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Ukrainian Mathematical Journal, Vol. 61, No. 2, 2009

APPROXIMATION OF UNBOUNDED OPERATORS

BY BOUNDED OPERATORS IN A HILBERT SPACE

V. F. Babenko

1

,

2

and R. O. Bilichenko

1

UDC 517.5

We determine the best approximation of an arbitrary power A

k

of an unbounded self-adjoint

operator A in a Hilbert space H on the class xDA Ax

rr

∈≤

{}

(): 1, k < r.

The problem of the best approximation of an unbounded operator by linear bounded operators on a class of

elements of a Banach space was formulated by Stechkin in 1965 [1]. The statement of the problem for operators

of differentiation of low order, the first important results for this problem, and its solution were presented by

Stechkin in [2]. For a survey of subsequent results in this direction and references, see [3].

Below, we present the general statement of the problem of the best approximation of an unbounded opera-

tor by linear bounded operators (see, e.g., [1 – 3] and [4], Sec. 7.1).

Let X and Y be Banach spaces, let A : X → Y be a certain (not necessarily linear) operator with domain

of definition DA() 傺 X, let

L()N =

L(N ; X, Y) be the set of linear bounded operators T from X into Y

whose norms

T = T

XY→

do not exceed the number N > 0, and let Q 傺 DA( ) be a certain class of

elements. The value

UT( ) =

sup – :Ax Tx x Q

Y

∈

{}

is called the deviation of the operator T ∈

L()N from the operator A on the class Q, and the value

EN( ) = EN AQ( ; , ) =

inf ( ): ( )UT T N∈

{}

L (1)

is called the best approximation of the operator A by the set of bounded operators

L()N on the class Q.

The Stechkin problem of the best approximation of the operator A on the class Q is to determine the

value EN( ) and to find the extremal operator, i.e., the operator that realizes the greatest lower bound on the

right-hand side of (1).

The Stechkin problem is closely related to the problem of determination of the modulus of continuity of the

operator A on the class of elements Q, which, in fact, is an abstract version of the Kolmogorov problem of es-

timation of the intermediate derivative. The modulus of continuity ωδ( ) of an operator A on a class Q is de-

fined as the following function of a real variable δ ∈

0, ∞

[

)

:

ωδ( ) =

sup : ,Ax x Q x

YX

∈≤

{}

δ . (2)

1

Dnepropetrovsk National University, Dnepropetrovsk, Ukraine.

2

Institute of Applied Mathematics and Mechanics, Ukrainian National Academy of Sciences, Donetsk, Ukraine.

Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 61, No. 2, pp. 147–153, February, 2009. Original article submitted May 28,

2008.

0041–5995/09/6102–0179 © 2009 Springer Science+Business Media, Inc. 179

180 V. F. BABENKO AND R. O. BILICHENKO

The Stechkin theorem [2] presented below gives an efficient lower bound for the best approximation (1) of

an operator in terms of its modulus of continuity.

Theorem 1. If A is a homogeneous (in particular, linear) operator and Q is a centrally symmetric

convex set from the domain of definition of the operator A, then the following inequalities are true:

EN() ≥ sup ( ) –

δ

ωδ δ

>

{}

0

N , N ≥ 0, (3)

ωδ() ≤ inf ( )

N

EN N

≥

+

{}

0

δ , δ ≥ 0. (4)

We now present several known results for the operator of differentiation in the space H = L

2

()R . Let

L

r

22,

()R ,

r ∈N , denote the space of all functions

xL∈

2

()R whose (r – 1) th derivative is locally absolutely

continuous and whose

r th derivative belongs to the space

L

2

()R . Let

W

r

22,

()R denote the set x

{

∈

L

r

22,

()R :

x

r()

2

≤ 1

}

.

Let r,

k ∈N, k < r. For functions

xL

r

∈

22,

()R , the following unimprovable Hardy–Littlewood–Pólya in-

equality is known [5]:

x

k()

2

≤ xx

k

r

r

k

r

2

1

2

–

()

. (5)

Relation (5) yields the following estimate for the modulus of continuity of the operator of differentiation

ddx

kk

/

on the class W

r

22,

()R :

ωδ() ≤

δ

1–

k

r

. (6)

In fact, the equality is realized in (6):

ωδ( ) =

δ

1–

k

r

. (7)

Remark 1. The proof of relation (7) is essentially based on the fact that, together with any function x, the

space L

r

22,

()R contains any function of the form ax bt( ), where a,

b ∈R and b > 0.

The best approximation of the operator A = ddt

kk

/ on the class of functions Q = W

r

22,

was determined

by Subbotin and Taikov in [6]. They proved that, in this case, one has

EN( ) =

k

r

k

r

N

rk

k

rk

k

1–

–

–

⎛

⎝

⎞

⎠

.

APPROXIMATION OF UNBOUNDED OPERATORS BY BOUNDED OPERATORS IN A HILBERT SPACE 181

Let H be a Hilbert space with scalar product ( , )xy and norm x = ( , )

/

xx

12

, let A be a linear un-

bounded self-adjoint operator in H, let DA( ) be the domain of definition of this operator, and let k and r be

natural numbers (k < r). We consider the problem of determination of the modulus of continuity and the

problem of approximation by bounded operators for the powers A

k

of the operator A and the class Q =

WD A

r

() = x

{

∈ DA

r

() : Ax

r

≤ 1

}

.

We now present some facts from the spectral theory of self-adjoint operators (see, e.g., Secs.

75 and 88 in

[7]).

A resolution of the identity is a one-parameter family of projection operators E

t

: H → H defined on a fi-

nite (or infinite) segment αβ,

[]

(if the segment αβ,

[]

is infinite, then, by definition, we set E

–∞

= lim

–t

t

E

→∞

and E

∞

= lim

t

t

E

→∞

in the sense of strong convergence) and satisfying the following conditions:

(a)

EE

u v

= E

s

∀u, v ∈ αβ,

[]

, where s =

min ,u v

{}

;

(b) in the sense of strong convergence, one has

E

t –0

= E

t

, α < t < β ;

(c) E

α

= 0 and E

β

= I (I is the identity operator, i.e., Ix = x ∀∈xH ).

We set E

t

= 0 for t ≤ α and E

t

= I for t ≥ β.

It follows from the definition that, for any xH∈ , the function

σ()t = ( , )Ex x

t

, – ∞ < t < ∞,

is a left-continuous nondecreasing function of bounded variation for which

σα( ) = 0, σβ( ) = (x, x).

According to the spectral theorem, every self-adjoint operator A is associated with a resolution of the iden-

tity E

t

, t ∈R, such that the vector x belongs to DA( ) if and only if

–

,

∞

∞

∫

()

tdExx

t

2

< ∞,

and if xDA∈ ( ), then

Ax =

–∞

∞

∫

td E x

t

.

Moreover,

Ax

2

=

–

,

∞

∞

∫

()

tdExx

t

2

< ∞.

182 V. F. BABENKO AND R. O. BILICHENKO

A function ϕ()A of an operator A is defined as the operator defined by the formula

ϕ()Ax =

–

()

∞

∞

∫

ϕ tdEx

t

on all vectors xH∈ for which

–

() ,

∞

∞

∫

()

ϕ tdExx

t

2

< ∞.

Moreover,

ϕ()Ax

2

=

–

() ,

∞

∞

∫

()

ϕ tdExx

t

2

.

In particular, for xDA

k

∈ ( ),

k ∈N, we have

Ax

k

=

–∞

∞

∫

tdEx

k

t

and

Ax

k

2

=

–

,

∞

∞

∫

()

tdExx

k

t

2

.

The theorem below gives an analog of equality (7) for self-adjoint operators that act in a Hilbert space H.

Theorem 2. Let k,

r ∈N , k < r, let A be an unbounded self-adjoint operator in the Hilbert space H,

and let ωδ() be the modulus of continuity of the operator A

k

on the class Q = WD A

r

( ). Then

ωδ() ≤

δ

1–

k

r

. (8)

If the operator A is such that

(–)EEDA

ts

r2

()

≠ θ

{}

, 0 ≤ s < t ≤ ∞, (9)

then

ωδ( ) =

δ

1–

k

r

. (10)

APPROXIMATION OF UNBOUNDED OPERATORS BY BOUNDED OPERATORS IN A HILBERT SPACE 183

Remark 2. In the general case, condition (9) for the operator A is replaced by the property of the spaces

L

r

22,

indicated in Remark 1.

Proof. For any xDA

r

∈ ( ), one has (see, e.g., Sec. 5.1 in [4])

Ax

k

≤ xAx

k

r

r

k

r

1–

For xWDA

r

∈ ( ) such that x = δ, this yields

Ax

k

≤ δ

1–

k

r

,

whence

ωδ() ≤ δ

1–

k

r

,

which proves (8).

We now prove that, under condition (9), one has

ωδ() ≥

δ

1–

k

r

.

Let δ > 0 be given. For this δ and an arbitrary ε ∈

(0, 1), we set

t =

1

1

δ

⎛

⎝

⎞

⎠

r

and s = (1 – ε)

1

1

δ

⎛

⎝

⎞

⎠

r

.

We choose an element x ∈ (E

t

–

E

s

) DA

r2

()

such that x = δ. The chosen element x belongs to the class

WD A

r

()

. Indeed,

Ax

r

2

= Ax Ax

rr

,

()

= Axx

r2

,

()

=

s

t

r

u

udExx

∫

2

(,) ≤ tx

r2

2

≤ 1.

For the chosen element x, we also have

Ax

k

2

= AxAx

kk

,

()

= Axx

k2

,

()

=

s

t

k

u

udExx

∫

2

(,).

Since x ∈

(E

t

–

E

s

) DA

r2

()

, we get

x

2

=

s

t

u

dExx

∫

(,).

184 V. F. BABENKO AND R. O. BILICHENKO

Therefore,

Ax

k

2

≥ sx

k2

2

= (–)

–

1

2

21

εδ

k

k

r

⎛

⎝

⎞

⎠

.

Thus, for any ε > 0, there exists x ∈

WD A

r

()

, x = δ, such that

Ax

k

≥ (– )

–

1

1

εδ

k

k

r

.

By virtue of the arbitrariness of ε, we get

Ax

k

≥ δ

1–

k

r

.

Using this result and relation (8), we obtain (10).

The theorem is proved.

The theorem below gives a solution of the Stechkin problem of approximation of the operator A

k

by oper-

ators bounded on the class WD A

r

()

.

Theorem 3. Let k ,

r ∈N , k < r, and let A be an unbounded self-adjoint operator in the Hilbert

space H. Then, for any N > 0, one has

EN() ≤

k

r

k

r

N

rk

k

rk

k

1–

–

–

⎛

⎝

⎞

⎠

. (11)

If the operator A is such that condition (9) is satisfied, then, for any N > 0, one has

EN( ) =

k

r

k

r

N

rk

k

rk

k

1–

–

–

⎛

⎝

⎞

⎠

. (12)

The extremal approximating operator from

L()N is the function ϕ

N

A() of the operator A, where

ϕ

N

t( ) =

ttt

k

r

k

r

N

tN

k

r

k

r

tN

k

r

k

r

krkr

rk

k

rk

k

k

kr k

k

kr k

–

–

,–,

,–.

–

–

–

–

–

–

–

sign

{}

⎛

⎝

⎞

⎠

≤

⎛

⎝

⎞

⎠

⎛

⎝

⎞

⎠

≥

⎛

⎝

⎞

⎠

⎛

⎝

⎞

⎠

⎧

⎨

⎪

⎪

⎪

⎩

⎪

⎪

⎪

1

1

01

1

11

1

11

(13)

APPROXIMATION OF UNBOUNDED OPERATORS BY BOUNDED OPERATORS IN A HILBERT SPACE 185

Proof. To prove the theorem, we use the arguments presented in [6]. As the approximating operator T,

we take the function of the operator ϕ

N

A( ) , where the function ϕ

N

is defined by (13). First, we verify that

ϕ

N

A() ∈

L()N . We have

ϕ

N

Ax()

2

=

–

() ( , )

∞

∞

∫

ϕ

Nt

tdExx

2

≤ max ( ) ( , )

–

t

Nt

tdExxϕ

2

∞

∞

∫

= max ( )

t

N

txϕ

2

2

.

Since

max ( )

t

N

tϕ

2

= N

2

,

we get

ϕ

N

Ax()

2

≤ Nx

2

2

,

i.e.,

ϕ

N

A() ≤ N.

We now consider

Ax

k

– ϕ

N

Ax( ) for x ∈ WD A

r

()

. We have

Ax Ax

k

N

–()ϕ =

–

–()

∞

∞

∫

()

t t dE x

k

Nt

ϕ =

–

–()(,)

∞

∞

∫

()

⎛

⎝

⎜

⎞

⎠

⎟

ttdExx

k

Nt

ϕ

2

1

2

≤ max

–()

(,)

–

t

k

N

r

r

t

tt

t

tdExx

ϕ

∞

∞

∫

⎛

⎝

⎜

⎞

⎠

⎟

2

1

2

.

One can directly verify that

max

–()

t

k

N

r

tt

t

ϕ

=

k

r

k

r

N

rk

k

rk

k

1–

–

–

⎛

⎝

⎞

⎠

.

Furthermore, the following relation holds for x ∈

WD A

r

()

:

–

(,)

∞

∞

∫

⎛

⎝

⎜

⎞

⎠

⎟

tdExx

r

t

2

1

2

≤ 1.

Thus, for x ∈ WD A

r

()

, we have

186 V. F. BABENKO AND R. O. BILICHENKO

Ax Ax

k

N

–()ϕ ≤

k

r

k

r

N

rk

k

rk

k

1–

–

–

⎛

⎝

⎞

⎠

,

whence

EN() ≤ sup – ( )

xQ

k

N

Ax Ax

∈

ϕ ≤

k

r

k

r

N

rk

k

rk

k

1–

–

–

⎛

⎝

⎞

⎠

.

Relation (11) is proved.

If condition (9) is satisfied, then, according to Theorem 2,

ωδ( ) =

δ

1–

k

r

.

Therefore, by virtue of Theorem 1,

EN() ≥ sup –

–

δ

δδ

>

⎧

⎨

⎪

⎩

⎪

⎫

⎬

⎪

⎭

⎪

0

1

k

r

N =

k

r

k

r

N

rk

k

rk

k

1–

–

–

⎛

⎝

⎞

⎠

.

Using this result and inequality (11), we obtain (12).

The theorem is proved.

As an example, we consider the operator of differentiation A

: L

2

()R → L

2

()R , Ax u( ) = i

dx

du

. This oper-

ator is associated with a resolution of the identity E

t

such that, for any s, t, s < t (see, e.g., Sec. 89 in [7]), one

has

(–)()EExu

ts

=

1

2π

–

(–) (–)

–

(–)

()

∞

∞

∫

ee

iz u

xzdz

itzu iszu

.

It is obvious that the operator A satisfies condition (9), which leads to the result obtained by Subbotin and

Taikov in [6].

Another important example is the operator of multiplication by the independent variable, namely, A

:

L

2

()R →

L

2

()R , Ax u( ) = ux(u). This operator is associated with a resolution of the identity such that, for any

s, t, s < t (see, e.g., Sec.

89 in [7]), one has

(–)()EExu

ts

=

X

st

xu

,

()

[]

,

where

X

st,

[]

is the characteristic function of the segment st,

[]

.

It is obvious that this operator satisfies the conditions of Theorem 2, and, hence, relation (12) is true for it.

APPROXIMATION OF UNBOUNDED OPERATORS BY BOUNDED OPERATORS IN A HILBERT SPACE 187

REFERENCES

1. S. B. Stechkin, “Inequalities between norms of derivatives of an arbitrary function,” Acta Sci. Math., 26, No. 3–4, 225–230

(1965).

2. S. B. Stechkin, “Best approximation of linear operators,” Mat. Zametki, 1, No. 2, 231–244 (1967).

3. V. V. Arestov, “Approximation of unbounded operators by bounded operators and related extremal problems,” Usp. Mat. Nauk,

51, No. 6, 88–124 (1996).

4. V. F. Babenko, N. P. Korneichuk, V. A. Kofanov, and S. A. Pichugov, Inequalities for Derivatives and Their Applications [in

Russian], Naukova Dumka, Kiev (2003).

5. G. H. Hardy, J. E. Littlewood, and G. Pólya, Inequalities, Cambridge University, Cambridge (1934).

6. Yu. N. Subbotin and L. V. Taikov, “Best approximation of the operator of differentiation in the space L

2

,” Mat. Zametki, 3,

No. 2, 157–164 (1968).

7. N. I. Akhiezer and I. M. Glazman, Theory of Linear Operators in a Hilbert Space [in Russian], Nauka, Moscow (1966).