CUBO A Mathematical Journal
Vol.14, No¯ 01, (111–117). March 2012

More on Approximate Operators

Philip J. Maher

Mathematics And Statistics Group,

Middlesex University, Hendon Campus, The Burrough,

London Nw4 4 Bt, United Kingdom.

email: p.maher@mdx.ac.uk

and

Mohammad Sal Moslehian

Department Of Pure Mathematics,

Centre Of Excellence In Analysis On Algebraic Structures,

(CEAAS), Ferdowsi University Of Mashhad, P.O. Box 1159, Mashhad 91775, Iran.

email: moslehian@ferdowsi.um.ac.ir, moslehian@member.ams.org

ABSTRACT

This note is a continuation of the work on (p, �)–approximate operators studied by

Mirzavaziri, Miura and Moslehian. [4]. We investigate approximate partial isometries

and approximate generalized inverses. We also prove that if T is an invertible contrac-

tion satisfying ‖TT∗T − T‖ < � <
2

3
√
3
. Then there exists a partial isometry V such

that ‖T − V‖ < K� for K > 0.

RESUMEN

Esta trabajo es una continuación del trabajo sobre operadores (p, �)–aproximados es-

tudiados por Mirzavaziri, Miura y Moslehian [4]. Investigamos isometŕıas parciales



112 Philip J. Maher and Mohammad Sal Moslehian CUBO
14, 1 (2012)

aproximadas e inversas aproximadas generalizadas. También probamos que si T es

una contracción invertible que satisface ‖TT∗T − T‖ < � <
2

3
√
3

entonces existe una

isometŕıa parcial V tal que ‖T − V‖ < K� para K > 0.

Keywords and Phrases: Hilbert space; approximation; unitary; partial isometry; polar decom-

position; (p, �)-approximate operator

2010 AMS Mathematics Subject Classification: Primary 47A55; secondary 39B52.

1 Introduction

This note is a continuation of the work on (p, �)–approximate operators and operator approxima-

tion studied in [4]. Mirzavaziri et al investigated (p, �)–approximate (co) isometries and (p, �)–

approximate unitaries. For example, a (p, �)–approximate isometry is defined as an operator T in

L(H) for which
‖ [T∗T − I] f‖ ≤ �‖f‖p (1.1)

where p is a real number and � a fixed positive number. They also proved, for example, the

following result on unitary approximation: if to each 0 < � < 1 an operator T in L(H) satisfies
‖T∗T − I‖ ≤ � and ‖TT∗ − I‖ ≤ � there corresponds a unitary operator U such that ‖T − U‖ < �.

In section 2 we investigate approximate partial isometries and approximate generalized in-

verses. In section 3 we investigate operator approximation. We prove (Theorem 3.2 below) that

an invertible contraction T satisfying ‖TT∗T − T‖ < � <
2

3
√
3

can be approximated by a partial

isometry.

Recall that a contraction T in L(H) is an operator such that ‖T‖ ≤ 1. Recall that the polar
decomposition of an operator T says that T can be expressed uniquely as T = U|T |, provided

KerU = Ker|T |, where U is a partial isometry. By definition, a partial isometry U is a isometric on

(KerU)+; and |T | denotes the positive square root of T∗T.

2 Approximate Operators

In (1.1) (the example of a (p, �)–approximate isometry) there is no question of letting � → 0; for
otherwise, the subject would collapse into triviality. For fixed � the upshot of this section is that

the (p, �)–approximate operators considered here coincide with their ordinary (exact) counterparts

provided p 6= 1. In the cases studied here the operator T we are concerned with must satisfy an
operator equation of the form

F(T, T∗, T−) = 0



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More on Approximate Operators 113

(where T− is a generalized inverse of T: see Example 2.2 below). Our results hinge on the following

lemma.

Lemma 2.1. Let p be a real number such that p 6= 1 and let � > 0. If

‖F(T, T∗, T−)‖ ≤ �‖f‖p (2.1)

then ‖F(T, T∗, T−)‖ = 0.

Proof. In (2.1) substitute rf for f where r > 0. Then, by the linearity of T,

‖F(T, T∗, T−)‖ ≤ �rp−1‖f‖p. (2.2)

If p < 1 so that rp−1 = r−k where k > 0 then �rp−1‖f‖p =
�‖f‖p

rk
→ 0 as r → ∞. If p > 1 then

�rp−1‖f‖p → 0 as r → 0.
Example 2.2. (Partial isometries). There is the following (equivalent) algebraic definition of

a partial isometry: T is a partial isometry if T = TT∗T [3, Problem 127, Corollary 3]. Given a real

number p and � > 0, a (p, �)–approximate partial isometry is an operator T in B(H) for which

‖ [TT∗T − T] f‖ ≤ ‖f‖p.

Let F(T, T∗) = TT∗T − T; if p 6= 1 then, by Lemma 2.1, F(T, T∗) = 0 i.e. T is an (exact) partial
isometry.

Counterexample 2.3. This shows that the condition p 6= 1 in Lemma 2.1 cannot be dropped.
Let

T =




0 0
√
�

0
√
� 0

√
� 0 0


 ∈ M3(C).

Then, for 1 < � < 2,

‖ [TT∗T − T] f‖ = |� − 1|
√
� ‖f‖ <

√
� ‖f‖ < � ‖f‖

yet T is not a partial isometry since ‖Tf‖ =
√
� ‖f‖ for all f in H.

A (p, �)–approximate normal partial isometry is an operator T in B(H) for which∥∥[T∗T2 − T] f∥∥ ≤ �‖f‖p (a)
and

‖
[
T2T∗ − T

]
f‖ ≤ �‖f‖p (b)



114 Philip J. Maher and Mohammad Sal Moslehian CUBO
14, 1 (2012)

for given � > 0 and a real number p. Let F1(T, T
∗) = T∗T2 − T and F2(T, T

∗) = T2T∗ − T; then if

p 6= 1 Lemma 2.1 applied to F1 and F2 yields (a) T∗T2 = T and (b) T2T∗ = T. Therefore, from (a),
T∗T2T∗ = TT∗ and, from (b), T∗T2T∗ = T∗T. Thus, T is normal and hence by, say (a), TT∗T = T.

Example 2.4. (Generalized inverses). An operator T− is said to be a generalized inverse of

the operator T if TT−T = I. An operator T in B(H) has a generalized inverse if and only if RanT is
closed [7, p. 261]. For an operator T, with closed range, its Moore – Penrose inverse T+ has range

RanT+ = (KerT)⊥ and satisfies

TT+T = T (i)

T+TT+ = T+ (ii)

(TT+)∗ = TT+ (iii)

(T+T)∗ = T+T (iv)




(MP)

and, further, T+ is uniquely determined by these properties. If an operator T− satisfies properties

(i), (iii) [(i), (iv)] it will be called a (i), (iii) [(i), (iv)] inverse of T. A (p, �)–approximate generalized

inverse of T is an operator T− in B(H) for which

‖[TT−T − T]f‖ ≤ �‖f‖p

for � > 0 and real p. Let F1(T, T
+) = TT+T − T, F2(T, T

+) = T+TT+ − T+, F3(T, T
+) = (TT+)∗ −

TT∗ − T and F4(T, T
+) = (T+T)∗ − T+T; then a (p, �)–approximate Moore–Penrose inverse pf T is

an operator T+ in B(H) for which

‖Fi(T < T+)f‖ ≤ �‖f‖p

for i = 1, . . . , 4 and for � > 0 and real p. Let F(T, T−) = TT−T − T; then if p 6= 1, by Lemma
2.1, F(T, T−) = 0 i.e. T− is a (exact) generalized inverse of T; and, for p 6= 1, applying Lemma 2.1
successively to F1, F2, F3 and F4 yields F1 = F2 = F3 = F4 = 0 i.e. T

+ satisfies (MP) so that T+ is

the (exact) Moore – Penrose inverse of T.

Counterexample 2.5. Again, we cannot drop the condition p 6= 1 in Lemma 2.1. Take T = �S
where

S =


12 12

1
2

1
2




and 0 < � ≤ 1. Let T ′ = T. Then

‖[TT ′T − T]f‖ = |�3 − �|‖Sf‖ ≤ �|�2 − 1|‖S‖‖f‖
= �|�2 − 1|‖f‖ ≤ �‖f‖

yet T ′ = �

[
1
2

1
2

1
2

1
2

]
is not a generalized inverse of T (except, as can be verified, if � = 1) for, e.g.,

if � = 1
2
then TT ′T = 1

4
T.



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More on Approximate Operators 115

Does the algebraic structure of approximate operators mirror that of their exact counterparts?

For approximate isometries the answer is “ yes”. The product of two (exact) isometries is an (ex-

act) isometry. The same is true for approximate isometries.

Proposition 2.6. The product of two (p, �)–approximate isometries is a (p, �′)–approximate isom-

etry.

Proof. For p 6= 1, by Lemma 2.1, a (p, �)–approximate isometry is an (exact) isometry. Therefore,
we need to prove this result in the case of p = 1. Accordingly, let T1 and T2 be two approximate

isometries such that

‖[T∗1 T1 − I]f‖ ≤ �1‖f‖ and ‖[T
∗
2 T2 − I]f‖ ≤ �2‖f‖

for �1 > 0, �2 > 0 for all f in H. Assertion: if ‖[T∗T − I]f‖ ≤ �‖f‖ for � > 0 and for all f in H then
‖T‖2 ≤ � + 1. Proof of assertion:

‖T∗Tf‖ = ‖[T∗T − I]f + f‖ ≤ ‖[T∗T − I]f‖ + ‖f‖ ≤ (� + 1)‖f‖

whence the result ‖T‖2 = ‖T∗T‖ ≤ � + 1 follows by taking supremum over unit vectors.

Now,

‖[(T1T2)∗(T1T2) − I]f‖ = ‖[T∗2 (T
∗
1 T1 − I)T2 − I + T

∗
2 T2]f‖

≤ ‖T∗2 ‖‖[T
∗
1 T1 − I]T2f‖ + ‖[T

∗
2 T2 − I]f‖

≤ ‖T∗2 ‖�1‖T2f‖ + �2‖f‖
≤ ((�2 + 1)�1 + �2)‖f‖ = (�1 + �1�2 + �2)‖f‖.

We cannot expect a similar result about product of approximate partial isometries since it is

not true that the product of two (exact) partial isometries is an (exact) partial isometry.

3 Approximating Contractions

We need the following lemma.

Lemma 3.1. Let T ≤ 0, ‖T‖ ≤ 1 and ‖T3 − T‖ < � <
2

3
√
3
. Then there is a self–adjoint partial

isometry S such that ‖T − S‖ < K� for a certain constant K > 0.

Proof. The conditions T ≤ 0, ‖T‖ ≤ 1 imply that sp(T) ⊆ [−1, 0]. Let δ1, δ2(δ1 < δ2) be the
solutions of polynomial equation t3 − t = � in [−1, 0]. Then |t3 − t| < � for all sp(T), whence

t ∈ sp(T) ⊆ [−1, δ1] ∪ [δ2, 0].



116 Philip J. Maher and Mohammad Sal Moslehian CUBO
14, 1 (2012)

- t

6

2

3
√
3

s = t3−t

−1√
3

δ1 δ2

s = �

s

Therefore,

ϕ(t) =

{
−1 t ∈ [−1, δ1]
0 t ∈ [δ2, 0]

is a continuous function on sp(T). Using the functional calculus, we observe that S = ϕ(T) satisfies

S∗ = S and SS∗S = S and

‖T − S‖ = sup
t∈sp(T)

|ϕ(t) − t| = max{1 + δ1, |δ2|} < K�,

for certain K > 0.

Now we are ready to proof our next result.

Theorem 3.2. Let T be an invertible contraction and let ‖TT∗T − T‖ < � <
2

3
√
3
. Then there

exists a partial isometry V such that ‖T − V‖ < K� for a certain constant K > 0.

Proof. Let T = U|T | be the polar decomposition of T. It is known that U is unitary, since T is

invertible. Then

‖|T |3 − |T |‖ = ‖U|T |T∗T − U|T |‖ = ‖TT∗T − T‖ < �

since the operator norm is unitarily invariant in the sense that ‖VXW‖ = ‖X‖ for all arbitrary
operators X and all unitaries V, W in B(H). Utilizing Lemma 3.1 for −|T | we get a self-adjoint
partial isometry S such that ‖|T | − S‖ < K� for a certain positive number K. Hence

‖T − US‖ = ‖U|T | − US‖ = ‖|T | − S‖ < K�

Since US(US)∗US = US, the operator US turns into a partial isometry V.

If T acts on a finite dimensional Hilbert space H, then the partial isometry U appeared in the
polar decomposition of T is a unitary. So the proof of Theorem 3.2 above follows the following

fact.



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More on Approximate Operators 117

Corollary 3.3. Let A be an m × m contractive matrix such that ‖AA∗A − A‖ < � <
2

3
√
3
. Then

there exists a partial isometry V such that ‖A − V‖ < K� for a certain constant K > 0.

Acknowledgement. The second author was supported by a grant from Ferdowsi University of

Mashhad (No, MP89163MOS).

Received: June 2011. Revised: August 2011.

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