Some Vector Inequalities for Continuous Functions of Self Adjoint Operators in Hilbert Spaces

Volume 2011, Article ID 564836,14pages doi:10.1155/2011/564836

Research Article

Some Vector Inequalities for Continuous Functions

of Self-Adjoint Operators in Hilbert Spaces

S. S. Dragomir

1_{Mathematics, School of Engineering & Science, Victoria University, P.O. Box 14428, Melbourne City,} VIC 8001, Australia

2_{School of Computational & Applied Mathematics, University of the Witwatersrand, Private Bag-3,} Johannesburg 2050, South Africa

Correspondence should be addressed to S. S. Dragomir,[email protected]

Received 24 November 2010; Accepted 21 February 2011

Academic Editor: Paolo E. Ricci

Copyrightq2011 S. S. Dragomir. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

On utilizing the spectral representation of self-adjoint operators in Hilbert spaces, some inequalities for the composite operatorfM1H−fAfA−fm1H, where SpA⊆m, M

and for various classes of continuous functionsf : m, M → Care given. Applications for the power function and the logarithmic function are also provided.

1. Introduction

LetUbe a self-adjoint operator on the complex Hilbert space H,·,·with the spectrum SpUincluded in the intervalm, Mfor some real numbers m < Mand let{Eλ}_λ be its spectral family. Then, for any continuous functionf :m, M → C, it is well known that we have the followingspectral representation in terms of the Riemann-Stieltjes integral:

fU

M

m−0fλdEλ, 1.1

which in terms of vectors can be written as

fUx, y

_M

m−0

fλdEλx, y

for anyx, y∈H. The functiongx,yλ:Eλx, yis ofbounded variationon the intervalm, M and

gx,ym−0 0, gx,yM

x, y, 1.3

for anyx, y ∈H. It is also well known thatgxλ: Eλx, xismonotonic nondecreasingand right continuousonm, M.

Utilising the spectral representation from 1.2, we have established the following Ostrowski-type vector inequality1.

Theorem 1.1. LetAbe a self-adjoint operator in the Hilbert spaceHwith the spectrum SpA ⊆

m, Mfor some real numbersm < Mand let{Eλ}_λbe its spectral family. Iff :m, M → Cis a continuous function of bounded variation onm, M, then one has the inequality

fsx, y−fAx, y

≤ Esx, x1/2

Esy, y

1/2s

m

f

1H−Esx, x1/2

1H−Esy, y

1/2M

s

f

≤ x y

1 2

M

m

f1

2

s

m

f−

M

s

f

≤ x y M

m

f,

1.4

for anyx, y∈H and for anys∈m, M.

Another result that compares the function of a self-adjoint operator with the integral mean is embodied in the following theorem2.

Theorem 1.2. With the assumptions inTheorem 1.1one has the inequalities

x, y· 1 M−m

_M

m

fsds−fAx, y

≤M m

f max t∈m,M

M−t

M−mEtx, x

1/2_E

ty, y

1/2 t−m

M−m1H−Etx, x

1/2₁

H−Ety, y

1/2

≤ x y M

m

f,

1.5

for anyx, y∈H.

Theorem 1.3. With the assumptions inTheorem 1.1one has the inequalities

fM ₂ fm·x, y−fAx, y

≤ 1 2λ∈maxm,M

Eλx, x1/2

Eλy, y

1/2

1H−Eλx, x1/2

1H−Eλy, y

1/2M

m

f

≤ 1

2x y M

m

f,

1.6

for anyx, y∈H.

For various inequalities for functions of self-adjoint operators, see4–8. For recent results see1,9–12.

In this paper, we investigate the quantity

_fM1H−fA

fA−fm1H

x, y, 1.7

wherex, yare vectors in the Hilbert spaceHandAis a self-adjoint operator with SpA ⊆

m, M, and provide different bounds for some classes of continuous functionsf:m, M → C. Applications for some particular cases including the power and logarithmic functions are provided as well.

2. Some Vector Inequalities

The following representation in terms of the spectral family is of interest in itself.

Lemma 2.1. Let Abe a self-adjoint operator in the Hilbert space H with the spectrumSpA ⊆

m, Mfor some real numbersm < Mand let{Eλ}_λbe its spectral family. Iff :m, M → Cis a continuous function onm, MwithfM/fm, then one has the representation

1

fM−fm2

fM1H−fA

fA−fm1H

1

fM−fm

M

m−0

Et− 1 fM−fm

M

m−0Esdfs

Et−1 21H

dft.

Proof. We observe

1 fM−fm

M

m−0

Et− 1 fM−fm

M

m−0Esdfs

Et−1 21H

dft

1

fM−fm

M

m−0E 2

tdft

− 1

fM−fm

M

m−0Esdfs

1 fM−fm

M

m−0Etdft

−1 2

M

m−0Etdft

1 2

M

m−0Esdfs

1

fM−fm

M

m−0

E2_tdft−

1 fM−fm

M

m−0 Etdft

2

,

2.2

which is an equality of interest in itself.

SinceEtare projections, we haveE2t Etfor anyt∈m, Mand then we can write

1 fM−fm

M

m−0

E_t2dft−

1 fM−fm

M

m−0

Etdft

2

1

fM−fm

M

m−0Etdft−

1 fM−fm

M

m−0Etdft 2

1

fM−fm

M

m−0Etdft

1H− 1

fM−fm

M

m−0Etdft

.

2.3

Integrating by parts in the Riemann-Stieltjes integral and utilizing the spectral repre-sentation1.1, we have

_M

m−0

Etdft fM1H−fA,

1H−

1 fM−fm

_M

m−0Etdft

fA−fm1H fM−fm ,

2.4

which together with2.3and2.2produce the desired result2.1.

Corollary 2.2. With the assumptions ofLemma 2.1one has the equality

fM1H−fA

fA−fm1H

x, y

fM−fm M

m−0

Et− 1 fM−fm

M

m−0Esdfs

x,

Et− 1 21H

y

dft,

2.5

for anyx, y∈m, M.

The following result that provides some bounds for continuous functions of bounded variation may be stated as well.

Theorem 2.3. LetAbe a self-adjoint operator in the Hilbert spaceHwith the spectrum SpA ⊆

m, Mfor some real numbersm < M, and let{Eλ}_λbe its spectral family. Iff :m, M → Cis a continuous function of bounded variation onm, MwithfM/fm, then we have the inequality

fM1H−fA

fA−fm1H

x, y

≤ 1

2 y fM−fm M

m

f

× sup t∈m,M

Etx−

1 fM−fm

_M

m−0Esdfs

≤ 1

2x y _M

m

f

2

,

2.6

for anyx, y∈H.

Proof. It is well known that ifp : a, b → Cis a bounded function,v : a, b → Cis of bounded variation, and the Riemann-Stieltjes integral_abptdvtexists, then the following inequality holds:

b

a

ptdvt

≤_tsup_∈a,bpt

b

a

v, 2.7

Utilising this property and the representation2.5, we have by the Schwarz inequality in Hilbert spaceHthat

_fM1H−fA

fA−fm1H

x, y

≤fM−fm M

m

f

× sup t∈m,M

Et− 1 fM−fm

M

m−0Esdfs

x,

Et−1 21H

y

≤fM−fm M

m

f

× sup t∈m,M

Etx−

1 fM−fm

_M

m−0Esxdfs Ety−

1 2y

,

2.8

for anyx, y∈m, M.

SinceEtare projections, in this case we have

Ety−

1 2y

2 Ety 2−

Ety, y

1

4 y

2

E2_ty, y−Ety, y

1

4 y

2 1

4 y

2

,

2.9

then from2.8, we deduce the first part of2.6.

Now, by the same property2.7for vector-valued functionspwith values in Hilbert spaces, we also have

fM−fmEtx−

M

m−0Esxdfs

_M

m−0

Etx−Esxdfs

≤M

m

f sup

s∈m,M

Etx−Esx,

2.10

for anyt∈m, Mandx∈H.

Since 0 ≤ Et ≤ 1H in the operator order, then −1H ≤ Et−Es ≤ 1 which gives that −x2 ≤ Et−Esx, x ≤ x2, that is,|Et−Esx, x| ≤ x2for anyx∈H, which implies thatEt−Es ≤1 for anyt, s∈m, M. Therefore, sup_s∈m,MEtx−Esx ≤ xwhich together with2.10prove the last part of2.6.

Theorem 2.4. LetAbe a self-adjoint operator in the Hilbert spaceHwith the spectrum SpA ⊆

m, Mfor some real numbersm < M, and let{Eλ}_λbe its spectral family. Iff :m, M → Cis a Lipschitzian function with the constantL >0onm, Mand withfM/fm, then one has the inequality

_fM1H−fA

fA−fm1H

x, y

≤ 1

2L y fM−fm

M

m−0

Etx− 1 fM−fm

M

m−0Esxdfs dt

≤ 1 2L

2 _y

M

m−0Etx−Esxds dt

≤ √

2 2 L

2 _{y M−mAx−mx, Mx−Ax}1/2

≤ √

2 4 L

2 _{y xM−m}2_,

2.11

for anyx, y∈H.

Proof. Recall that ifp : a, b → Cis a Riemann integrable function andv : a, b → Cis Lipschitzian with the constantL >0, that is,

fs−ft≤L|s−t| for anyt, s∈a, b, 2.12

then the Riemann-Stieltjes integral_abptdvtexists and the following inequality holds:

b

a

ptdvt≤L b

a

ptdt. 2.13

Now, on applying this property of the Riemann-Stieltjes integral, then we have from the representation2.5that

_fM1H−fA

fA−fm1H

x, y

≤fM−fm

M

m−0

Et− 1 fM−fm

M

m−0Esdfs

x,

Et−1 21H

y

dft,

≤LfM−fm

M

m−0

Etx− 1 fM−fm

M

m−0Esxdfs

Ety−1 2y

dt

1

2L y fM−fm

M

m−0

Etx− 1 fM−fm

M

m−0Esxdfs dt,

2.14

Further, observe that

fM−fm

_M

m−0 Etx−

1 fM−fm

_M

m−0Esxdfs dt

_M

m−0

fM−fmEtx−

_M

m−0Esxdfs dt

_M

m−0 M

m−0Etx−Esxdfs dt,

2.15

for anyx∈H.

If we use the vector-valued version of the property2.13, then we have

M

m−0 M

m−0Etx−Esxdfs dt≤L

M

m−0Etx−Esxds dt, 2.16

for anyx∈Hand the second part of2.11is proved.

Further on, by applying the double-integral version of the Cauchy-Buniakowski-Schwarz inequality, we have

M

m−0Etx−Esxds dt≤M−m

M

m−0Etx−Esx 2_{ds dt}

1/2

, 2.17

for anyx∈H.

Now, by utilizing the fact thatEs are projections for eachs∈m, M, then we have

_M

m−0Etx−Esx 2_{ds dt}

2 ⎡

⎣M−m

_M

m−0Etx 2_dt−

_M

m−0Etxdt 2

⎤ ⎦

2 ⎡

⎣M−m

_M

m−0Etx, xdt− M

m−0Etxdt 2

⎤ ⎦,

2.18

If we integrate by parts and use the spectral representation1.2, then we get

M

m−0Etx, xdtMx−Ax, x,

M

m−0EtxdtMx−Ax 2.19

and by2.18, we then obtain the following equality of interest:

_M

m−0

Etx−Esx2ds dt2Ax−mx, Mx−Ax, 2.20

for anyx∈H.

On making use of2.20and2.17, we then deduce the third part of2.11. Finally, by utilizing the elementary inequality in inner product spaces

Rea, b ≤ 1 4ab

2_{, a, b∈H, 2.21}

we also have that

Ax−mx, Mx−Ax ≤ 1

4M−m

2_x2_{, 2.22}

for anyx∈H, which proves the last inequality in2.11.

The case of nondecreasing monotonic functions is as follows.

Theorem 2.5. LetAbe a self-adjoint operator in the Hilbert spaceHwith the spectrum SpA ⊆

m, Mfor some real numbersm < M, and let{Eλ}_λbe its spectral family. Iff :m, M → Ris a monotonic nondecreasing function onm, M, then one has the inequality

fM1H−fA

fA−fm1H

x, y

≤ 1

2 y fM−fm M

m−0

Etx−

1 fM−fm

_M

m−0

Esxdfs

dft

≤ 1

2 y fM−fm

fM1H−fA

fA−fm1H

x, x1/2

≤ 1

4 y x

fM−fm2,

2.23

Proof. From the theory of Riemann-Stieltjes integral, it is also well known that ifp :a, b → Cis of bounded variation andv :a, b → Ris continuous and monotonic nondecreasing, then the Riemann-Stieltjes integrals_abptdvtand_ab|pt|dvtexist and

_b

a

ptdvt≤ _b

a

ptdvt. 2.24

Now, on applying this property of the Riemann-Stieltjes integral, we have from the representation2.5that

_fM1H−fA

fA−fm1H

x, y

≤fM−fm M

m−0

Et−

1 fM−fm

_M

m−0

Esdfs

x,

Et−

1 21H

ydft,

≤fM−fm M

m−0

Etx− 1 fM−fm

M

m−0Esxdfs

Ety−1 2y

dft

1

2 y fM−fm M

m−0 Etx−

1 fM−fm

_M

m−0Esxdfs dft,

2.25

for anyx, y∈H, which proves the first inequality in2.23.

On utilizing the Cauchy-Buniakowski-Schwarz-type inequality for the Riemann-Stieltjes integral of monotonic nondecreasing integrators, we have

M

m−0

Etx− 1 fM−fm

M

m−0Esxdfs dft

≤

M

m−0dft 1/2⎡

⎣M

m−0 Etx−

1 fM−fm

_M

m−0Esxdfs 2dft

⎤ ⎦

1/2

,

2.26

Observe that

M

m−0

Etx− 1 fM−fm

M

m−0Esxdfs 2dft

_M

m−0

Etx2−2 Re

Etx,

1 fM−fm

_M

m−0Esxdfs

1

fM−fm

_M

m−0Esxdfs 2

⎤ ⎦dft

fM−fm ⎡

⎣ 1

fM−fm

M

m−0Etx

2_dft− 1

fM−fm

M

m−0Esxdfs 2

⎤ ⎦

2.27

and, integrating by parts in the Riemann-Stieltjes integral, we have

M

m−0Etx 2_dft

M

m−0Etx, Etxdft

M

m−0Etx, xdft

fMx2−

M

m−0ftdEtx, x

fMx2−fAx, xfM1H−fA

x, x,

_M

m−0

Esxdfs fMx−fAx,

2.28

for anyx∈H.

On making use of the equalities2.28, we have

1 fM−fm

M

m−0Etx

2_dft− 1

fM−fm

M

m−0Esxdfs 2

1

fM−fm2

fM−fmfM1H−fA

x, x− fMx−fAx 2

fM−fmfM1H−fA

x, x−fMx−fAx, fMx−fAx

fM−fm2

fM−fmfM1H−fA

x, x−fMx−fAx, fMx−fAx

fM−fm2

fMx−fAx, fAx−fmx

fM−fm2 ,

2.29

Therefore, we obtain the following equality of interest in itself as well:

1 fM−fm

_M

m−0 Etx−

1 fM−fm

_M

m−0Esxdfs 2dft

fMx−fAx, fAx−fmx

fM−fm2

fM1H−fA

fA−fm1H

x, x

fM−fm2 ,

2.30

for anyx∈H

On making use of the inequality2.26, we deduce the second inequality in2.23. The last part follows by2.21, and the details are omitted.

3. Applications

We consider the power functionft:tp_{, wherep∈R\ {0}andt >0. The following power} inequalities hold.

Proposition 3.1. LetAbe a self-adjoint operator in the Hilbert spaceHwith the spectrumSpA⊆

m, Mfor some real numbers with0≤m < M. Ifp >0, then for anyx, y∈H,

Mp1H−ApAp−mp1Hx, y

≤ √

2 2 B

2

p y M−mAx−mx, Mx−Ax1/2

≤ √

2 4 B

2

p y xM−m2,

3.1

where

Bpp×

⎧ ⎪ ⎨ ⎪ ⎩

Mp−1_{, if p≥1,}

mp−1_{, if 0< p <1, m >0,}

_A−p_−M−p₁

Hm−p1H−A−px, y

≤ √

2 2 C

2

p y M−mAx−mx, Mx−Ax1/2

≤ √

2 4 C

2

p y xM−m2,

where

Cppm−p−1, m >0. 3.3

The proof follows fromTheorem 2.4applied for the power function.

Proposition 3.2. LetAbe a self-adjoint operator in the Hilbert spaceHwith the spectrumSpA⊆

m, Mfor some real numbers with0≤m < M. Ifp >0, then for anyx, y∈H

_Mp₁

H−ApAp−mp1Hx, y

≤ 1

2 y M

p_−mp_Mp₁

H−ApAp−mp1Hx, x1/2

≤ 1

4 y xM

p_−mp2_{, A}−p_−M−p₁

H

m−p1H−A−p

x, y

≤ 1 2 y m

−p_−M−p_A−p_−M−p₁ H

m−p1H−A−p

x, x1/2

≤ 1

4 y x

m−p_−M−p2_.

3.4

The proof follows fromTheorem 2.5.

Now, consider the logarithmic functionft lnt, t >0. We have the following

Proposition 3.3. LetAbe a self-adjoint operator in the Hilbert spaceHwith the spectrumSpA⊆

m, Mfor some real numbers with0< m < M. Then one has the inequalities

lnM1H−lnAlnA−lnm1Hx, y

≤ √

2

2m2 y M−mAx−mx, Mx−Ax 1/2

≤ √

2

4 y x

M m −1

2

,

lnM1H−lnAlnA−lnm1Hx, y

≤ 1

2 y lnM1H−lnAlnA−lnm1Hx, x

1/2_ln

M m

≤ 1

4 y x

ln

M m

2

.

3.5

References

1 S. S. Dragomir, “Some Ostrowski’s type vector inequalities for functions of selfadjoint operators in Hilbert spaces,”RGMIA Research Report Collection, vol. 13, no. 2, article 7, 2010.

2 S. S. Dragomir, “Comparison between functions of selfadjoint operators in Hilbert spaces and integral means,”RGMIA Research Report Collection, vol. 13, no. 2, article 10, 2010.

3 S. S. Dragomir, “Approximating n-time differentiable functions of selfadjoint operators in Hilbert spaces by two point Taylor’s type expansion,”RGMIA Research Report Collection, vol. 13, no. 2, article 13, 2010.

4 T. Furuta, J. Mi´ci´c Hot, J. Peˇcari´c, and Y. Seo,Mond-Peˇcari´c Method in Operator Inequalities. Inequalities for Bounded Selfadjoint Operators on a Hilbert Space, Element, Zagreb, 2005.

5 A. Matkovi´c, J. Peˇcari´c, and I. Peri´c, “A variant of Jensen’s inequality of Mercer’s type for operators with applications,”Linear Algebra and Its Applications, vol. 418, no. 2-3, pp. 551–564, 2006.

6 B. Mond and J. E. Peˇcari´c, “Convex inequalities in Hilbert space,”Houston Journal of Mathematics, vol. 19, no. 3, pp. 405–420, 1993.

7 B. Mond and J. E. Peˇcari´c, “Classical inequalities for matrix functions,”Utilitas Mathematica, vol. 46, pp. 155–166, 1994.

8 J. Peˇcari´c, J. Mi´ci´c, and Y. Seo, “Inequalities between operator means based on the Mond-Peˇcari´c method,”Houston Journal of Mathematics, vol. 30, no. 1, pp. 191–207, 2004.

9 S. S. Dragomir, “ ˇCebyˇsev’s type inequalities for functions of selfadjoint operators in Hilbert spaces,”

Linear and Multilinear Algebra, vol. 58, no. 7-8, pp. 805–814, 2010.

10 S. S. Dragomir, “Gr ¨uss’ type inequalities for functions of selfadjoint operators in Hilbert spaces,”

RGMIA Research Report Collection, vol. 11, article 11, 2008.

11 S. S. Dragomir, “Inequalities for the ˇCebyˇsev functional of two functions of selfadjoint operators in Hilbert spaces,”RGMIA Research Report Collection, vol. 11, article 17, 2008.