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EXTRACTA

MATHEMATICAE

Volumen 33, Número 2, 2018

instituto de investigación de matemáticas de la

universidad de extremadura

EXTRACTA MATHEMATICAE

Vol. 35, Num. 2 (2020), 197 – 204

doi:10.17398/2605-5686.35.2.197

Available online October 20, 2020

On angular localization of spectra of perturbed operators

M.I. Gil’

Department of Mathematics, Ben Gurion University of the Negev
P.O. Box 653, Beer-Sheva 84105, Israel

gilmi@bezeqint.net

Received June 29, 2020 Presented by Manuel González
Accepted September 17, 2020

Abstract: Let A and à be bounded operators in a Hilbert space. We consider the following problem:

let the spectrum of A lie in some angular sector. In what sector the spectrum of à lies if A and Ã

are “close”? Applications of the obtained results to integral operators are also discussed.

Key words: Operators, spectrum, angular location, perturbations, integral operator.

AMS Subject Class. (2010): 47A10, 47A55, 47B10.

1. Introduction and preliminaries

Let H be a complex separable Hilbert space with a scalar product (. , .),
the norm ‖.‖ =

√
(. , .) and unit operator I. By B(H) we denote the set of

bounded operators in H. For an A ∈ B(H), A∗ is the adjoint operator, ‖A‖
is the operator norm and σ(A) is the spectrum.

We consider the following problem: let A and à be “close” operators and
σ(A) lie in some angular sector. In what sector σ(Ã) lies?

Not too much works are devoted to the angular localizations of spectra.
The papers [5, 6, 7, 8] should be mentioned. In particular, in the papers by
E.I. Jury, N.K. Bose and B.D.O. Anderson [5, 6] it is shown that the test to de-
termine whether all eigenvalues of a complex matrix of order n lie in a certain
sector can be replaced by an equivalent test to find whether all eigenvalues of
a real matrix of order 4n lie in the left half plane. The results from [5] have
been applied by G.H. Hostetter [4] to obtain an improved test for the zeros
of a polynomial in a sector. In [7] M.G. Krein announces two theorems con-
cerning the angular localization of the spectrum of a multiplicative operator
integral. In the paper [8] G.V. Rozenblyum studies the asymptotic behavior of
the distribution functions of eigenvalues that appear in a fixed angular region
of the complex plane for operators that are close to normal. As applications,
he calculates the asymptotic behavior of the spectrum of two classes of oper-

ISSN: 0213-8743 (print), 2605-5686 (online)

c© The author(s) - Released under a Creative Commons Attribution License (CC BY-NC 3.0)

https://doi.org/10.17398/2605-5686.35.2.197
mailto:gilmi@bezeqint.net
https://www.eweb.unex.es/eweb/extracta/
https://creativecommons.org/licenses/by-nc/3.0/


198 m.i. gil’

ators: elliptic pseudo-differential operators acting on the sections of a vector
bundle over a manifold with a boundary, and operators of elliptic boundary
value problems for pseudo-differential operators. It should be noted that in
the just pointed papers the perturbations of an operator whose spectrum lie
in a given sector are not considered. Below we give bounds for the spectral
sector of a perturbed operator.

Without loss of the generality it is assumed that

β(A) := inf Re σ(A) > 0. (1.1)

If this condition does not hold, instead of A we can consider perturbations of
the operator A1 = A + Ic with a constant c > |β(A)|.

For a Y ∈B(H) we write Y > 0 if Y is positive definite, i.e., infx∈H,‖x‖=1
(Y x,x) > 0. Let Y > 0. Define the angular Y -characteristic τ(A,Y ) of A by

cos τ(A,Y ) := inf
x∈H,‖x‖=1

Re(Y Ax,x)

|(Y Ax,x)|
.

The set

S(A,Y ) := {z ∈ C : |arg z| ≤ τ(A,Y )}

will be called the Y -spectral sector of A.

Lemma 1.1. For an A ∈B(H), let condition (1.1) hold and Y be a positive
definite operator, such that (Y A)∗+Y A > 0. Then σ(A) lies in the Y -spectral
sector of A.

Proof. Take a ray z = reit (0 < r < ∞) intersecting σ(A), and take the
point z0 = r0e

it on it with the maximum modulus. By the theorem on the
boundary point of the spectrum [1, Section I.4.3, p. 28] there exists a normed
sequence {xn}, such that Axn −z0xn → 0 , (n →∞). Hence,

Re(Y Axn,xn)

|(Y Axn,xn)|
=

Re r0e
it(Y xn,xn)

r0|(Y xn,xn)|
+ �n = cos t + �n

with �n → 0 as n →∞ . So z0 is in S(A,Y ). This proves the lemma.

Example 1.2. Let A = A∗ > 0. Then condition (1.1) holds. For any
Y > 0 commuting with A (for example Y = I) we have (Y A)∗ + Y A = 2Y A
and Re(Y Ax,x) = |(Y Ax,x)|. Thus cos τ(A,Y ) = 1 and S(A,Y ) = {z ∈ C :
arg z = 0}.



on angular localization of spectra 199

So Lemma 1.1 is sharp.

Remark 1.3. Suppose A has a bounded inverse. Recall that the quantity
dev(A) defined by

cos dev(A) := inf
x∈H,x 6=0

Re(Ax,x)

‖Ax‖‖x‖

is called the angular deviation of A, cf. [1, Chapter 1, Exercise 32]. For
example, for a positive definite operator A one has

cos dev(A) =
2
√
λMλm

λM + λm
,

where λm and λM are the minimum and maximum of the spectrum of A,
respectively (see [1, Chapter 1, Exercise 33]). Besides, in Exercise 32 it is
pointed that the spectrum of A lies in the sector |arg z| ≤ dev(A). Since
|(Ax,x)| ≤ ‖Ax‖‖x‖, Lemma 1.1 refines the just pointed assertion.

2. The main result

Let A be a bounded linear operator in H, whose spectrum lies in the open
right half-plane. Then by the Lyapunov theorem, cf. [1, Theorem I.5.1], there
exists a positive definite operator X ∈B(H) solving the Lyapunov equation

2 Re(AX) = XA + A∗X = 2I. (2.1)

So Re(XAx,x) = ((XA + A∗X)x,x)/2 = (x,x) (x ∈H) and

cos τ(A,X) = inf
x∈H,‖x‖=1

(x,x)

|(XAx,x)|
=

1

supx∈H,‖x‖=1 |(XAx,x)|
≥

1

‖AX‖
.

Put

J(A) = 2

∫ ∞
0
‖e−At‖2dt.

Now we are in a position to formulate our main result.

Theorem 2.1. Let A,Ã ∈B(H), condition (1.1) hold and X be a solution
of (2.1). Then with the notation q = ‖A− Ã‖ one has

cos τ(Ã,X) ≥ cos τ(A,X)
(1 −qJ(A))
(1 + qJ(A))

,

provided
qJ(A) < 1.



200 m.i. gil’

The proof of this theorem is based on the following lemma.

Lemma 2.2. Let A,Ã ∈ B(H), condition (1.1) hold and X be a solution
of (2.1). If, in addition,

q‖X‖ < 1, (2.2)

then

cos τ(Ã,X) ≥ cos τ(A,X)
(1 −‖X‖q)
(1 + ‖X‖q)

.

Proof. Put E = Ã − A. Then q = ‖E‖ and due to (2.1), with x ∈ H,
‖x‖ = 1, we obtain

Re(X(A + E)x,x) ≥ Re(XAx,x) −|(XEx,x)|
= (x,x) −|(XEx,x)|
≥ (x,x) −‖X‖‖E‖‖x‖2 = 1 −‖X‖q.

(2.3)

In addition,

|(X(A + E)x,x)| ≤ |(XAx,x)| + ‖X‖‖E‖‖x‖2

= |(XAx,x)|
(

1 +
‖X‖q

|(XAx,x)|

)
(‖x‖ = 1).

But
|(XAx,x)| ≥ |Re(XAx,x)| = Re(XAx,x) = (x,x) = 1.

Hence

|(X(A + E)x,x)| ≤ |(XAx,x)|
(

1 +
‖X‖q

Re(XAx,x)

)
≤ |(XAx,x)|(1 + ‖X‖q).

Now (2.3) yields.

Re(XÃx,x)

|(XÃx,x)|
≥

(1 −‖X‖q)
|(XAx,x)|(1 + ‖X‖q)

(‖x‖ = 1),

provided (2.2) holds. Since

cos τ(Ã,X) = inf
x∈B,‖x‖=1

Re(XÃx,x)

|(XÃx,x)|
,

we arrive at the required result.



on angular localization of spectra 201

Proof of Theorem 2.1 Note that X is representable as

X = 2

∫ ∞
0

e−A
∗te−Atdt

[1, Section 1.5]. Hence, we easily have ‖X‖ ≤ J(A). Now the latter lemma
proves the theorem.

3. Operators with Hilbert-Schmidt Hermitian components

In this section we obtain an estimate for J(A) (A ∈B(H)) assuming that
A ∈B(H) and

AI := (A−A∗)/i is a Hilbert-Schmidt operator, (3.1)

i.e., N2(AI) := (trace(A
2
I))

1/2 < ∞. Numerous integral operators satisfy this
condition. Introduce the quantity (the departure from normality)

gI(A) :=

[
2N22 (AI) − 2

∞∑
k=1

|Im λk(A)|2
]1/2

≤
√

2N2(AI),

where λk(A) (k = 1, 2, . . .) are the eigenvalues of A taken with their mul-
tiplicities and ordered as |Im λk+1(A)| ≤ |Im λk(A)|. If A is normal, then
gI(A) = 0, cf. [2, Lemma 9.3].

Lemma 3.1. Let conditions (1.1) and (3.1) hold. Then J(A) ≤ Ĵ(A),
where

Ĵ(A) :=

∞∑
j,k=0

g
j+k
I (A)(k + j)!

2j+kβj+k+1(A)(j! k!)3/2
.

Proof. By [2, Theorem 10.1] we have

‖e−At‖≤ exp
[
−β(A)t

] ∞∑
k=0

gkI (A)t
k

(k!)3/2
(t ≥ 0).



202 m.i. gil’

Then

J(A) ≤ 2
∫ ∞

0
exp[−2β(A)t]

(
∞∑
k=0

gkI (A)t
k

(k!)3/2

)2
dt

= 2

∫ ∞
0

exp[−2β(A)t]


 ∞∑
j,k=0

g
k+j
I (A)t

k+j

(j!k!)3/2


dt

=

∞∑
j,k=0

2(k + j)!g
j+k
I (A)

(2β(A))j+k+1(j! k!)3/2
,

as claimed.

If A is normal, then gI(A) = 0 and with 0
0 = 1 we have Ĵ(A) = 1

β(A)
.

The latter lemma and Theorem 2.1 imply

Corollary 3.2. Let A,Ã ∈B(H) and let the conditions (1.1), (3.1) and
qĴ(A) < 1 hold. Then

cos τ(Ã,X) ≥
(1 −qĴ(A))
(1 + qĴ(A))

cos τ(A,X).

4. Integral operators

As usually L2 = L2(0, 1) is the space of scalar-valued functions h defined
on [0, 1] and equipped with the norm

‖h‖ =
[∫ 1

0
|h(x)|2dx

]1/2
.

Consider in L2(0, 1) the operator à defined by

(Ãh)(x) = a(x)h(x) +

∫ 1
0
k(x,s)h(s)ds (h ∈ L2,x ∈ [0, 1]), (4.1)

where a(x) is a real bounded measurable function with

a0 := inf a(x) > 0, (4.2)

and k(x,s) is a scalar kernel defined on 0 ≤ x,s ≤ 1, and∫ 1
0

∫ 1
0
|k(x,s)|2ds dx < ∞. (4.3)



on angular localization of spectra 203

So the Volterra operator V defined by

(V h)(x) =

∫ 1
x
k(x,s)h(s)ds (h ∈ L2,x ∈ [0, 1]),

is a Hilbert-Schmidt one. Define operator A by

(Ah)(x) = a(x)h(x) +

∫ 1
x
k(x,s)h(s)ds (h ∈ L2,x ∈ [0, 1]).

Then A = D + V, where D is defined by (Dh)(x) = a(x)h(x). Due to Lemma
7.1 and Corollary 3.5 from [3] we have σ(A) = σ(D). So σ(A) is real and
β(A) = a0. Moreover,

N2(AI) = N2(VI) ≤ N2(V ) =
[∫ 1

0

∫ 1
x
|k(x,s)|2ds dx

]1/2
.

Here VI = (V −V ∗)/2i. Thus,

gI(A) ≤ gV :=
√

2N2(V )

and

‖A− Ã‖≤ q0 :=
[∫ 1

0

∫ x
0
|k(x,s)|2ds dx

]1/2
.

Simple calculations show that under consideration

Ĵ(A) ≤ Ĵ0 :=
∞∑

j,k=0

g
j+k
V (k + j)!

2j+ka
j+k+1
0 (j! k!)

3/2
.

Making use of Corollary 3.2 and taking into account that in the considered
case cos τ(A,X) = 1, we arrive at the following result.

Corollary 4.1. Let à be defined by (4.1) and the conditions (4.2) and
(4.3) hold. If, in addition, q0Ĵ0 < 1, then σ(Ã) lies in the angular sector{

z ∈ C : |arg z| ≤ arccos
(1 −q0Ĵ0)
(1 + q0Ĵ0)

}
.

Example 4.2. To estimate the sharpness of our results consider in L2(0,1)
the operators

(Ah)(x) = 2h(x) and (Ãh)(x) = (2 + i)h(x) (h ∈ L2,x ∈ [0, 1]).



204 m.i. gil’

σ(A) consists of the unique point λ = 2 and so cos(A,X) = cos arg λ = 1.
We have

J(A) = 2

∫ ∞
0

e−4tdt = 1/2 and q = 1.

By Corollary 3.2

cos τ(Ã,X) ≥
1 − 1/2
1 + 1/2

= 1/3.

Compare this inequality with the sharp result: σ(Ã) consists of the unique
point λ̃ = 2 + i. So tan(arg λ̃) = 1/2, and therefore cos(arg λ̃) = 2/(

√
5).

Acknowledgements

I am very grateful to the referee of this paper for his (her) deep and
helpful remarks.

References

[1] Yu.L. Daleckii, M.G. Krein, “Stability of Solutions of Differential Equa-
tions in Banach Space”, Vol. 43, American Mathematical Society, Providence,
R. I., 1974.

[2] M.I. Gil’, “Operator Functions and Operator Equations”, World Scientific
Publishing Co. Pte. Ltd., Hackensack, New Jersey, 2018.

[3] M.I. Gil’, Norm estimates for resolvents of linear operators in a Banach space
and spectral variations, Adv. Oper. Theory 4 (1) (2019), 113 – 139.

[4] G.H. Hostetter, An improved test for the zeros of a polynomial in a sector,
IEEE Trans. Automatic Control AC-20 (3) (1975), 433 – 434.

[5] E.I. Jury, N.K. Bose, B.D.O. Anderson, A simple test for zeros of
a complex polynomial in a sector, IEEE Trans. Automatic Control AC-19
(1974), 437 – 438.

[6] E.I. Jury, N.K. Bose, B.D.O. Anderson, On eigenvalues of complex
matrices in a sector, IEEE Trans. Automatic Control AC-20 (1975), 433 –
434.

[7] M.G. Krein, The angular localization of the spectrum of a multiplicative
integral in Hilbert space (in Russian) Funkcional. Anal. i Prilozhen 3 (1)
(1969), 89 – 90.

[8] G.V. Rozenblyum, Angular asymptotics of the spectrum of operators that
are close to normal, J. Soviet Math. 45 (3) (1989), 1250 – 1261.


	Introduction and preliminaries
	The main result
	Operators with Hilbert-Schmidt Hermitian components
	Integral operators