−1 , we have Rλ (A) = (I + (I − D−1 λ)−1 D−1 W )−1 (I − D−1 λ)−1 D−1 .
(2.3)
But D is normal. Therefore, (I − D−1 λ)−1 ≤
1 . w(λ, D)
Moreover, due to Lemma 7.3.3, (I − D−1 λ)−1 V is a Volterra operator. So ν(λ)−1
(I + (I − D−1 λ)−1 V )−1 ≤
((I − D−1 λ)−1 V )k ≤
k=0 ν(λ)−1
k=0
θk |((I − D−1 λ)−1 V )k |Y ≤
11. Relatively P -Triangular Operators
166
ν(λ)−1
θk w−k (λ, D)|V |kY = w(λ, D)JY (V, ν(λ), w(λ, D)).
k=0
Thus (2.3) yields Rλ (A) ≤ (I + (I − D−1 λ)−1 V )−1 D−1 (I − D−1 λ)−1 ≤ JY (V, ν(λ), w(λ, D))rl−1 (D). Therefore (2.1) is proved. Moreover, thanks to (2.3) Rλ (A)D ≤ (I + (I − D−1 λ)−1 V )−1 (I − D−1 λ)−1 ≤ JY (V, ν(λ), w(λ, 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 operator satisfies the inequality (A + Z)−1 ≤ Proof:
JY (V ) . ρl (D)(1 − JY (V )m(Z))
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
11.4
167
Resolvents of Perturbed RPTO
Recall that JY (V, ν(λ), w(λ, D)) is defined in Section 11.2. Theorem 11.4.1 Under conditions (1.1)-(1.5) and (3.1), for a λ ∈ σ(D), let m(Z)JY (V, ν(λ), w(λ, D)) < 1. (4.1) Then λ is a regular point of operator A + Z. Moreover, Rλ (A + Z) ≤ Proof:
JY (V, ν(λ), w(λ, D)) . ρl (D)(1 − m(Z)JY (V, ν(λ), w(λ, D)))
Due to Lemma 11.2.1, Rλ (A)Z = Rλ (A)DD−1 Z ≤ m(Z)JY (V, ν(λ), w(λ, D)).
But Rλ (A + Z) − Rλ (A) = −Rλ (A)ZRλ (A + Z). Hence, Rλ (A + Z) ≤ Rλ (A) + Rλ (A)ZRλ (A + Z) ≤ Rλ (A) + m(Z)JY (V, ν(λ), w(λ, D))Rλ (A + Z). Due to (4.1) Rλ (A + Z) ≤ Rλ (A)(1 − m(Z)JY (V, ν(λ), w(λ, 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 0 |) ≥ 1.
11.5
(4.2)
Relative Spectral Variations
Definition 11.5.1 Let A and B be linear operators in H. Then the quantity rsvA (B) := sup
inf |1 −
µ∈σ(B) λ∈σ(A)
µ | λ
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 Hausdorff distance between the spectra of A and B.
11. Relatively P -Triangular Operators
168
Put ν˜0 = sup
λ∈σ(D)
ν(λ) = sup
λ∈σ(D)
ni ((D − λI)−1 W ).
In the sequel one can replace ν˜0 by ∞. Theorem 11.5.2 Under conditions (1.1)-(1.5) and (3.1), the equation m(Z)JY (V, ν˜0 , 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 inequality (5.3) rsvD (A + Z) ≤ δY (A, Z), where δY (A, Z) ≡ 2 max
j=1,2,...
j θj−1 |V |j−1 Y Z.
Due to (5.3), for any µ ∈ σ(A + Z), there is a µ0 ∈ σ(D), such that 1 ≤ |µµ−1 0 | + δY (A, Z). 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 inequality rl (A + Z) ≥ rl (D) max{0, 1 − δY (A, Z)}.
11.6 11.6.1
Operators with von Neumann-Schatten Relatively Nilpotent Parts 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
Jp (V ) =
k=0
where (p) θk =
(p)
k θk N2p (V ),
1 [k/p]!
and [.] means the integer part. Due to Corollary 6.9.4, (p)
j (V ) (j = 1, 2, ...). V j ≤ θj N2p
(6.2)
Then Lemma 11.1.2 implies that under (1.1)-(1.4) and (6.1), A−1 ≤ rl−1 (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, (A + Z)−1 ≤
Jp (V ) . (1 − m(Z)Jp (V ))rl (D)
Under (6.1) for a z > 0, put ψp (V, z) = a0
p−1 j N2p (V ) j=0
z j+1
2p b0 N2p (V ) exp [ ], z 2p
where constants a0 , b0 can be taken from the relations c , b0 = c/2 for a c > 1, a0 = c−1 or
a0 = e1/2 , b0 = 1/2.
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, (A + Z)−1 ≤
ψp (V, 1) . rl (D) (1 − m(Z)ψp (V, 1))
(6.4)
(6.5)
(6.6)
170
Proof:
11. Relatively P -Triangular Operators
Due to Theorem 6.7.3, (I − V )−1 ≤ ψp (V, 1).
(6.7)
But A−1 = (I + V )−1 D−1 . So −1 D ≤ ψp (V, 1). A−1 ≤ ρ−1 l (D)ψp (V, 1) and A
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(λ) is defined in Section 11.2. Relation (6.2) and Lemma 11.2.1, under (6.1) imply Rλ (A) ≤ rl−1 (D)J˜p (V, ν(λ), w(λ, D)) and Rλ (A)D ≤ J˜p (V, ν(λ), w(λ, D)) (λ ∈ σ(A)), where J˜p (V, m, z) =
m−1 k=0
(p)
k θk N2p (V ) (z > 0). z k+1
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)J˜p (V, ν(λ), w(λ, D)) < 1. Then λ is a regular point of operator A + Z. Moreover, Rλ (A + Z) ≤ Put
J˜p (V, ν(λ), w(λ, D)) . rl (D) (1 − m(Z)J˜p (V, ν(λ), w(λ, D))
V˜ (λ) = (I − D−1 λ)−1 V.
Thanks to (2.3) Rλ (A)D = (I + V˜ (λ))−1 (I − D−1 λ)−1 . Taking into account (6.7), we get Rλ (A) ≤ ψp (V˜ (λ), 1)w(λ, D). Hence, we arrive at the inequalities Rλ (A) ≤ rl−1 (D)ψp (V, w(λ, D)) and Rλ (A)D ≤ ψp (V, w(λ, D)) (6.8) for all λ ∈ 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(λ, D)) < 1. Then λ is a regular point of operator A + Z. Moreover, Rλ (A + Z) ≤
11.6.3
ψp (V, w(λ, D)) . rl (D) (1 − m(Z)ψp (V, w(λ, D))
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 m(Z)ψp (V, |1 −
µ |) ≥ 1. µ0
(6.9)
Theorem 11.6.6 Under conditions (1.1)-(1.4), (3.1) and (6.1), the equation p−1 2p k N2p (V ) N2p (V ) m(Z) exp [(1 + )/2] = 1 (6.10) k+1 2p z z 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 satisfies the inequality rl (A + Z) ≥ rl (D) max{0, 1 − zp (V, Z)}. Proof: 2
Comparing (6.9) and (6.10), we have w(D, µ) ≤ zp (V, Z), as claimed.
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 ∆p (V, Z) :=
m(Z)pe N2p (V )[ln (N2p (V )/m(Z)p)]−1/2p
if N2p (V ) ≤ m(Z)pe . 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)}.
172
11. Relatively P -Triangular Operators
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 infinite matrices, Mathematical Physics, Analysis and Geometry, 4, 379-394 (2002).