Approximation of unbounded operators by bounded operators in a Hilbert space

Abstract

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 {x ∈ D(A^r) : ∥A^rx∥ ≤ 1}, k r.

Full text

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).