On the Numerical Range and Numerical Radius of the Volterra Operator Текст научной статьи по специальности «Физика»

КРАТКИЕ СООБЩЕНИЯ

giltei х:|аайййЫ

Серия «Математика»

2018. Т. 24. С. 102-108

Онлайн-доступ к журналу: http: / / mathizv.isu.ru

УДК 517.98 MSG 47А10, 47А30

DOI https://doi.org/10.26516/1997-7670.2018.24.102

On the Numerical Range and Numerical Radius of the Volterra Operator

L. Khadkhuu

National University of Mongolia, Ulaanbaatar, Mongolia D. Tsedenbayar

Mongolian University of Science and Technology, Ulaanbaatar, Mongolia

Abstract. In this paper, we investigated the numerical range and the numerical radius of the classical Volterra operator on the complex space i2 [0,1]. In particular, we determined the numerical range, the numerical radius of real and imaginary part of the Volterra operator.

Keywords: Volterra operator, numerical range, numerical radius.

Let. H be a complex Hilbert space equipped with the inner product (•,•), which induces the norm || • ||. Denote by B(H) the Banach algebra of bounded linear operators acting on H with the operator norm defined by

1. Introduction

||A|| = sup {||At|| : x € H}, AeB(H).

||.t|| = 1

Recall that for an operator A the spectrum

a(A) = {A € С : such that A — XI is not invert.ible}

is a non-empty compact subset of the complex plane.

For a bounded linear operator A on a complex Hilbert space H, the numerical range W(A) is the image of the unit sphere of H under the quadratic form x —> (Ax, x) associated with the operator. More precisely,

W(A) = {(Ax,x) :xeH, ||a:|| = 1}.

It is well known that numerical range of an operator is convex (The Toep-litz-Hausdorf theorem) and spectrum is contained in the closure of its numerical range. Note that A is a self-adjoint if and only if W(A) c R. The numerical radius of an operator A is defined by

u(A) = sup{|A| : A € W(A)} and the following inequalities

hold (see [2]).

Denote by V the classical Volterra operator

X

(Vf)(x) = jf(t)dt, /a2[0,i]. 0

The adjoint of the Volterra operator is

l

(V*f)(x) = J f(t)dt.

x

The Volterra operator is compact and quasinilpotent. The Volterra operator pencils studies were made in several directions concerning it, see e.g. [5-8] and the references therein.

2. The Results

We will need the following theorem.

Theorem 1. [1, page 268] If A is a bounded operator on H and 9 € [-7r,7r], put \e = m&Ko(Be), where Be = \(e~ieA + eieA*) = B*d. Then

W(A)= p| He

0&[-tt,TT]

where the half-space Hq is defined by

He = {ze C : Re(e~idz) < Xe}.

Remark 1. According to Theorem 1 and under the assumption € Cl[—7r,7r], we have

x cos 9 + y sin 9 =

which is envelope curve. Because, if 0 < 9 < it, then sin0 > 0 and y < — a; cot 9. Similarly, if — it < 9 < 0, then sin 9 < 0 and y > — a; cot 9.

Proposition 1. The operator zV (z £ C) is accretive on L2[0,1] if and only if Rez > 0 and Imz = 0.

Proof (If) Let zV (z £ C) be accretive, that is

«zV + zV*)f,f)> 0 for all f£L2[0,1]. We choose fk(x) = etk7TX for all k £ Z. Then

[Vfk){x) = Г eikntdt = -L(eifc™ - 1) Jo

1

r-l

1

l-(-l)A k42

and

kit ' k2ir2

for all k £ Z. Thus, Rez > 0 and Imz = 0.

(Only if) Assume that Rez > 0 and Imz = 0. Then

((,zV + zV*)f,f) = z / f(t)dt

Remark 2. The operator V is accretive.

> 0.

□

Proposition 2. The numerical range of V is the set lying between the curves

1 — cos ip (p — sin (fi

9 ^ 9

where <p £ [0,27r]. (see [4], Problem 150) Proof. The identity

MVf,f) = ±

f(t)dt

<\j\f{t)\4t

implies that W(V) ç {z : 0 < Rez < Put A = V as in Theorem 1, then

A* =

2 J sin0

29

or

x

(see [1, page 270])

Remark 1 implies that, envelope curve is

sin 0

xcosd + ys'md = —— (2.1)

20

for 0 € [—n, n].

Moreover, (2.1) implies that the boundary of numerical range is

sin 0

x cos 9 + y sin 9 = —— 20

0 cos 0 - sin 0 -x sin 9 + y cos 9 =-29^-

sin2 0 1 — cos 20 1 — cos p ~W = (:29)2 = ^ 0 — sin 0 cos 0 20 — sin 20 p — sin p 202 = (20)2 =

for p = 20 € [0,2n].

If z = (Vf, f) € W(V) then (Vf, f) = z € W(V). We have

1 — cos p

X ~

p2

, p — sin p

y = ±-2-

p2

where p € [0,2n]. Put fv(t) = ei1tp, then

(^/^-^(¿p-l + e"^)

T iMf f\ p — sin p

y = lm(Vfv, fv) =--2-

p2

Therefore, the numerical range of V is the set lying between the curves

1 — cos p . p — sin p

p2 p2

where p € [0,27r]. □

Proposition 3. Let V be the Volterra operator on L2[0,1]. Then

i) oj(V) = (For on real space see [3, Theorem 6.1])

ii) W(ReV) = [0,±],

iii) w(ImV) = \-±,±\.

Proof, i). It is easy to see that

f(<p) = x2 + y2 = -ij (2 - 2 cos <p - sin Lp + Lp2)

Lp

and

/'M = -^(^cos|-2sin|)2<0.

We have uj\V)= sup /(p) =

0<ip<2iT

0 »0 \ y Lp2 / y p2 / / 4'

or

ii). Re F is self-adjoint, bounded convex subset of the real line. Note that

((ReF)cos7ra;,cos7ra;) = 0, and ((ReF)l,l) = -

and {0, i} € dW(ReV). We have W(ReV) = [0,

iii). ImV is self-adjoint and max ——S*D^ = — imply that

^€[0,27T] <fi2, 7T

W{lmV) =

1 1

7Г ' 7Г

□

Remark 3. From Proposition 3, we have

ReV) = I and ImV) = -. 2 7T

3. Conclusion

We investigated the numerical range and the numerical radius of the classical Volterra operator. In particular, we give new proof of the numerical range and the numerical radius of the classical Volterra operator on the complex space L2[0,1] (see [3, page 984], [4, page 113]).

References

1. Davies E. Brian. Linear Operators and Their Spectra. Cambridge Studies in Advanced Mathematics, 2007, no. 106. https://doi.org/10.1017/CBO9780511618864

2. Gustafson K.E. and Rao D.K.M. Numerical Range. Springer-Verlag New York, Inc., 1997.

3. Gurdal M., Garayev M.T. and Saltan S. Some concrete operators and their properties. Turkhish Journal of Mathematics, 2015, vol. 39, pp. 970-989. https://doi.org/10.3906/mat-1502-48

4. Halmos P.R. A Hilbert Space Problem Book. Springer-Verlag New York Inc., 1982. https://doi.org/10.1007/978-1-4684-9330-6

5. Khadkhuu L., Tsedenbayar D. A note about Volterra operator. Mathematica Slovaca, 2017, Accepted.

6. Khadkhuu L., Tsedenbayar D. Some norm one functions of the Volterra operator. Mathematica Slovaca, 2015, vol. 65, Issue 6, pp. 1505-1508. https://doi.org/10.1515/ms-2015-0102

7. Khadkhuu L., Tsedenbayar D., Zemanek J. Operator Theory: Advanced and, Applications, 2015, vol. 250, pp. 281-285.

8. Tsedenbayar D. On the power boundedness of certain Volterra operator pencils, Studia Mathematica, 2003, no. 156, pp. 59-66. https://doi.org/10.4064/sm156-1-4

Lkhamjav Khadkhuu, Candidate of Science (Physics and Mathematics), Associate Professor, Department of Mathematics, National University of Mongolia, P. O. Box 46/145, National University Street-3, Ulaanbaatar, Mongolia, tel.: (976) 99080815 (e-mail: [email protected])

Dashdondog Tsedenbayar, Doctor of Science (Physics and Mathematics), Professor, Department of Mathematics, Mongolian University of Science and Technology, P. O. Box 46/520, Baga toiruu -6, Sukhbaatar district, Ulaanbaatar-14191, Mongolia, tel.: (976) 99716122 (e-mail: cdnbr@yahoo. com)

Received 26.04.2018

Числовая область и числовой радиус оператора Вольтерра

Л. Хадхуу

Национальный университет Монголии, Улан-Батор, Монголия Д. Цэдэнбаяр

Монгольский университет науки и технологии, Улан-Батор, Монголия

Аннотация. Обозначим через V классический оператор Вольтерра на комплексном пространстве Ь2[0,1]. Мы определили числовую область и числовой радиус классического оператора Вольтерра, при этом доказательства соответствующих утверждений отличаются от известных. В частности, определены числовая область и числовой радиус действительной (комплексной) части оператора Вольтерра.

Ключевые слова: Оператор Вольтерра, числовая область, числовой радиус.

Лхамжав Хадхуу, кандидат физико-математических наук, Национальный университет Монголии, Монголия, Улан-Батор, Округ Сухэ-Батора, ул. Бага Тойру, 4, тел.: 976-99080815, (e-mail: [email protected])

Дашдондог Цэдэнбаяр, доктор физико-математических наук, профессор, Монгольский университет науки и технологии, Монголия, 14191, Улан-Батор, Округ Сухэ-Батора, ул. Бага Тойру, 6, тел.: 97699716122, (e-mail: [email protected])

Поступила в редакцию 26.04-2018