Ratio Mathematica Volume 41, 2021, pp. 173-180

Tikhonov type regularization for
unbounded operators

E Shine Lal*
P Ramya†

Abstract

In this paper, we introduce a Tikhonov type regularization method for
an ill-posed operator equation Tx = y, where T is a closed densely
defined unbounded operator on a Hilbert space H.
Keywords: densely defined operator, closed operator, Tikhonov type
regularization.
2020 AMS subject classifications: 47A10, 47A52 1

*E Shine Lal, (Department of Mathematics, University College, Thiruvananthapuram, Kerala,
India -695034); shinelal.e@gmail.com

†P Ramya, (Department of Mathematics, N.S.S College, Nemmara, Kerala, India-678508);
ramyagcc@gmail.com
1Received on September 20, 2021. Accepted on December 10, 2021. Published on December 31,
2021. doi: 10.23755/rm.v41i0.663. ©The Authors. This paper is published under the CC-BY
licence agreement.

173



E Shine Lal, P Ramya

1 Introduction

Most of the problems arise in the field of science and engineering can be mod-
elled as an operator equation

Tx = y (1)

where T : X → Y is a bounded linear map from a normed linear space X to a
normed linear space Y . In most of the cases (1) is Ill-posed. Certain regularization
procedures are known for solving ill-posed operator equation (1). For example
Tikhonov regularization, Mollifier method, Ritz method [5, 3]. In this paper we
introduce a Tikhonov type regularization method for solving an ill-posed operator
equation (1), where T is a closed densely defined operator on a Hilbert space H
and we study the order of convergence.

2 Preliminaries

Let L(H),C(H) and B(H) denote the space of all linear, closed linear and
bounded linear operators on a Hilbert space H respectively. For T ∈ L(H),
the domain, range of T are denoted by D(T),N(T) respectively. An operator
T ∈ L(H) is said to be densely defined if D(T) = H.
For example let T : l2(N) → l2(N) defined by

T(x1,x2,x3, ....,xn, ....) = (x1, 2x2, 3x3, .....,nxn, ....)

with domain

D(T) = {(x1,x2,x3, ....,xn, ....) ∈ H : Σ∞j=1|jxj|
2 < ∞}.

Then T is closed and unbounded. Since c00 ⊆ D(T) and c00 is dense in l2(N),
D(T) is dense in l2(N).

Proposition 2.1. Let T ∈ C(H) be a densely defined operator. Then there exist a
unique operator T∗ ∈ C(H) such that

〈Tx,y〉 = 〈x,T∗y〉 ∀x ∈ D(T), ∀y ∈ D(T∗).

Proof. Let D(T∗) = {y ∈ H : 〈Tx,y〉 is continuous for every x ∈ D(T) }. For
y ∈ D(T), define f : D(T) → C by f(x) = 〈Tx,y〉 ∀x ∈ D(T). Extend f to
f0 : H → C by f0(x) = lim

n→∞
〈Txn,y〉 where (xn) is a sequence in D(T) such

that xn → x.

174



Tikhonov type regularization for unbounded operators

Next we prove that f0 is well defined. For, let (xn) and (yn) be two sequences
in D(T) converges to x. Since T is closed, T(xn−yn) → 0. If 〈Txn,y〉→ 〈x,y〉,
then

|〈Tyn,y〉−〈x,y〉| = |〈Tyn −Txn + Txn −x,y〉|
≤ ‖T(yn −xn)‖‖y‖ + |〈Txn −x,y〉|
→ 0 as n →∞

Hence f0 is well defined.
Since f0 is a bounded linear functional on the Hilbert space H, by Riesz rep-

resentation theorem there exist a unique y∗ ∈ H such that f0(x) = 〈x,y∗〉. Thus
〈Tx,y〉 = 〈x,y∗〉 ∀x ∈ D(T). Define T∗ : D(T∗) → H by T∗y = y∗. Then T∗
is well-defined. Also 〈Tx,y〉 = 〈x,T∗y〉 ∀x ∈ D(T), ∀y ∈ D(T∗).

Consider an ill-posed operator equation

Tx = y (2)

where T is a closed densely defined operator on H.

Definition 2.1. [7]
Let T ∈ C(H) be densely defined. Then there exist a unique densely defined

operator T† ∈ C(H) with domain D(T†) = R(T)⊕R(T)⊥ satisfies the following
properties

(i) TT†y = P
R(T)

y for all y ∈ D(T†),
(ii) T†Tx = QN(T)⊥x for all x ∈ D(T).
(iii) N(T†) = R(T)⊥.
where P and Q are the orthogonal projection on to R(T) and N(T⊥) respec-

tively. The operator T† is called the Moore-Penrose inverse of T .

For y ∈ D(T†), let Sy = {x ∈ D(T) : ‖Tx−y‖ ≤ ‖Tu−y‖ ∀u ∈ D(T)}.
Then u ∈ Sy is called least square solution of the operator equation (2). Note that
‖T†y‖ ≤ ‖x‖ ∀x ∈ Sy, is called least square solution of minimal norm and is
denoted by x̂ [7].

If R(T) is not closed, then T† is not continuous. Now we introduce a Tikhonov
type regularization procedure for finding an approximate solution for T†y.

3 Tikhonov type regularization
In this section we introduce a Tikhonov type regularization procedure for solv-

ing (2).

175



E Shine Lal, P Ramya

Lemma 3.1. Let T ∈ C(H) be densely defined and α > 0. Then T∗T + αI and
TT∗+αI are bijective closed densely defined operators on H. Also (TT∗+αI)−1

and (T∗T + αI)−1 are bounded, self adjoint operators on H.

Proof. Let T ∈ C(H) and α > 0. By proposition 2.1, we have T∗ ∈ C(H).
Hence, (TT∗ + αI) and (T∗T + αI) are closed densely defined operators on
H. Since 〈(TT∗ + αI)x,x〉 = 〈T∗x,T∗x〉 + α〈x,x〉 ≥ 0,∀x ∈ D(T∗), we have
(TT∗+αI) is a positive operator. Similarly (T∗T +αI) is also a positive operator.
Since T∗T + αI is positive,

‖(T∗T + αI)x‖‖x‖≥ 〈(T∗T + αI)x,x〉
= 〈T∗Tx,x〉 + α‖x‖2

≥ α‖x‖2 ∀x ∈ H

Thus
‖(T∗T + αI)x‖≥ α‖x‖∀x ∈ H (3)

Since T∗T + αI is bounded below, it is one-one and its inverse from the range is
continuous. Also R(T∗T + αI) is closed. Since T∗T + αI is also self adjoint,
R(T∗T + αI) = N(T∗T + αI)⊥ = H. Hence T∗T + αI is onto. Therefore
(T∗T + αI)−1 ∈ B(H). Similary (TT∗ + αI)−1 ∈ B(H). From (3),
‖(T∗T + αI)−1‖≤

1

α
.

Theorem 3.1. Let T ∈ C(H) be densely defined. Then T∗(TT∗ + αI)−1 and
T(T∗T + αI)−1 are bounded operators on H. Also ‖ T∗(TT∗ + αI)−1 ‖≤ 1√

α
and ‖ T(T∗T + αI)−1 ‖≤ 1√

α
.

Proof. We have (T∗T + αI)−1T∗T = I −α(T∗T + αI)−1
Since 〈(T∗T + αI)−1x,x〉≥ 0 ∀x ∈ H,

〈(T∗T + αI)−1T∗Tx,x〉 = 〈I −α(T∗T + αI)−1x,x〉
= 〈x,x〉−α〈(T∗T + αI)−1x,x〉≤ 〈x,x〉.

Since (T∗T + αI)−1T∗T self adjoint, ‖(T∗T + αI)−1T∗T‖≤ 1.
Let x ∈ H.

‖T∗(TT∗ + αI)−1x‖2 = 〈T∗(TT∗ + αI)−1x,T∗(TT∗ + αI)−1x〉
= 〈TT∗(TT∗ + αI)−1x, (TT∗ + αI)−1x〉
= 〈(TT∗ + αI)−1TT∗x, (TT∗ + αI)−1x〉
≤ ‖(TT∗ + αI)−1TT∗x‖‖(TT∗ + αI)−1x‖

≤
1

α
‖x‖2

176



Tikhonov type regularization for unbounded operators

we have ‖T∗(TT∗ + αI)−1x‖2 ≤
1

α
‖x‖2 ∀x ∈ H.

Thus ‖T∗(TT∗ + αI)−1‖≤
1
√
α

. Hence T∗(TT∗ + αI)−1 is bounded. Similarly

T(T∗T + αI)−1 is bounded.

Lemma 3.2. [7]
Let T ∈ C(H) be densely defined. Then

(i) (TT∗ + I)−1T ⊆ T(T∗T + I)−1
(ii) (T∗T + I)−1T∗ ⊆ T∗(TT∗ + I)−1

Remark 3.1. From Theorem 3.1 , we have T∗(TT∗ + αI)−1 and T(T∗T + αI)−1
are bounded. Therefore from Lemma 3.2, we have (TT∗ + αI)−1T and
(T∗T + αI)−1T∗ are bounded.

Lemma 3.3. Let T ∈ C(H) be densely defined. For every x ∈ D(T) ∩ N(T)⊥
‖α(T∗T + αI)−1x‖−→ 0,as α → 0.

Proof. Let Tα = α(T∗T + αI)−1, α > 0.

From (3.1) we have ‖(T∗T + αI)−1‖≤
1

α
. Hence ‖Tα‖≤ 1 for every α > 0. Let

u ∈ R(T∗T) then there exist v ∈ D(T∗T) such that T∗Tv = u.

‖Tαu‖ = ‖TαT∗Tv‖
= α‖(T∗T + αI)−1T∗Tv‖
≤ α‖(T∗T + αI)−1T∗T‖‖v‖
≤ α‖v‖

Hence ‖Tαu‖≤ α‖v‖∀u ∈ R(T∗T).
Thus for every u ∈ R(T∗T), ‖α(T∗T + αI)−1u‖ −→ 0 as α −→ 0. Since
R(T∗T) = N(T)⊥, ‖α(T∗T + αI)−1x‖−→ 0, ∀x ∈ D(T) ∩N(T)⊥.

Theorem 3.2. Let T ∈ C(H) be densely defined and Rα = (T∗T + αI)−1T∗.
Then {Rα}α>0 is a regularization family for (2).

Proof. Let y ∈ D(T∗). Then (T∗T + αI)x̂ = T∗y + αx̂.
Hence x̂ = (T∗T + αI)−1(T∗y + αx̂). Thus

T†y −Rαy = x̂− (T∗T + αI)−1T∗y
= (T∗T + αI)−1(T∗y + αx̂) − (T∗T + αI)−1T∗y
= (T∗T + αI)−1αx̂

177



E Shine Lal, P Ramya

Hence ‖T†y −Rαy‖ = α‖(T∗T + αI)−1x̂‖.
Since x̂ ∈ D(T) ∩N(T)⊥, by Lemma 3.3, ‖T†y −Rαy‖−→ 0 as α −→ 0.
Thus {Rα}α>0 is a regularization family for (2).

4 Order estimate
In this section we find an error estimate for the regularization family

Rα = (T
∗T + αI)−1T∗, where T is a closed densely defined operator. We use the

following lemmas.

Lemma 4.1. [7]
For T ∈ C(H) we have the following
(i) If µ ∈ C and λ ∈ σ(T) then λ + µ ∈ σ(T + µI)
(ii) If α ∈ C and λ ∈ σ(T) then αλ ∈ σ(αT )
(iii) σ(T2) = {λ2 : λ ∈ σ(T)}

Lemma 4.2. [7]
Let T ∈ L(H) be a positive operator. Then the following results bold.
(i) T† is positive.
(ii) σ(T) = σa(T)
(iii) 0 /∈ σ(I + T) that is (I + T)−1 ∈ B(H)
(iv) If 0 /∈ σ(T) then 0 6= λ ∈ σ(T) if and only if

1

λ
∈ σ(T−1)

Theorem 4.1. Suppose T ∈ C(H) is densely defined positive operator. Then for
every α > 0

σ
(

(T + αI)−2T
)

=
{ λ

(λ + α)2
: λ ∈ σ(T)

}
Proof. Since T is positive, T + αI is bijective.
Also (T + αI)−2T = (T + αI)−1 −α(T + αI)−2.
From Lemmas 4.1, 4.2 for α,λ > 0, we have
λ ∈ σ(T) if and only if (λ + α)−1 ∈ σ

(
(T + αI)−1

)
.

Hence

σ
(

(T + αI)−2T
)

=
{
µ−αµ2 : µ ∈ σ

(
(T + αI)−1

)}
=
{ 1
λ + α

−
α

(λ + α)2
: λ ∈ σ(T)

}
=
{ λ

(λ + α)2
: λ ∈ σ(T)

}
.

178



Tikhonov type regularization for unbounded operators

Corolary 4.1. Let T ∈ C(H) be densely defined and α > 0.

Then ‖(T∗T + αI)−1T∗‖ = sup
{ √λ
λ + α

: λ ∈ σ(T∗T)
}
6

1

2
√
α

Proof. We have Rα = (T∗T + αI)−1T∗. Hence R∗αRα = T(T
∗T + αI)−2T∗.

From Lemma 2.2 in [2], we have R∗αRα = (TT
∗ + αI)−2TT∗.

Since R∗αRα is self adjoint and bounded, ‖Rα‖2 = ‖R∗αRα‖
= Sup

{
|k| : k ∈ σ(R∗αRα)

}
= Sup

{ λ
(λ + α)2

: λ ∈ σ(TT∗)
}

‖Rα‖ = Sup
{ √λ
λ + α

: λ ∈ σ(TT∗)
}
.

Since 2
√
αλ(λ + α)−1 6 1 for λ,α > 0, we have ‖Rα‖ 6

1

2
√
α
.

Now we find an order estimate for Rα.

Corolary 4.2. Let T ∈ C(H) is densely defined and Rα = (T∗T + αI)−1T∗.
For every α > 0 and δ > 0, let yδ ∈ H be such that ‖y − yδ‖ 6 δ. Then
‖Rαy −Rαyδ‖ 6

δ

2
√
α
.

Proof. For ‖y −yδ‖ 6 δ,

‖Rαy −Rαyδ‖ 6 ‖Rα‖‖y −yδ‖

6
1

2
√
α
‖y −yδ‖

6
δ

2
√
α

Theorem 4.2. Let T ∈ C(H) is densely defined and Rα = (T∗T +αI)−1T∗. Then
‖x̂−Rαyδ‖ 6 ‖x̂−Rαy‖ +

δ

2
√
α

. If α = α(δ) is chosen such that α(δ) −→ 0

and
δ

√
α(δ)

−→ 0 as δ −→ 0, then ‖x̂−Rδα(δ)‖−→ 0 as δ −→ 0.

Proof. ‖x̂−Rαyδ‖≤‖x̂−Rαy‖ + ‖Rαy −Rαyδ‖
≤‖x̂−Rαy‖ +

δ

2
√
α

by Theorem 3.6, ‖x̂−Rαy‖−→ 0 as α −→ 0.

179



E Shine Lal, P Ramya

References
[1] P Mathe B Hofmann and H. R.V Weizsacker. Regularization in Hilbert space

under Unbounded operators and general source conditions. IOP Publishing
Inverse Problems, 2009.

[2] C. W Groetsch. Spectral methods for linear inverse problems with unbounded
operators. Journal of approximation theory, 70:16–28, 1992.

[3] S. H Kulkarni and M. T Nair. A characterization of closed range operators.
Indian J.Pure Appl. Math., 31:353–361, 2000.

[4] S. H Kulkarni and G Ramesh. Projection method for inversion of unbounded
operators. Indian J.Pure Appl. Math, 39:185–202, 2008.

[5] A. K Louis and P Maass. A mollifier method for linear operator equations of
the first kind. inverse problem, 1990.

[6] A. G Rahim. Ill-posed problems with unbounded operators. Journal of math-
ematical analysis and application, 325:490–495, 2007.

[7] M. T Nair S. H Kulkarni and G Ramesh. Some properties of unbounded
operators with closed range. Proc.Indian Acadamy of science, 118:613–625,
2008.

180