Accepted Manuscript

On Young and Heinz inequalities for τ -measurable operators

Jingjing Shao

PII: S0022-247X(14)00024-9DOI: 10.1016/j.jmaa.2014.01.017Reference: YJMAA 18188

To appear in: Journal of Mathematical Analysis and Applications

Received date: 2 July 2013

Please cite this article in press as: J. Shao, On Young and Heinz inequalities for τ -measurableoperators, J. Math. Anal. Appl. (2014), http://dx.doi.org/10.1016/j.jmaa.2014.01.017

This is a PDF file of an unedited manuscript that has been accepted for publication. As a service toour customers we are providing this early version of the manuscript. The manuscript will undergocopyediting, typesetting, and review of the resulting proof before it is published in its final form.Please note that during the production process errors may be discovered which could affect thecontent, and all legal disclaimers that apply to the journal pertain.

On Young and Heinz inequalities for

τ-measurable operators ∗

Jingjing Shao

College of Mathematics and System Sciences, Xinjiang University, Urumqi 830046, China

E-mail: liangjingjing0868@126.com

Abstract The purpose of this article is to prove Young and Heinz inequalities

for τ -measurable operators.

Keywords von Neumann algebras, positive operators, Young inequality, Heinz

inequality

2010 Mathematics Subject Classification 47A30; 47L05; 47L50

1 Introduction

Let Mn(C) be the space of n×n complex matrices. Let ||| · ||| denote any unitarily

invariant (or symmetric) norm on Mn(C), that is to say, |||UAV ||| = |||A||| for allA ∈ Mn(C) and for all unitary matrices U , V ∈ Mn(C). In 1979, McIntosh [4] proved

that for any unitary invariant norm Heinz inequality for matrices holds. Using the

refinements of the classical Young inequality for positive real numbers, Kittaneh and

Manasrab [3] established improved Young and Heinz inequalities for matrices.

In this paper we consider the noncommutative Lp-spaces of τ -measurable op-

erators affiliated with a semi-finite von Neumann algebra equipped with a normal

faithful semi-finite trace τ . We use the method of Kittaneh and Manasrab, via

the notion of generalized singular value studied by Fack and Kosaki ([2]), to obtain

generalizations of results in [3] for τ -measurable operators case.

∗Partially supported by NSFC grant No. 11071204 and XJUBSCX-2012002

1

2 Preliminaries

Throughout the paper we denote by M a semi-finite von Neumann algebra acting

on the Hilbert space H, with a normal faithful semi-finite trace τ . We denote the

identity in M by 1 and let P denote the projection lattice of M. A closed densely

defined linear operator x in H with domain D(x) ⊆ H is said to be affiliated with

M if u∗xu = x for all unitary u which belong to the commutant M′ of M. If

x is affiliated with M, then x is said to be τ -measurable if for every ε > 0 there

exists a projection e ∈ M such that e(H) ⊆ D(x) and τ(1 − e) < ε. The set of

all τ -measurable operators will be denoted by L0(M, τ), or simply L0(M). The

set L0(M) is a ∗-algebra with sum and product being the respective closures of

the algebraic sum and product. A closed densely defined linear operator x admits

a unique polar decomposition x = u|x|, where u is a partial isometry such that

u∗u = (kerx)⊥ and uu∗ = imx (with imx = x(D(x))). We call r(x) = (kerx)⊥

and l(x) = imx the left and right supports of x, respectively. Thus l(x) ∼ r(x).

Moreover, if x is self-adjoint, we let s(x) = r(x), the support of x.

LetM+ be the positive part ofM. Set S+(M) = {x ∈ M+ : τ(s(x)) < ∞} andlet S(M) be the linear span of S+(M), we will often abbreviate S+(M) and S(M)

respectively as S+ and S. Let 0 < p < ∞, the noncommutative Lp-space Lp(M, τ)

is the completion of (S, ‖ · ‖p), where ‖x‖p = τ(|x|p) 1p < ∞, ∀ x ∈ Lp(M, τ). In

addition, we put L∞(M, τ) = M and denote by ‖ · ‖∞(= ‖ · ‖) the usual operator

norm. It is well known that Lp(M, τ) are Banach spaces under ‖ · ‖p for 1 ≤ p < ∞and they have a lot of expected properties of classical Lp-spaces (see [5] or [6]).

Let x be a τ -measurable operator and t > 0. The ”tth singular number (or

generalized s-number) of x” is defined by

μt(x) = inf{‖xe‖ : e ∈ P , τ(1− e) ≤ t}.

See [2] for basic properties and detailed information on the generalized s-numbers.

To achieve one of our main results, we state for easy reference the following fact

obtaining from [7] that will be applied below.

Lemma 2.1 Let x, y ∈ Lp(M) be positive operators with 1 ≤ p < ∞ and let

2

z ∈ M, then

‖xvzy1−v‖p ≤ ‖xz‖vp · ‖zy‖1−vp , 0 ≤ v ≤ 1.

3 Main results

First, we generalize the improved Young inequality in [3] for positive τ -measurable

operators case.

Theorem 3.1 Let x, y ∈ L1(M) be positive operators and let 0 ≤ v ≤ 1, then

τ(xvy1−v) + r0((τ(x))12 − (τ(y))

12 )2 ≤ τ(vx+ (1− v)y),

where r0 = min{v, 1− v}.

Proof. By Theorem 2.1 of [3] we have

vμt(x) + (1− v)μt(y) ≥ μt(x)vμt(y)

1−v + r0(μt(x)12 − μt(y)

12 )2, ∀ t > 0.

Thus, together Lemma 4.2 of [2] with Holder type inequality we get

τ(vx+ (1− v)y) = vτ(x) + (1− v)τ(y)

=

∫ ∞

0

[vμt(x) + (1− v)μt(y)]dt

≥∫ ∞

0

μt(x)vμt(y)

1−vdt+ r0

∫ ∞

0

[μt(x) + μt(y)− 2μ(x12 )μt(y

12 )]dt

=

∫ ∞

0

μt(x)vμt(y)

1−vdt+ r0[τ(x) + τ(y)− 2

∫ ∞

0

μt(x12 )μt(y

12 )dt]

≥∫ ∞

0

μt(xvy1−v)dt+ r0[τ(x) + τ(y)

− 2(

∫ ∞

0

μt(x12 )2dt)

12 (

∫ ∞

0

μt(y12 )2dt)

12 ]

= τ(xvy1−v) + r0[τ(x) + τ(y)− 2τ(x)12 τ(y)

12 ]

= τ(xvy1−v) + r0((τ(x))12 − (τ(y))

12 )2.

�The following result is an improved arithmetic-geometric mean inequality for

the norm ‖ · ‖2.

3

Lemma 3.2 Let x, y ∈ L2(M) be positive operators and let z ∈ M, then

2‖x 12 zy

12‖2 + (‖xz‖

122 − ‖zy‖

122 )

2 ≤ ‖xz + zy‖2.

Proof. It is easy to get this lemma in the similar way as in Theorem 3.3 of [3].

Therefore, we may omit the proof. �To obtain the improved Heinz inequality for the norm ‖·‖2 we need the following

results.

Lemma 3.3 Let x, y ∈ L2(M) be positive operators and let z ∈ M, then the

function

f(t) = ‖x1+tzy1−t + x1−tzy1+t‖2is convex on the interval [−1, 1] and attains its minimum at t = 0.

Proof. (i) First we assume that τ is finite. By the density of M in L2(M), we

first consider the case x, y ∈ M+ and x, y are invertible. Since f is continuous and

f(t) = f(−t), to get both the conclusions, it suffices to prove that

f(t) ≤ 1

2[f(t+ s) + f(t− s)], whenever t± s are in [−1, 1].

For each t ∈ [−1, 1], let Mt : M+ → M+ be the mapping

Mt(a) =1

2(xtay−t + x−tayt).

Since

‖xzy‖22 = τ((xzy)∗(xzy))

= τ(y∗z∗x∗xzy)

= τ(zyy∗z∗x∗x)

≤ τ((x∗xz)∗(x∗xz))12 τ((zyy∗)∗(zyy∗))

12

= ‖x∗xz‖2‖zyy∗‖2,

we get

‖x∗xz + zyy∗‖22 = ‖x∗xz‖22 + ‖zyy∗‖22 + 2 < x∗xz, zyy∗ >

≥ 4‖xzy‖22.

4

Thus for every a ∈ M+ we obtain

‖a‖2 = ‖xt(x−tay−t)yt)‖2≤ 1

2‖xt · xt(x−tay−t) + x−tay−t · yt · yt‖2

=1

2‖xtay−t + x−tayt‖2.

It follows that

‖a‖2 ≤ ‖Mt(a)‖2.Hence, ‖Mt(xzy)‖2 ≤ ‖MsMt(xzy)‖2, for all s, t with t± s are in [-1, 1]. Apply

this to a = Mt(xzy), using the identity 2Ms(Mt(a)) = Mt+s(a) +Mt−s(a), and the

result is

2f(t) = 4‖Mt(xzy)‖2 ≤ 4‖Ms(Mt(xzy))‖2≤ 2(‖Mt+s(xzy)‖2 + ‖Mt−s(xzy)‖2)= f(t+ s) + f(t− s).

For the general case, namely, for any x, y ∈ L2(M), there exist xn, yn ∈ M+

such that xn, yn are invertible and xn → x, yn → y in L2(M). Moreover, we have

fn(t) = ‖x1+tn zy1−t

n +x1−tn zy1+t

n ‖2 is convex for all t ∈ [−1, 1] and attains its minimum

at t = 0. Since x1+tn zy1−t

n → x1+tzy1−t, x1−tn zy1+t

n → x1−tzy1+t in L2(M), we obtain

fn(t) → f(t), n → ∞. Hence, f(t) is convex on [−1, 1] and attains its minimum at

t = 0.

(ii) In the general case when τ is semi-finite, there exists an increasing family

(ei)i∈I ∈ P such that τ(ei) < ∞ for every i ∈ I and such that ei converges to 1 in

the strong operator topology (see [5] or [6]). Thus, eiMei is finite for each i ∈ I.Let x, y ∈ �L2(M)+, then eixei, eiyei ∈ L2(eiMei)+. Write xi = eixei, yi = eiyei,

it follows from the case (i) that the function gi(t) = ‖x1+ti zy1−t

i + x1−ti zy1+t

i ‖2 is

convex on [−1, 1] and attains its minimum at t = 0. In view of the fact that xi → x,

yi → y in L2(M), by a simple computation we get limi gi(t) = f(t). Therefore, f(t)

is convex on [−1, 1] and attains its minimum at t = 0. �

Corollary 3.4 Let x, y ∈ L2(M) be positive operators and let z ∈ M, then the

function

g(v) = ‖xvzy1−v + x1−vzyv‖2

5

is convex on [0, 1].

Proof. This is an immediate result of Lemma 3.3 by replacing x, y respectively

with x12 , y

12 and putting v = 1+t

2. �

Corollary 3.5 Let x, y ∈ L2(M) be positive operators and z ∈ M, then

‖xvzy1−v + x1−vzyv‖2 ≤ ‖xz + zy‖2, 0 ≤ v ≤ 1.

Proof. Let g(v) be the function defined in Corollary 3.4. Note that g(1) = g(0). So

the assertion follows from the convexity of g. �In the next result, we give an improved Heinz inequality for the norm ‖ · ‖2.

Theorem 3.6 Let x, y ∈ L2(M) be positive operators and let z ∈ M, then

‖xvzy1−v + x1−vzyv‖2 + 2r0(‖xz‖122 − ‖zy‖

122 )

2 ≤ ‖xz + zy‖2, 0 ≤ v ≤ 1,

where r0 = min{v, 1− v}.

Proof. Let ϕ(v) = ‖xvzy1−v + x1−vzyv‖2, 0 ≤ v ≤ 1. By Lemma 3.3 we know ϕ is

a continuous convex function on [0,1]. Moreover, ϕ is twice differentiable on (0, 1)

almost everywhere. Write f(v) = ‖xz + zy‖2 − ‖xvzy1−v + x1−vzyv‖2, it is easy to

see that f(v) = f(1− v) and f(0) = f(1) = 0, moreover, applying the approach to

prove Lemma 3.3 we know that f is concave on [0,1], and from Corollary 3.5 we see

that f(12) ≥ 0.

Let g(v) = 1min{v,1−v}f(v) for 0 < v < 1. Then g can be written as follows:

g(v) =

⎧⎪⎨⎪⎩

f(v)v, if 0 < v ≤ 1

2,

f(1−v)1−v

, if 12< v < 1.

(3.1)

Thus

g′(v) =

⎧⎪⎨⎪⎩

vf ′(v)−f(v)v2

, if 0 < v < 12,

−(1−v)f ′(1−v)+f(1−v)(1−v)2

, if 12< v < 1.

(3.2)

Consider the function ψ(v) = vf ′(v)− f(v), 0 ≤ v ≤ 1. Then ψ(0) = 0 and ψ′(v) =

f ′(v) + vf ′′(v) − f ′(v) = vf ′′(v) ≤ 0, which implies that ψ(v) ≤ ψ(0) = 0. Hence,

vf ′(v) ≤ f(v) for 0 ≤ v ≤ 12. Similarly, we can get that f(1−v) ≥ (1−v)f ′(1−v) for

6

12≤ v ≤ 1, which means g′(v) ≥ 0 for 1

2≤ v ≤ 1. This indicates that g is decreasing

on (0, 12) and increasing on (1

2, 1). In view of the continuity of g we obtain that g

attains its minimum at v = 12. Therefore, g(v) ≥ g(1

2). It follows from Lemma 3.2

that

g(v) ≥ 2(‖xz + zy‖2 − 2‖x 12 zy

12‖2)

≥ 2(‖xz‖122 − ‖zy‖

122 )

2.

Consequently,

‖xz + zy‖2 − ‖xvzy1−v + x1−vzyv‖2 ≥ 2min{v, 1− v}(‖xz‖122 − ‖zy‖

122 )

2.

Set r0 = min{v, 1− v} and we prove the theorem. �Our next result provides another improvement of the Heinz inequality for the

norm ‖ · ‖2.

Theorem 3.7 Let x, y ∈ L2(M) be positive operators and let z ∈ M, then

‖xvzy1−v + x1−vzyv‖22 + 2r0‖xz − zy‖22 ≤ ‖xz + zy‖22, 0 ≤ v ≤ 1,

where r0 = min{v, 1− v}.

Proof. Let φ(v) = ‖xvzy1−v+x1−vzyv‖22, then we know that φ is a continuous convex

function on [0,1]. Moreover, φ is twice differentiable on (0, 1) almost everywhere.

Setting h(v) = ‖xz + zy‖22 − ‖xvzy1−v + x1−vzyv‖22, we see that h(v) = h(1 − v),

h(0) = h(1). Also applying the method to prove Lemma 3.3 we get that h is concave

on [0, 1], moreover, from Corollary 3.5 we have h(12) ≥ 0. Put g1(v) =

1min{v,1−v}h(v)

for 0 < v < 1, thus g1 can be written as follows:

g1(v) =

⎧⎪⎨⎪⎩

h(v)v, if 0 < v ≤ 1

2,

h(1−v)1−v

, if 12< v < 1.

(3.3)

Similarly to the proof of Theorem 3.6 we obtain that g1 gets its minimum at v = 12.

Hence, g1(v) ≥ g1(12), i.e.,

g1(v) ≥ 2(‖xz + zy‖22 − 4‖x 12 zy

12‖22)

= 2(‖xz‖22 + ‖zy‖22 + 2 < xz, zy > −4 < xz, zy >)

= 2(‖xz − zy‖22).

7

Therefore,

‖xz + zy‖22 − ‖xvzy1−v + x1−vzyv‖22 ≥ 2min{v, 1− v}‖xz − zy‖22.

Write r0 = min{v, 1− v} and we show the theorem. �From Theorem 2.1 of [3] we can improve this inequality and generalize the

inequality in Lemma 3.2. To attain this, we need the following results.

Theorem 3.8 Let x, y ∈ Lp(M) be positive operators with 1 ≤ p < ∞ and let

z ∈ M, then

‖xvzy1−v‖p + r0(‖xz‖12p − ‖zy‖

12p )

2 ≤ v‖xz‖p + (1− v)‖zy‖p,

where r0 = min{v, 1− v}.

Proof. Using Lemma 2.1 and Theorem 2.1 of [3] we get

‖xvzy1−v‖p + r0(‖xz‖12p − ‖zy‖

12p )

2 ≤ ‖xz‖vp · ‖zy‖1−vp + r0(‖xz‖

12p − ‖zy‖

12p )

2

≤ v‖xz‖p + (1− v)‖zy‖p.

�In the same way, Lemma 2.1 together with inequality (2.1) of [3] yield the

following result related to Theorem 3.6.

Theorem 3.9 Let x, y ∈ Lp(M) be positive operators with 1 ≤ p < ∞ and let

z ∈ M, then

‖xvzy1−v + x1−vzyv‖p + 2r0(‖xz‖12p − ‖zy‖

12p )

2 ≤ ‖xz‖p + ‖zy‖p, 0 ≤ v ≤ 1,

where r0 = min{v, 1− v}.

Proof. Applying Theorem 2.1 of [3] twice and using Lemma 2.1 we have

‖xvzy1−v + x1−vzyv‖p ≤ ‖xvzy1−v‖p + ‖x1−vzyv‖p≤ ‖xz‖vp · ‖zy‖1−v

p + ‖xz‖1−vp · ‖zy‖vp

≤ ‖xz‖p + ‖zy‖p − 2r0(‖xz‖12p − ‖zy‖

12p )

2.

�

8

Acknowledgements

The author is grateful to the editor and the anonymous referee for their useful

suggestions on the quality improvement of the manuscript.

References

[1] T. Fack, Type and cotype inequalities for non commutative Lp-spaces, J.

Operator Theory, 17 (1987) 255-279.

[2] T. Fack, H. Kosaki, Generalized s-numbers of τ -measurable Operators, Pac J

Math. 123 (1986) 269-300.

[3] F. Kittaneh, Y. Manasrah, Improved Young and Heinz inequalities for matrices,

J. Math. Anal. Appl. 361 (2010) 262-269.

[4] A. McIntosh, Heinz inequalities and perturbation of sprectral families, Mac-

quarie Mathmatics Reports, Macquarie Univ., 1979.

[5] G. Pisier, Q. Xu, Noncommutative Lp− spaces, in: Handbook of the Geometry

of Banach spaces, vol. 2, 2003, pp. 1459-1517.

[6] Q. Xu, Noncommutative Lp− spaces, Preprint.

[7] Zhou Jia, Wang Yunxia, Wu Tianfeng, A Schwarz inequality for τ -measurable

operator A∗XB∗, Journal of Xinjiang University (Natural Science Edition), 1

(2009) 69-73.

9