Hindawi Publishing Corporation
Journal of Inequalities and Applications
Volume 2011, Article ID 564836, 14 pages
doi:10.1155/2011/564836
Research Article
Some Vector Inequalities for Continuous Functions
of Self-Adjoint Operators in Hilbert Spaces
S. S. Dragomir1, 2
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, sever.dragomir@vu.edu.au
Received 24 November 2010; Accepted 21 February 2011
Academic Editor: Paolo E. Ricci
Copyright q 2011 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 operator fM1H − fAfA − fm1H , where SpA ⊆ m, M
and for various classes of continuous functions f : m, M → C are given. Applications for the
power function and the logarithmic function are also provided.
1. Introduction
Let U be a self-adjoint operator on the complex Hilbert space H, ·, · with the spectrum
SpU included in the interval m, M for some real numbers m < M and let {Eλ }λ be its
spectral family. Then, for any continuous function f : m, M → C, it is well known that we
have the following spectral representation in terms of the Riemann-Stieltjes integral:
fU M
fλdEλ ,
1.1
fλd Eλ x, y ,
1.2
m−0
which in terms of vectors can be written as
fUx, y M
m−0
2
Journal of Inequalities and Applications
for any x, y ∈ H. The function gx,y λ : Eλ x, y is of bounded variation on the interval m, M
and
gx,y M x, y ,
gx,y m − 0 0,
1.3
for any x, y ∈ H. It is also well known that gx λ : Eλ x, x is monotonic nondecreasing and
right continuous on m, M.
Utilising the spectral representation from 1.2, we have established the following
Ostrowski-type vector inequality 1.
Theorem 1.1. Let A be a self-adjoint operator in the Hilbert space H with the spectrum SpA ⊆
m, M for some real numbers m < M and let {Eλ }λ be its spectral family. If f : m, M → C is a
continuous function of bounded variation on m, M, then one has the inequality
fs x, y − fAx, y s
1/2 ≤ Es x, x1/2 Es y, y
f
m
1.4
M
1/2 1H − Es x, x1/2 1H − Es y, y
f
≤ x y
M
s
M
1 1
f f −
f 2m
2 m
s
s
≤ x y
M
f ,
m
for any x, y ∈ H and for any s ∈ m, M.
Another result that compares the function of a self-adjoint operator with the integral
mean is embodied in the following theorem 2.
Theorem 1.2. With the assumptions in Theorem 1.1 one has the inequalities
M
1
fsds − fAx, y x, y ·
M−m m
≤
M
f max
m
t∈m,M
≤ x y
1/2
1/2
t−m
M−t
Et x, x1/2 Et y, y
1H − Et x, x1/2 1H − Et y, y
M−m
M−m
M
f ,
m
1.5
for any x, y ∈ H.
The trapezoid version of the above result has been obtained in 3 and is as follows.
Journal of Inequalities and Applications
3
Theorem 1.3. With the assumptions in Theorem 1.1 one has the inequalities
fM fm ·
x,
y
−
fAx,
y
2
≤
M
1/2
1/2 1
max Eλ x, x1/2 Eλ y, y
1H − Eλ x, x1/2 1H − Eλ y, y
f
2 λ∈m,M
m
≤
M
1
f ,
x y
2
m
1.6
for any x, y ∈ H.
For various inequalities for functions of self-adjoint operators, see 4–8. For recent
results see 1, 9–12.
In this paper, we investigate the quantity
fM1H − fA fA − fm1H x, y ,
1.7
where x, y are vectors in the Hilbert space H and A is a self-adjoint operator with SpA ⊆
m, M, and provide different bounds for some classes of continuous functions f : 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 A be a self-adjoint operator in the Hilbert space H with the spectrum SpA ⊆
m, M for some real numbers m < M and let {Eλ }λ be its spectral family. If f : m, M → C is a
continuous function on m, M with fM /
fm, then one has the representation
1
2 fM1H − fA fA − fm1H
fM − fm
1
fM − fm
M
m−0
Et −
1
fM − fm
M
Es dfs
m−0
1
Et − 1H dft.
2
2.1
4
Journal of Inequalities and Applications
Proof. We observe
1
fM − fm
M
1
Et −
fM − fm
m−0
1
fM − fm
M
m−0
M
1
Es dfs
Et − 1H dft
2
m−0
Et2 dft
M
M
1
1
Es dfs
Et dft
fM − fm m−0
fM − fm m−0
1 M
1 M
−
Et dft Es dfs
2 m−0
2 m−0
2
M
M
1
1
2
E dft −
Et dft ,
fM − fm m−0 t
fM − fm m−0
−
2.2
which is an equality of interest in itself.
Since Et are projections, we have Et2 Et for any t ∈ m, M and then we can write
1
fM − fm
M
m−0
Et2 dft
1
−
fM − fm
2
M
Et dft
m−0
2
M
1
Et dft −
Et dft
fM − fm m−0
m−0
M
M
1
1
Et dft 1H −
Et dft .
fM − fm m−0
fM − fm m−0
1
fM − fm
M
2.3
Integrating by parts in the Riemann-Stieltjes integral and utilizing the spectral representation 1.1, we have
M
Et dft fM1H − fA,
m−0
1
1H −
fM − fm
M
fA − fm1H
,
Et dft fM − fm
m−0
which together with 2.3 and 2.2 produce the desired result 2.1.
The following vector version may be stated as well.
2.4
Journal of Inequalities and Applications
5
Corollary 2.2. With the assumptions of Lemma 2.1 one has the equality
fM1H − fA fA − fm1H x, y
fM − fm
M m−0
1
Et −
fM − fm
1
Es dfs x, Et − 1H y dft,
2
m−0
2.5
M
for any x, y ∈ m, M.
The following result that provides some bounds for continuous functions of bounded
variation may be stated as well.
Theorem 2.3. Let A be a self-adjoint operator in the Hilbert space H with the spectrum SpA ⊆
m, M for some real numbers m < M, and let {Eλ }λ be its spectral family. If f : m, M → C is a
continuous function of bounded variation on m, M with fM /
fm, then we have the inequality
fM1H − fA fA − fm1H x, y ≤
M
1
y fM − fm
f
2
m
× sup
t∈m,M
1
≤ x y
2
1
Et x −
fM − fm
M
2.6
Es dfs
m−0
2
M
,
f
m
for any x, y ∈ H.
Proof. It is well known that if p : a, b → C is a bounded function, v : a, b → C is of
b
bounded variation, and the Riemann-Stieltjes integral a ptdvt exists, then the following
inequality holds:
b
b
ptdvt ≤ sup pt v,
t∈a,b
a
a
where
b
a v
denotes the total variation of v on a, b.
2.7
6
Journal of Inequalities and Applications
Utilising this property and the representation 2.5, we have by the Schwarz inequality
in Hilbert space H that
fM1H − fA fA − fm1H x, y M
f
≤ fM − fm
m
M
1
1
Et −
× sup Es dfs x, Et − 1H y fM
−
fm
2
m−0
t∈m,M
2.8
M
≤ fM − fm
f
× sup
t∈m,M
m
1
Et x −
fM − fm
M
1
Et y − y
2
Es xdfs
m−0
,
for any x, y ∈ m, M.
Since Et are projections, in this case we have
1
Et y − y
2
2
1
2
y
− Et y, y 4
1
y
Et2 y, y − Et y, y 4
Et y
2
2.9
2
1
y
4
2
,
then from 2.8, we deduce the first part of 2.6.
Now, by the same property 2.7 for vector-valued functions p with values in Hilbert
spaces, we also have
fM − fm Et x −
M
Es xdfs
m−0
M
m−0
M
f sup Et x − Es x,
Et x − Es xdfs ≤
m
2.10
s∈m,M
for any t ∈ m, M and x ∈ H.
Since 0 ≤ Et ≤ 1H in the operator order, then −1H ≤ Et − Es ≤ 1 which gives that
−x2 ≤ Et − Es x, x ≤ x2 , that is, |Et − Es x, x| ≤ x2 for any x ∈ H, which implies
that Et − Es ≤ 1 for any t, s ∈ m, M. Therefore, sups∈m,M Et x − Es x ≤ x which together
with 2.10 prove the last part of 2.6.
The case of Lipschitzian functions is as follows.
Journal of Inequalities and Applications
7
Theorem 2.4. Let A be a self-adjoint operator in the Hilbert space H with the spectrum SpA ⊆
m, M for some real numbers m < M, and let {Eλ }λ be its spectral family. If f : m, M → C is
a Lipschitzian function with the constant L > 0 on m, M and with fM /
fm, then one has the
inequality
fM1H − fA fA − fm1H x, y ≤
1
L y fM − fm
2
M
m−0
Et x −
1
fM − fm
M
Es xdfs dt
m−0
M
1 2
L y
Et x − Es xds dt
2
m−0
√
2 2
L y M − mAx − mx, Mx − Ax1/2
≤
2
√
2 2
L y xM − m2 ,
≤
4
≤
2.11
for any x, y ∈ H.
Proof. Recall that if p : a, b → C is a Riemann integrable function and v : a, b → C is
Lipschitzian with the constant L > 0, that is,
fs − ft ≤ L|s − t| for any t, s ∈ a, b,
then the Riemann-Stieltjes integral
b
a
2.12
ptdvt exists and the following inequality holds:
b
b
ptdvt ≤ L ptdt.
a
a
2.13
Now, on applying this property of the Riemann-Stieltjes integral, then we have from
the representation 2.5 that
fM1H − fA fA − fm1H x, y M
M 1
1
Et −
Es dfs x, Et − 1H y dft,
≤ fM − fm
fM
−
fm
2
m−0
m−0
≤ LfM − fm
M
m−0
1
L y fM − fm
2
1
Et x −
fM − fm
M
m−0
Et x −
M
Es xdfs
m−0
1
fM − fm
1
Et y − y dt
2
M
Es xdfs dt,
m−0
2.14
for any x, y ∈ H and the first inequality in 2.11 is proved.
8
Journal of Inequalities and Applications
Further, observe that
fM − fm
M
m−0
M
fM − fm Et x −
Es xdfs dt
m−0
M
m−0
M
M
1
fM − fm
Et x −
2.15
Es xdfs dt
m−0
M
m−0
Et x − Es xdfs dt,
m−0
for any x ∈ H.
If we use the vector-valued version of the property 2.13, then we have
M
m−0
M
Et x − Es xdfs dt ≤ L
M
m−0
Et x − Es xds dt,
2.16
m−0
for any x ∈ H and the second part of 2.11 is proved.
Further on, by applying the double-integral version of the Cauchy-BuniakowskiSchwarz inequality, we have
M
Et x − Es xds dt ≤ M − m
m−0
M
1/2
Et x − Es x2 ds dt
,
2.17
m−0
for any x ∈ H.
Now, by utilizing the fact that Es are projections for each s ∈ m, M, then we have
M
Et x − Es x2 ds dt
m−0
⎡
2⎣M − m
M
Et x2 dt −
m−0
⎡
2⎣M − m
M
m−0
for any x ∈ H.
M
2
⎤
Et xdt ⎦
m−0
Et x, xdt −
M
m−0
2
⎤
Et xdt ⎦,
2.18
Journal of Inequalities and Applications
9
If we integrate by parts and use the spectral representation 1.2, then we get
M
M
Et x, xdt Mx − Ax, x,
m−0
Et xdt Mx − Ax
2.19
m−0
and by 2.18, we then obtain the following equality of interest:
M
Et x − Es x2 ds dt 2Ax − mx, Mx − Ax,
2.20
m−0
for any x ∈ H.
On making use of 2.20 and 2.17, we then deduce the third part of 2.11.
Finally, by utilizing the elementary inequality in inner product spaces
Rea, b ≤
1
a b2 ,
4
a, b ∈ H,
2.21
1
M − m2 x2 ,
4
2.22
we also have that
Ax − mx, Mx − Ax ≤
for any x ∈ H, which proves the last inequality in 2.11.
The case of nondecreasing monotonic functions is as follows.
Theorem 2.5. Let A be a self-adjoint operator in the Hilbert space H with the spectrum SpA ⊆
m, M for some real numbers m < M, and let {Eλ }λ be its spectral family. If f : m, M → R is a
monotonic nondecreasing function on m, M, then one has the inequality
fM1H − fA fA − fm1H x, y 1
y fM − fm
≤
2
M
m−0
1
Et x −
fM − fm
M
Es xdfs dft
m−0
≤
1/2
1
y fM − fm fM1H − fA fA − fm1H x, x
2
≤
2
1
y x fM − fm ,
4
for any x, y ∈ H.
2.23
10
Journal of Inequalities and Applications
Proof. From the theory of Riemann-Stieltjes integral, it is also well known that if p : a, b →
C is of bounded variation and v : a, b → R is continuous and monotonic nondecreasing,
b
b
then the Riemann-Stieltjes integrals a ptdvt and a |pt|dvt exist and
b
b
ptdvt ≤ ptdvt.
a
a
2.24
Now, on applying this property of the Riemann-Stieltjes integral, we have from the
representation 2.5 that
fM1H − fA fA − fm1H x, y ≤ fM − fm
≤ fM − fm
M
M 1
1
Et −
Es dfs x, Et − 1H y dft,
fM
−
fm
2
m−0
m−0
M
1
fM − fm
Et x −
m−0
1
y fM − fm
2
M
Et x −
m−0
M
Es xdfs
m−0
1
fM − fm
1
Et y − y dft
2
M
Es xdfs dft,
m−0
2.25
for any x, y ∈ H, which proves the first inequality in 2.23.
On utilizing the Cauchy-Buniakowski-Schwarz-type inequality for the RiemannStieltjes integral of monotonic nondecreasing integrators, we have
M
m−0
1
Et x −
fM − fm
≤
M
m−0
for any x, y ∈ H.
M
Es xdfs dft
m−0
1/2 ⎡ M
⎣
dft
m−0
Et x −
1
fM − fm
M
2
Es xdfs
m−0
2.26
⎤1/2
dft⎦
,
Journal of Inequalities and Applications
11
Observe that
2
M
1
Es xdfs dft
fM − fm m−0
m−0
M
M
1
2
Es xdfs
Et x − 2 Re Et x,
fM − fm m−0
m−0
⎤
2
M
1
Es xdfs ⎦dft
fM − fm m−0
M
Et x −
⎡
fM − fm ⎣
1
fM − fm
M
1
Et x2 dft −
fM
− fm
m−0
M
2
⎤
⎦
Es xdfs
m−0
2.27
and, integrating by parts in the Riemann-Stieltjes integral, we have
M
M
M
Et x, Et xdft Et x, xdft
Et x2 dft m−0
m−0
fMx2 −
ftdEt x, x
m−0
fMx − fAx, x fM1H − fA x, x ,
2
M
m−0
M
2.28
Es xdfs fMx − fAx,
m−0
for any x ∈ H.
On making use of the equalities 2.28, we have
1
fM − fm
M
1
Et x dft −
fM − fm
m−0
2
M
2
Es xdfs
m−0
1
2 fM − fm fM1H − fA x, x − fMx − fAx
fM − fm
2
fM − fm fM1H − fA x, x − fMx − fAx, fMx − fAx
2
fM − fm
fM − fm fM1H − fA x, x − fMx − fAx, fMx − fAx
2
fM − fm
fMx − fAx, fAx − fmx
,
2
fM − fm
for any x ∈ H.
2.29
12
Journal of Inequalities and Applications
Therefore, we obtain the following equality of interest in itself as well:
1
fM − fm
M
m−0
1
Et x −
fM − fm
fMx − fAx, fAx − fmx
2
fM − fm
M
2
Es xdfs
dft
m−0
2.30
fM1H − fA fA − fm1H x, x
,
2
fM − fm
for any x ∈ H
On making use of the inequality 2.26, we deduce the second inequality in 2.23.
The last part follows by 2.21, and the details are omitted.
3. Applications
We consider the power function ft : tp , where p ∈ R \ {0} and t > 0. The following power
inequalities hold.
Proposition 3.1. Let A be a self-adjoint operator in the Hilbert space H with the spectrum SpA ⊆
m, M for some real numbers with 0 ≤ m < M.
If p > 0, then for any x, y ∈ H,
Mp 1H − Ap Ap − mp 1H x, y √
2 2
≤
B y M − mAx − mx, Mx − Ax1/2
2 p
√
2 2
B y xM − m2 ,
≤
4 p
3.1
where
Bp p ×
⎧
p−1
⎪
⎨M ,
if p ≥ 1,
⎪
⎩mp−1 ,
if 0 < p < 1, m > 0,
−p
A − M−p 1H m−p 1H − A−p x, y √
2 2
C y M − mAx − mx, Mx − Ax1/2
≤
2 p
√
2 2
C y xM − m2 ,
≤
4 p
3.2
Journal of Inequalities and Applications
13
where
Cp pm−p−1 ,
m > 0.
3.3
The proof follows from Theorem 2.4 applied for the power function.
Proposition 3.2. Let A be a self-adjoint operator in the Hilbert space H with the spectrum SpA ⊆
m, M for some real numbers with 0 ≤ m < M.
If p > 0, then for any x, y ∈ H
Mp 1H − Ap Ap − mp 1H x, y ≤
1
y Mp − mp Mp 1H − Ap Ap − mp 1H x, x1/2
2
1
y xMp − mp 2 ,
4
−p
A − M−p 1H m−p 1H − A−p x, y ≤
≤
1/2
1
y m−p − M−p A−p − M−p 1H m−p 1H − A−p x, x
2
≤
2
1
y x m−p − M−p .
4
3.4
The proof follows from Theorem 2.5.
Now, consider the logarithmic function ft ln t, t > 0. We have the following
Proposition 3.3. Let A be a self-adjoint operator in the Hilbert space H with the spectrum SpA ⊆
m, M for some real numbers with 0 < m < M. Then one has the inequalities
ln M1H − ln Aln A − ln m1H x, y √
2
≤
y M − mAx − mx, Mx − Ax1/2
2m2
√
2
M
2
y x
−1 ,
≤
4
m
ln M1H − ln Aln A − ln m1H x, y 3.5
1
M
1/2
≤
y ln M1H − ln Aln A − ln m1H x, x ln
2
m
2
M
1
y x ln
.
≤
4
m
The proof follows from Theorems 2.4 and 2.5 applied for the logarithmic function.
14
Journal of Inequalities and Applications
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