International Journal of Analysis and Applications
ISSN 2291-8639
Volume 5, Number 1 (2014), 1-9
http://www.etamaths.com

GENERALIZED NORMS INEQUALITIES FOR ABSOLUTE

VALUE OPERATORS

ILYAS ALI∗, HU YANG, ABDUL SHAKOOR

Abstract. In this article, we generalize some norms inequalities for sums,

differences, and products of absolute value operators. Our results based on

Minkowski type inequalities and generalized forms of the Cauchy-Schwarz in-
equality. Some other related inequalities are also discussed.

1. Introduction

In this article, notations are same as in [3], for reader convenience we recall that
let H be a complex separable Hilbert space and B(H) denote the C∗-algebra of all
bounded linear operators on H. Let |A| denote the absolute value of A ∈ B(H),
and is defined as |A| = (A∗A)

1
2 , where A∗ is the adjoint operator of A. If A is

compact operator on complex separable Hilbert space H, then the singular values
of A enumerated as s1(A) ≥ s2(A) ≥ ... which are the eigenvalues of positive
operator |A|. A norm |||.||| stand for untarily invariant norm i.e., a norm with
the property that |||UAV ||| = |||A||| for all A and for all unitary operators U,
V in B(H). Operator norm and Schatten p-norms are denoted as ||.|| and ||.||P
respectively. Except the operator norm, which is defined on all of B(H), each
unitarily invariant norm is defined on an ideal in B(H). When we use the symbol
|||A||| it is implicit understood that operator A is in this ideal.

For 0 < p < 1, a norm ‖.‖p defines a quasi-norm. For this norm it is well-known
that

‖A + B‖p ≤ 2
1
p
−1 (‖A‖p + ‖B‖p) .(1.1)

By the definition of the Schatten p-norm, we have

‖ | A |r ‖p ≤‖A‖prp,(1.2)

where r,p are real numbers. Also, since the singular values of | A |r and | A∗ |r are
same, so

||| | A |r ||| = ||| | A∗ |r |||.(1.3)

The unitarily invariant norms for differences of the absolute values of Hilbert
space operators have attracted the attention of several mathematicians. It has been

2000 Mathematics Subject Classification. 47A30; 47A63; 47B10.
Key words and phrases. Unitarily invariant norm; Schatten p-norm; Cauchy-Schwarz inequal-

ity; Minkowski inequality; Absolute value operators.

c©2014 Authors retain the copyrights of their papers, and all
open access articles are distributed under the terms of the Creative Commons Attribution License.

1



2 ALI, YANG AND SHAKOOR

proved by K. Shebrawi and H. Albadawi in [3] that if Ai,Bi,Xi (i = 1, 2, ...,n) be
operators in B(H) such that Xi is self adjoint operator and 0 < r ≤ 1 . Then

|||
n∑
i=1

| A∗iXiBi + B
∗
i XiAi |

r |||

≤ 2n1−
r
2

n∑
i=1

|||(A∗i | Xi | Ai)
r |||

1
2 |||(B∗i | Xi | Bi)

r |||
1
2 ,(1.4)

which leads to the following inequality

||| | A |2r − | B |2r ||| ≤ 21−r||| | A + B |2r |||
1
2 ||| | A−B |2r |||

1
2 .(1.5)

Inequality (1.5) generalize the result presented by Bhatia in [5] as follows:

||| | A | − | B | ||| ≤
√

2||| | A + B | |||
1
2 ||| | A−B | |||

1
2 .(1.6)

K. Shebrawi and H. Albadawi also proved in [3] that if A,B,X be operators in
B(H) such that X is self adjoint operator and 0 < r ≤ 1

2
, 1 ≤ p ≤ 2, then

‖ | A∗XB + B∗XA |r ‖p

≤ 2
1
p
−r+ 1

2‖(A∗ | X | A)r ‖
1
2
p‖(B∗ | X | B)

r ‖
1
2
p ,(1.7)

this leads to the following inequality

‖ | A |2r − | B |2r ‖p ≤ 2
1
p
−2r+ 1

2‖ | A + B |2r ‖
1
2
p‖ | A−B |2r ‖

1
2
p .(1.8)

Inequality (1.8) generalize the following result in [5]

‖ | A | − | B | ‖p ≤ 2
1
p
−1

2‖ | A + B | ‖
1
2
p‖ | A−B | ‖

1
2
p ,(1.9)

where 1 ≤ p ≤ 2.
This article we have organized as: In Section 2, we generalize the inequality

(1.5) and also we discuss some other related results. In Section 3, we present some
Schatten p-norms inequalities, one of which generalize the inequality (1.8).

2. Generalized unitarily invariant norms inequalities for absolute
value operators

In this section, we generalize some unitarily invariant norms inequalities for ab-
solute value operators. Our results based on several lemmas. First two lemmas
contain norm inequalities of Minkowski type and generalized forms of the Cauchy-
Schwarz inequality, see [4] and [2] respectively.

Lemma 2.1. Let Ai,Bi ∈ B(H), i = 1, 2, ...,n. Then

n
1
2
−1
r |||

n∑
i=1

| Ai + Bi |r |||
1
r ≤ 2

1
r
−1

(
|||

n∑
i=1

| Ai |r |||
1
r + |||

n∑
i=1

| Bi |r |||
1
r

)
(2.1)

for 0 < r ≤ 1,

n−|
1
r
−1

2
||||

n∑
i=1

| Ai + Bi |r |||
1
r ≤ |||

n∑
i=1

| Ari | |||
1
r + |||

n∑
i=1

| Bi |r |||
1
r(2.2)



GENERALIZED NORMS INEQUALITIES FOR ABSOLUTE VALUE OPERATORS 3

for r ≥ 1, and

n−(1−
1
p
)/r|||

n∑
i=1

| Ai + Bi |r |||
1
r
p ≤ |||

n∑
i=1

| Ari | |||
1
r
p + |||

n∑
i=1

| Bi |r |||
1
r
p(2.3)

for 1 ≤ p,r < ∞.

Lemma 2.2. For A,B,X ∈ B(H), for all unitarily invariant norms and for all
positive real numbers µ1, µ2 and r such that µ

−1
1 + µ

−1
2 = 1, we have

||| | A∗XB |r ||| ≤ ||| | AA∗X |
µ1r
2 |||

1
µ1 ||| | XBB∗ |

µ2r
2 |||

1
µ2 ,(2.4)

and also, if f and g are nonnegative continuous functions on [0,∞) satisfying
f(t)g(t) = t, for all t ∈ [0,∞), then we have

||| | A∗XB |r ||| ≤ |||
(
A∗f2(| X∗ |)A

)µ1r
2 |||

1
µ1 |||

(
B∗g2(| X |)B

)µ2r
2 |||

1
µ2 .(2.5)

For following two lemmas see [1] and [6, pp. 293, 294].

Lemma 2.3. Let A be a positive operator in B(H). Then for every normalized
unitarily invariant norm (i.e.,|||diag(1, 0, 0, ..., 0)||| = 1), we have

|||A|||r ≤ |||Ar|||(2.6)

for 0 ≤ r ≤ 1 and

|||Ar||| ≤ |||A|||r(2.7)

for r ≥ 1.

Lemma 2.4. Let A and B be a positive operator in B(H). Then

|||Ar −Br||| ≤ ||| | A−B |r |||(2.8)

for 0 ≤ r ≤ 1 and

||| | A−B |r ||| ≤ |||Ar −Br|||(2.9)

for r ≥ 1.

Last lemma is a consequence of the concavity (convexity) of the function f(t) =
tr, 0 ≤ r ≤ 1 (r ≥ 1).

Lemma 2.5. Let a and b be two positive real numbers

(a + b)r ≤ ar + br(2.10)

for 0 ≤ r ≤ 1 and

(a + b)r ≤ 2r−1(ar + br)(2.11)

for r ≥ 1.

Theorem 2.1. Let Ai,Bi,Xi (i = 1, 2, ...,n) be operators in B(H) such that Xi is
self adjoint operator and if µ1, µ2 are positive real numbers, such that µ

−1
1 +µ

−1
2 = 1



4 ALI, YANG AND SHAKOOR

and 0 < r ≤ 1 . Then

|||
n∑
i=1

| A∗iXiBi + B
∗
i XiAi |

r |||

≤ 2n1−
r
2

n∑
i=1

||| | AiA∗iXi |
µ1r
2 |||

1
µ1 ||| | XiBiB∗i |

µ2r
2 |||

1
µ2 ,(2.12)

also, if f and g are nonnegative continuous functions on [0,∞) satisfying f(t)g(t) =
t, for all t ∈ [0,∞), then we have

|||
n∑
i=1

| A∗iXiBi + B
∗
i XiAi |

r |||

≤ 2n1−
r
2

n∑
i=1

|||
(
A∗if

2(| Xi |)Ai
)µ1r

2 |||
1
µ1 |||

(
B∗i g

2(| Xi |)Bi
)µ2r

2 |||
1
µ2 .

(2.13)

Proof. By applying (2.1), the triangle inequality, (1.3) and (2.4), respectively, we
obtain

|||
n∑
i=1

| A∗iXiBi + B
∗
i XiAi |

r |||
1
r

≤ 2
1
r
−1n

1
r
−1

2

(
|||

n∑
i=1

| A∗iXiBi |
r |||

1
r + |||

n∑
i=1

| B∗i XiAi |
r |||

1
r

)

≤ 2
1
r
−1n

1
r
−1

2


( n∑

i=1

||| | A∗iXiBi |
r |||

)1
r

+

(
n∑
i=1

||| | B∗i XiAi |
r |||

)1
r




≤ 2
1
r n

1
r
−1

2

(
n∑
i=1

||| | A∗iXiBi |
r |||

)1
r

≤ 2
1
r n

1
r
−1

2

(
n∑
i=1

||| | AiA∗iXi |
µ1r
2 |||

1
µ1 ||| | XiBiB∗i |

µ2r
2 |||

1
µ2

)1
r

.

The proof is completed. By applying (2.5) and the proof of the first part of Theo-
rem 2.1, we can obtain (2.13). �

Replace Ai, Bi by Ai + Bi, Ai −Bi respectively and also take f(t) = g(t) = t
1
2

in Theorem 2.1, then, we obtain the following result.

Corollary 2.1. Let Ai,Bi,Xi (i = 1, 2, ...,n) be operators in B(H) such that Xi is
self adjoint operator and if µ1, µ2 are positive real numbers, such that µ

−1
1 +µ

−1
2 = 1

and 0 < r ≤ 1. Then

2r−1n
r
2
−1|||

n∑
i=1

| A∗iXiAi −B
∗
i XiBi |

r |||

≤
n∑
i=1

||| | (Ai + Bi)(Ai + Bi)∗Xi |
µ1r
2 |||

1
µ1 ||| | Xi(Ai −Bi)(Ai −Bi)∗ |

µ2r
2 |||

1
µ2 ,



GENERALIZED NORMS INEQUALITIES FOR ABSOLUTE VALUE OPERATORS 5

and

2r−1n
r
2
−1|||

n∑
i=1

| A∗iXiAi −B
∗
i XiBi |

r |||

≤
n∑
i=1

|||((Ai + Bi)∗ | Xi | (Ai + Bi))
µ1r
2 |||

1
µ1 |||((Ai −Bi)∗ | Xi | (Ai −Bi))

µ2r
2 |||

1
µ2 .

By similar way applying to the proof of theorem 2.1, based on the inequality
(2.2), we can obtain the following result.

Theorem 2.2. Let Ai,Bi,Xi (i = 1, 2, ...,n) be operators in B(H) such that Xi is
self adjoint operator and if µ1, µ2 are positive real numbers, such that µ

−1
1 +µ

−1
2 = 1

and r ≥ 1. Then

|||
n∑
i=1

| A∗iXiBi + B
∗
i XiAi |

r |||

≤ 2rn|1−
r
2
|
n∑
i=1

||| | AiA∗iXi |
µ1r
2 |||

1
µ1 ||| | XiBiB∗i |

µ2r
2 |||

1
µ2 ,(2.14)

also, if f and g are nonnegative continuous functions on [0,∞) satisfying f(t)g(t) =
t, for all t ∈ [0,∞), then we have

|||
n∑
i=1

| A∗iXiBi + B
∗
i XiAi |

r |||

≤ 2rn|1−
r
2
|
n∑
i=1

|||
(
A∗if

2(| Xi |)Ai
)µ1r

2 |||
1
µ1 |||

(
B∗i g

2(| Xi |)Bi
)µ2r

2 |||
1
µ2 .

(2.15)

Corollary 2.2. Let Ai,Bi,Xi (i = 1, 2, ...,n) be operators in B(H) such that Xi is
self adjoint operator and if µ1, µ2 are positive real numbers, such that µ

−1
1 +µ

−1
2 = 1

and r ≥ 1. Then

n−|1−
r
2
||||

n∑
i=1

| A∗iXiAi −B
∗
i XiBi |

r |||

≤
n∑
i=1

||| | (Ai + Bi)(Ai + Bi)∗Xi |
µ1r
2 |||

1
µ1 ||| | Xi(Ai −Bi)(Ai −Bi)∗ |

µ2r
2 |||

1
µ2 ,

and

n−|1−
r
2
||||

n∑
i=1

| A∗iXiAi −B
∗
i XiBi |

r |||

≤
n∑
i=1

|||((Ai + Bi)∗ | Xi | (Ai + Bi))
µ1r
2 |||

1
µ1 |||((Ai −Bi)∗ | Xi | (Ai −Bi))

µ2r
2 |||

1
µ2 .

Remark 2.1. If we take f(t) = tα and g(t) = t(1−α) for α ∈ [0, 1], then from
the inequality (2.13) we can obtain important special case. Also, if we take f(t) =



6 ALI, YANG AND SHAKOOR

g(t) = t
1
2 then from (2.13) we have

|||
n∑
i=1

| A∗iXiBi + B
∗
i XiAi |

r |||

≤ 2n1−
r
2

n∑
i=1

|||(A∗i | Xi | Ai)
µ1r
2 |||

1
µ1 |||(B∗i | Xi | Bi)

µ2r
2 |||

1
µ2 ,

which is the more general form of the inequality (1.4). Similar remark we can give
for the inequality (2.15), which is more general form of the inequality (2.19) in [3].

Our following result contains a promised generalization of (1.5).

Corollary 2.3. Let A and B be operators in B(H) and µ1, µ2 are positive real
numbers such that µ−11 + µ

−1
2 = 1 then

||| | A |2r − | B |2r ||| ≤ 21−r||| | A + B |µ1r |||
1
µ1 ||| | A−B |µ2r |||

1
µ2(2.16)

for 0 < r ≤ 1, and

||| | A∗A−B∗B |r ||| ≤ ||| | A + B |µ1r |||
1
µ1 ||| | A−B |µ2r |||

1
µ2(2.17)

for r ≥ 1.
Proof. By (2.8) and from second result of Corollary (2.1), we have

||| | A |2r − | B |2r ||| ≤ ||| || A |2 − | B |2|r |||

≤ 21−r||| | A + B |µ1r |||
1
µ1 ||| | A−B |µ2r |||

1
µ2 ,

for 0 < r ≤ 1. and inequality (2.17) is a special case of the first result of Corollary
(2.2).

Remark 2.2. Special cases of second results in Corollary 2.1 and 2.2 respectively
are: Let A,B,X be operators in B(H) such that X is self adjoint operator and if
µ1, µ2 are positive real numbers such that µ

−1
1 + µ

−1
2 = 1, then

||| | A∗XA−B∗XB |r |||

≤ 21−r|||((A + B)∗ | X | (A + B))
µ1r
2 |||

1
µ1 |||((A−B)∗ | X | (A−B))

µ2r
2 |||

1
µ2 .

for 0 < r ≤ 1, and

||| | A∗XA−B∗XB |r |||

≤ |||((A + B)∗ | X | (A + B))
µ1r
2 |||

1
µ1 |||((A−B)∗ | X | (A−B))

µ2r
2 |||

1
µ2 .

for r ≥ 1.

Corollary 2.4. Let A be an operator in B(H) and if µ1, µ2 are positive real
numbers such that µ−11 + µ

−1
2 = 1, then

||| | A∗A−AA∗ |r ||| ≤ 21+r||| | ReA |µ1r |||
1
µ1 ||| | ImA |µ2r |||

1
µ2 ,

for 0 < r ≤ 1, and

||| | A∗A−AA∗ |r ||| ≤ 22r||| | ReA |µ1r |||
1
µ1 ||| | ImA |µ2r |||

1
µ2 ,

for r ≥ 1.



GENERALIZED NORMS INEQUALITIES FOR ABSOLUTE VALUE OPERATORS 7

3. Generalized norm inequalities for the Schatten p-norm

Schatten p-norm for absolute value operators are discussed in this section. Our
these results refine some of the results in Section 2 and also, our first result leads
to a generalization of (1.8).

Theorem 3.1. Let A,B,X be operators in B(H) such that X is self adjoint
operator and if µ1, µ2 are positive real numbers such that µ

−1
1 + µ

−1
2 = 1, then

‖ | A∗XB + B∗XA |r ‖p

≤ 2
1
p
−r+ 1

2‖ | AA∗X |
µ1r
2 ‖

1
µ1
p ‖ | XBB∗ |

µ2r
2 ‖

1
µ2
p ,(3.1)

also, if f and g are nonnegative continuous functions on [0,∞) satisfying f(t)g(t) =
t, for all t ∈ [0,∞), then we have

‖ | A∗XB + B∗XA |r ‖p

≤ 2
1
p
−r+ 1

2‖
(
A∗f2(| X |)A

)µ1r
2 ‖

1
µ1
p ‖

(
B∗g2(| X |)B

)µ2r
2 ‖

1
µ2
p .(3.2)

for 0 < r ≤ 1
2

and 1 ≤ p ≤ 2.
Proof. By applying (1.2), (1.1), (2.10), (1.3) and (2.4) respectively, we obtain

‖ | A∗XB + B∗XA |r ‖p = ‖A∗XB + B∗XA‖rrp

≤
(

2
1
rp
−1 (‖A∗XB‖rp + ‖B∗XA‖rp)

)r
≤ 2

1
p
−r (‖A∗XB‖2rrp + ‖B∗XA‖2rrp)12

≤ 2
1
p
−r (‖ | A∗XB |r ‖2p + ‖ | B∗XA |r ‖2p)12

= 2
1
p
−r+ 1

2‖ | A∗XB |r ‖p

≤ 2
1
p
−r+ 1

2‖ | AA∗X |
µ1r
2 ‖

1
µ1
p ‖ | XBB∗ |

µ2r
2 ‖

1
µ2
p .

The proof is completed. By applying (2.5) and the proof of the first part of Theo-
rem 3.1, we can obtain (3.2). �

Corollary 3.1. Let A,B,X be operators in B(H) such that X is self adjoint
operator and if µ1, µ2 are positive real numbers such that µ

−1
1 + µ

−1
2 = 1, then

‖ | A∗XA−B∗XB |r ‖p

≤ 2
1
p
−2r+ 1

2‖ | (A + B)(A + B)∗X |
µ1r
2 ‖

1
µ1
p ‖ | X(A−B)(A−B)∗ |

µ2r
2 ‖

1
µ2
p ,

and

‖ | A∗XA−B∗XB |r ‖p

≤ 2
1
p
−2r+ 1

2‖((A + B)∗ | X | (A + B))
µ1r
2 ‖

1
µ1
p ‖((A−B)∗ | X | (A−B))

µ2r
2 ‖

1
µ2
p ,

for 0 < r ≤ 1
2

and 1 ≤ p ≤ 2.

Remark 3.1. By using (2.8) and from second inequality in Corollary (3.1), we can
obtain

‖ | A |2r + | B |2r ‖p ≤ ‖ || A |2 − | B |2|r ‖p

≤ 2
1
p
−2r+ 1

2‖ | A + B |µ1r| ‖
1
µ1
p ‖ | A−B |µ2r ‖

1
µ2
p ,



8 ALI, YANG AND SHAKOOR

which is the generalized form of the inequality (1.8).
Similarly to the proof of Theorem 3.1, based on (2.11), we can obtain the fol-

lowing result.

Theorem 3.2. Let A,B,X be operators in B(H) such that X is self adjoint
operator and if µ1, µ2 are positive real numbers such that µ

−1
1 + µ

−1
2 = 1, then

‖ | A∗XB + B∗XA |r ‖p

≤ 2r‖ | AA∗X |
µ1r
2 ‖

1
µ1
p ‖ | XBB∗ |

µ2r
2 ‖

1
µ2
p ,(3.3)

also, if f and g are nonnegative continuous functions on [0,∞) satisfying f(t)g(t) =
t, for all t ∈ [0,∞), then we have

‖ | A∗XB + B∗XA |r ‖p

≤ 2r‖
(
A∗f2(| X |)A

)µ1r
2 ‖

1
µ1
p ‖

(
B∗g2(| X |)B

)µ2r
2 ‖

1
µ2
p ,(3.4)

for r ≥ 1
2

and p ≥ 2.

Corollary 3.2. Let A,B,X be operators in B(H) such that X is self adjoint
operator and if µ1, µ2 are positive real numbers such that µ

−1
1 + µ

−1
2 = 1, then

‖ | A∗XA−B∗XB |r ‖p

≤ ‖ | (A + B)(A + B)∗X |
µ1r
2 ‖

1
µ1
p ‖ | X(A−B)(A−B)∗ |

µ2r
2 ‖

1
µ2
p ,

and

‖ | A∗XA−B∗XB |r ‖p

≤ ‖((A + B)∗ | X | (A + B))
µ1r
2 ‖

1
µ1
p ‖((A−B)∗ | X | (A−B))

µ2r
2 ‖

1
µ2
p ,

for r ≥ 1
2

and p ≥ 2.

Similarly to the proof of Theorem 3.1, based on (2.3), we can also obtain the
following result.

Theorem 3.3. Let Ai,Bi,Xi (i = 1, 2, ...,n) be operators in B(H) such that Xi is
self adjoint operator and if µ1, µ2 are positive real numbers such that µ

−1
1 +µ

−1
2 = 1,

Then

‖
n∑
i=1

| A∗iXiBi + B
∗
i XiAi |

r ‖p

≤ 2rn1−
1
p

n∑
i=1

‖ | AiA∗iXi |
µ1r
2 ‖

1
µ1
p ‖ | XiBiB∗i |

µ2r
2 ‖

1
µ2
p ,(3.5)

also, if f and g are nonnegative continuous functions on [0,∞) satisfying f(t)g(t) =
t, for all t ∈ [0,∞), then we have

‖
n∑
i=1

| A∗iXiBi + B
∗
i XiAi |

r ‖p

≤ 2rn1−
1
p

n∑
i=1

‖
(
A∗if

2(| Xi |)Ai
)µ1r

2 ‖
1
µ1
p ‖

(
B∗i g

2(| Xi |)Bi
)µ2r

2 ‖
1
µ2
p ,(3.6)



GENERALIZED NORMS INEQUALITIES FOR ABSOLUTE VALUE OPERATORS 9

for r,p ≥ 1.

Corollary 3.3. Let Ai,Bi,Xi (i = 1, 2, ...,n) be operators in B(H) such that Xi is
self adjoint operator and if µ1, µ2 are positive real numbers such that µ

−1
1 +µ

−1
2 = 1,

then

n
1
p
−1‖

n∑
i=1

| A∗iXiAi −B
∗
i XiBi |

r ‖p

≤
n∑
i=1

||| | (Ai + Bi)(Ai + Bi)∗Xi |
µ1r
2 ‖

1
µ1
p ‖ | Xi(Ai −Bi)(Ai −Bi)∗ |

µ2r
2 ‖

1
µ2
p ,

and

n
1
p
−1‖

n∑
i=1

| A∗iXiAi −B
∗
i XiBi |

r ‖p

≤
n∑
i=1

‖((Ai + Bi)∗ | Xi | (Ai + Bi))
µ1r
2 ‖

1
µ1
p ‖((Ai −Bi)∗ | Xi | (Ai −Bi))

µ2r
2 ‖

1
µ2
p ,

for r,p ≥ 1.

Remark 3.2. For r ≤ 2 and p ≤ 2
r

or r ≥ 2 and p(4 − r) ≤ 2, the results in
Corollary 3.3 refine the results in Corollary 2.2 respectively.

Acknowledgements

The authors thank the referees for their careful reading of the manuscript. This
work was supported by the National Natural Science Foundation of China (No.
11171361).

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College of Mathematics and Statistics, Chongqing University, Chongqing 401331,

P.R.China

∗Corresponding author