vol22_pp161-178. 167KB Jun 04 2011 12:06:12 AM
Electronic Journal of Linear Algebra ISSN 1081-3810
A publication of the International Linear Algebra Society
Volume 22, pp. 161-178, March 2011
ELA
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VECTOR AND OPERATOR TRAPEZOIDAL TYPE INEQUALITIES
FOR CONTINUOUS FUNCTIONS OF SELFADJOINT OPERATORS
IN HILBERT SPACES∗
SEVER S. DRAGOMIR†
Abstract. Some vector and operator generalized trapezoidal inequalities for continuous functions of selfadjoint operators in Hilbert spaces are given. Applications for power and logarithmic
functions of operators are provided as well.
Key words. Selfadjoint operators, Functions of selfadjoint operators, Spectral representation,
Inequalities for selfadjoint operators.
AMS subject classifications. 47A63, 47A99.
1. Introduction. Let A be a selfadjoint linear operator on a complex Hilbert
space (H; h·, ·i). The Gelfand map establishes an isometric ∗-isomorphism Φ between
the C ∗ -algebra C (Sp (A)) of all continuous functions defined on the spectrum of
A, denoted by Sp (A), and the C ∗ -algebra C ∗ (A) generated by A and the identity
operator 1H on H (see for instance [10, p. 3]) with the property that Φ (f0 ) = 1H
and Φ (f1 ) = A, where f0 (t) = 1 and f1 (t) = t for t ∈ Sp (A).
With this notation, we define
f (A) := Φ (f ) for all f ∈ C (Sp (A))
and we call it the continuous functional calculus for a selfadjoint operator A.
If A is a selfadjoint operator and f is a real valued continuous function on Sp (A),
then f (t) ≥ 0 for any t ∈ Sp (A) implies that f (A) ≥ 0, i.e., f (A) is a positive
operator on H. Moreover, if both f and g are real valued functions on Sp (A), then
the following important property holds:
f (t) ≥ g (t) for any t ∈ Sp (A) implies that f (A) ≥ g (A)
(P)
in the operator order of B (H).
∗ Received
by the editors on November 29, 2010. Accepted for publication on February 17, 2011.
Handling Editor: Moshe Goldberg.
† School of Engineering & Science, Victoria University, PO Box 14481, Melbourne City, MC
8001, Australia ([email protected]), and School of Computational & Applied Mathematics,
University of the Witwatersrand, Private Bag-3, Wits-2050, Johannesburg, South Africa.
161
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S.S. Dragomir
For a recent monograph devoted to various inequalities for continuous functions
of selfadjoint operators, see [10] and the references therein.
For other recent results, see [5]–[7], [12]–[14] and [15].
Let U be a selfadjoint operator on the complex Hilbert space (H, h·, ·i) with the
spectrum Sp (U ) 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 :
hf (U ) x, yi =
Z
M
f (λ) d (hEλ x, yi) ,
(1.1)
m−0
for any x, y ∈ H. The function gx,y (λ) := hEλ x, yi is of bounded variation on the
interval [m, M ], and
gx,y (m − 0) = 0 and gx,y (M ) = hx, yi
for any x, y ∈ H. It is also well known that gx (λ) := hEλ x, xi is monotonic nondecreasing and right continuous on [m, M ].
With the notations introduced above, we have considered in the recent paper [8]
the problem of bounding the error
f (M ) + f (m)
· hx, yi − hf (A) x, yi
2
(m)
·hx, yi, where
in approximating hf (A) x, yi by the trapezoidal type formula f (M )+f
2
x, y are vectors in the Hilbert space H and f is a continuous function of the selfadjoint
operator A with the spectrum in the compact interval of real numbers [m, M ].
We recall here only two such results. The first deals with the case of continuous
functions of bounded variation and is incorporated in the following theorem [8]:
Theorem 1.1. Let A be a selfadjoint operator in the Hilbert space H with the
spectrum Sp (A) ⊆ [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 ],
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Vector and Operator Trapezoidal Type Inequalities
then we have the inequalities
f (M ) + f (m)
· hx, yi − hf (A) x, yi
2
h
1
1/2
1/2
max hEλ x, xi hEλ y, yi
≤
2 λ∈[m,M ]
+ h(1H − Eλ ) x, xi
1/2
(1.2)
1/2
h(1H − Eλ ) y, yi
M
≤
M
i_
(f )
m
_
1
kxk kyk (f )
2
m
for any x, y ∈ H.
The case of Lipschitzian functions is as follows [8]:
Theorem 1.2. Let A be a selfadjoint operator in the Hilbert space H with the
spectrum Sp (A) ⊆ [m, M ] for some real numbers m < M , and let {Eλ }λ be its spectral
family. If f : [m, M ] → C is Lipschitzian with the constant L > 0 on [m, M ], then we
have the inequalities
f (M ) + f (m)
·
hx,
yi
−
hf
(A)
x,
yi
2
Z M h
1
1/2
1/2
hEλ x, xi hEλ y, yi
≤ L
2
m−0
+ h(1H − Eλ ) x, xi
≤
1/2
1
(M − m) L kxk kyk
2
h(1H − Eλ ) y, yi
(1.3)
1/2
i
dλ
for any x, y ∈ H.
In order to provide error bounds in approximating hf (A) x, yi with the quantity
1
[f (m) (M hx, yi − hAx, yi) + f (M ) (hAx, yi − m hx, yi)] ,
M −m
where x, y ∈ H, which is a generalized trapezoid formula, we obtained in the recent
paper [9] the following results:
Theorem 1.3. Let A be a selfadjoint operator in the Hilbert space H with the
spectrum Sp (A) ⊆ [m, M ] for some real numbers m < M , and let {Eλ }λ be its spectral
family.
Electronic Journal of Linear Algebra ISSN 1081-3810
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S.S. Dragomir
1. If f : [m, M ] → C is continuous and of bounded variation on [m, M ], then
f (m) (M 1H − A) + f (M ) (A − m1H )
x, y − hf (A) x, yi
(1.4)
M −m
#M
"
t
M
_
M −t _
t−m _
E(·) x, y +
E(·) x, y
(f )
≤ sup
M −m t
t∈[m,M ] M − m m
m
≤
M
_
E(·) x, y
M
_
m
m
for any x, y ∈ H.
(f ) ≤ kxk kyk
M
_
(f )
m
2. If f : [m, M ] → C is Lipschitzian with the constant L > 0 on [m, M ], then
f (m) (M 1H − A) + f (M ) (A − m1H )
x, y − hf (A) x, yi
(1.5)
M −m
#
Z M "
t
M
M −t _
t−m _
dt
E(·) x, y +
E(·) x, y
≤L
M −m t
m−0 M − m m−0
≤ L (M − m)
M
_
m−0
for any x, y ∈ H.
E(·) x, y
≤ L (M − m) kxk kyk
3. If f : [m, M ] → R is continuous and monotonic nondecreasing on [m, M ], then
f (m) (M 1H − A) + f (M ) (A − m1H )
x,
y
−
hf
(A)
x,
yi
(1.6)
M −m
#
Z M "
t
M
M −t _
t−m _
df (t)
E(·) x, y +
E(·) x, y
≤
M −m t
m−0 M − m m−0
≤
M
_
m
for any x, y ∈ H.
E(·) x, y
[f (M ) − f (m)] ≤ kxk kyk [f (M ) − f (m)]
For other results of this type, see [9]. For scalar trapezoidal inequalities, see
[1]–[4].
The main aim of this paper is to provide more results of the above type as well
as bounds in the operator order for the selfadjoint linear operator
f (m) (M 1H − A) + f (M ) (A − m1H )
,
−
f
(A)
M −m
where f is a continuous function of bounded variation, Lipschitzian or convex on
[m, M ]. Applications for certain functions of large interest such as the power and
logarithmic functions are provided as well.
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Vector and Operator Trapezoidal Type Inequalities
165
2. Some new vector inequalities. The following result for general continuous
functions holds:
Theorem 2.1. Let A be a selfadjoint operator in the Hilbert space H with the
spectrum Sp (A) ⊆ [m, M ] for some real numbers m < M , and let {Eλ }λ be its spectral
family. If f : [m, M ] → R is continuous on [m, M ], then we have the inequalities
f (m) (M 1H − A) + f (M ) (A − m1H )
(2.1)
x,
y
−
hf
(A)
x,
yi
M −m
_
M
≤ max f (t) − min f (t)
E(·) x, y
t∈[m,M ]
≤
t∈[m,M ]
m
max f (t) − min f (t) kxk kyk
t∈[m,M ]
t∈[m,M ]
for any x, y ∈ H.
Proof. We observe that, by the spectral representation (1.1), we have the equality
f (m) (M 1H − A) + f (M ) (A − m1H )
x, y − hf (A) x, yi
(2.2)
M −m
Z M
Φf (t) d (hEt x, yi)
=
m−0
for any x, y ∈ H, where Φf : [m, M ] → R is given by
Φf (t) =
1
[(M − t) f (m) + (t − m) f (M )] − f (t) .
M −m
Since Φf is continuous on [m, M ] and Φf (m) = 0, we can replace in what follows
RM
RM
Φ (t) dv (t) by m Φf (t) dv (t) for every function v : [m, M ] → C of bounded
m−0 f
variation.
It is well known that if p : [a, b] → C is a continuous function and v : [a, b] → C
Rb
is of bounded variation, then the Riemann-Stieltjes integral a p (t) dv (t) exists and
the following inequality holds:
Z
b
b
_
p (t) dv (t) ≤ sup |p (t)| (v) ,
(2.3)
a
t∈[a,b]
a
where
b
_
(v) denotes the total variation of v on [a, b].
a
Now, if we denote by γ := mint∈[m,M ] f (t) and by Γ := maxt∈[m,M ] f (t), then we
have
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S.S. Dragomir
γ (M − t) ≤ (M − t) f (m) ≤ Γ (M − t) ,
γ (t − m) ≤ (t − m) f (M ) ≤ Γ (t − m)
and
− (M − m) Γ ≤ − (M − m) f (t) ≤ −γ (M − m)
for any t ∈ [m, M ]. If we add these three inequalities, then we get
− (M − m) (Γ − γ) ≤ (M − m) Φf (t) ≤ (M − m) (Γ − γ)
for any t ∈ [m, M ], which shows that
|Φf (t)| ≤ Γ − γ for any t ∈ [m, M ] .
(2.4)
Applying the inequality (2.3) for the representation (2.2), we have from (2.4) that
Z
M
M
_
Φf (t)d (hEt x, yi) ≤ (Γ − γ)
E(·) x, y
m
m
for any x, y ∈ H, which proves the first part of (2.1).
If P is a nonnegative operator on H, i.e., hP x, xi ≥ 0 for any x ∈ H, then the
following inequality is a generalization of the Schwarz inequality in H:
2
|hP x, yi| ≤ hP x, xi hP y, yi
(2.5)
for any x, y ∈ H.
Now, if d : m = t0 < t1 < · · · < tn−1 < tn = M is an arbitrary partition of the
interval [m, M ], then we have by Schwarz’s inequality for nonnegative operators that
M
_
m
E(·) x, y
(n−1
)
X
Et − Et x, y
= sup
i
i+1
d
≤ sup
d
i=0
(n−1
X h
i=0
Eti+1 − Eti
)
1/2
1/2 i
x, x
Eti+1 − Eti y, y
:= I.
By the Cauchy-Bunyakovsky-Schwarz inequality for sequences of real numbers, we
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Vector and Operator Trapezoidal Type Inequalities
also have that
I ≤ sup
d
≤ sup
d
=
"
n−1
X
i=0
"
n−1
X
i=0
"M
_
m
Eti+1 − Eti
x, x
Eti+1 − Eti
x, x
E(·) x, x
#1/2 " M
_
#1/2 "n−1
X
i=0
#1/2
E(·) y, y
m
sup
d
#1/2
Eti+1 − Eti
"
n−1
X
i=0
#1/2
y, y
Eti+1 − Eti
= kxk kyk
#1/2
y, y
for any x, y ∈ H. These prove the last part of (2.1).
When the generating function is of bounded variation, we have the following result
that complements Theorem 1.3:
Theorem 2.2. Let A be a selfadjoint operator in the Hilbert space H with the
spectrum Sp (A) ⊆ [m, M ] for some real numbers m < M , and let {Eλ }λ be its spectral
family. If f : [m, M ] → C is continuous and of bounded variation on [m, M ], then we
have the inequalities
f (m) (M 1H − A) + f (M ) (A − m1H )
x, y − hf (A) x, yi
M −m
#M
"
t
M
_
t−m _
M −t _
(f ) +
(f )
E(·) x, y
≤ max
M −m t
t∈[m,M ] M − m
m
m
≤
M
_
m
E(·) x, y
M
_
m
(f ) ≤
M
_
m
(2.6)
(f ) kxk kyk
for any x, y ∈ H.
Proof. First of all, observe that
(M − m) Φf (t) = (t − M ) [f (t) − f (m)] + (t − m) [f (M ) − f (t)]
for any t ∈ [m, M ], where here Φf : [m, M ] → C.
(2.7)
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Therefore,
t−m
M −t
|f (t) − f (m)| +
|f (M ) − f (t)|
M −m
M −m
t
M
M −t _
t−m _
≤
(f ) +
(f )
M −m m
M −m t
#
"_
t
M
_
M −t t−m
,
(f ) +
(f )
≤ max
M −m M −m
m
t
"
# M
_
1 t − m+M
2
(f )
=
+
2
M −m
m
|Φf (t)| ≤
for any t ∈ [m, M ], which implies that
"
#
t
M
M −t _
t−m _
max |Φf (t)| ≤ max
(f ) +
(f )
M −m t
t∈[m,M ]
t∈[m,M ] M − m
m
"
# M
M
_
_
1 t − m+M
2
≤ max
(f ) =
(f ) .
+
M −m
t∈[m,M ] 2
m
m
(2.8)
(2.9)
Applying the inequality (2.3) for the representation (2.2), we have from (2.9) that
Z
"
#M
t
M
M
_
M −t _
t−m _
Φf (t) d (hEt x, yi) ≤ max
(f ) +
(f )
E(·) x, y
t∈[m,M ] M − m m
m
M −m t
m
≤
M
_
m
(f )
M
_
m
E(·) x, y
for any x, y ∈ H, which implies the desired result (2.6).
The case of Lipschitzian functions is as follows:
Theorem 2.3. Let A be a selfadjoint operator in the Hilbert space H with the
spectrum Sp (A) ⊆ [m, M ] for some real numbers m < M , and let {Eλ }λ be its spectral
family. If f : [m, M ] → C is Lipschitzian with the constant L > 0 on [m, M ], then we
have the inequalities
f (m) (M 1H − A) + f (M ) (A − m1H )
x, y − hf (A) x, yi
(2.10)
M −m
M
_
M −t
t−m
|f (t) − f (m)| +
|f (M ) − f (t)|
E(·) x, y
max
≤
M −m
t∈[m,M ] M − m
m
≤
M
_
1
1
E(·) x, y ≤ (M − m) L kxk kyk
(M − m) L
2
2
m
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for any x, y ∈ H.
Proof. We have from the first part of the equality (2.7) that
M −t
t−m
|f (t) − f (m)| +
|f (M ) − f (t)|
M −m
M −m
2L
1
≤
(M − t) (t − m) ≤ (M − m) L
M −m
2
|Φf (t)| ≤
(2.11)
for any t ∈ [m, M ], which, by arguments similar to the arguments of the proof of
Theorem 2.2, yields the desired result (2.10). The details are omitted.
The following lemma may be stated.
Lemma 2.4. Let u : [a, b] → R and ϕ, Φ ∈ R be such that Φ > ϕ. The following
statements are equivalent:
(i) The function u− ϕ+Φ
2 ·e, where e (t) = t, t ∈ [a, b], is
(ii) It holds that
ϕ≤
1
2
(Φ − ϕ) −Lipschitzian;
u (t) − u (s)
≤ Φ for each t, s ∈ [a, b] with t 6= s;
t−s
(2.12)
(iii) It holds that
ϕ (t − s) ≤ u (t) − u (s) ≤ Φ (t − s) for each t, s ∈ [a, b] with t > s. (2.13)
Following [11], we can introduce the concept:
Definition 2.5. A function u : [a, b] → R which satisfies one of the equivalent
conditions (i)–(iii) is said to be (ϕ, Φ) −Lipschitzian on [a, b].
Notice that in [11], the definition was introduced on utilizing the statement (iii)
and only the equivalence (i) ⇔ (iii) was considered.
Utilizing Lagrange’s mean value theorem, we can state the following result that
provides practical examples of (ϕ, Φ) −Lipschitzian functions.
Proposition 2.6. Let u : [a, b] → R be continuous on [a, b] and differentiable on
(a, b). If
−∞ < γ := inf u′ (t) and
t∈(a,b)
sup u′ (t) =: Γ < ∞,
(2.14)
t∈(a,b)
then u is (γ, Γ) −Lipschitzian on [a, b].
The following corollary holds:
Corollary 2.7. Let A be a selfadjoint operator in the Hilbert space H with the
spectrum Sp (A) ⊆ [m, M ] for some real numbers m < M , and let {Eλ }λ be its spectral
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S.S. Dragomir
family. If l, L ∈ R are such that L > l and f : [m, M ] → R is (l, L) −Lipschitzian on
[m, M ], then we have the inequality
f (m) (M 1H − A) + f (M ) (A − m1H )
(2.15)
x, y − hf (A) x, yi
M −m
≤
M
_
1
1
E(·) x, y ≤ (M − m) (L − l) kxk kyk
(M − m) (L − l)
4
4
m
for any x, y ∈ H.
Proof. The proof follows by applying the inequality (2.10) to the 21 (L − l)–
Lipschitzian function f − 12 (l + L) e, where e (t) = t, t ∈ [m, M ]. The details are
omitted.
When the generating function is continuous convex, we can state the following
result as well:
Theorem 2.8. Let A be a selfadjoint operator in the Hilbert space H with the
spectrum Sp (A) ⊆ [m, M ] for some real numbers m < M , and let {Eλ }λ be its
spectral family. If f : [m, M ] → R is continuous convex on [m, M ] with finite lateral
′
′
(m), then we have the inequalities
(M ) and f+
derivatives f−
f (m) (M 1H − A) + f (M ) (A − m1H )
x, y − hf (A) x, yi
M −m
(2.16)
M
′
_
1
′
E(·) x, y
(M − m) f−
(M ) − f+
(m)
4
m
′
1
′
≤ (M − m) f− (M ) − f+ (m) kxk kyk
4
≤
for any x, y ∈ H.
Proof. By the convexity of f on [m, M ], we have
′
f (t) − f (M ) ≥ f−
(M ) (t − M )
for any t ∈ [m, M ]. If we multiply this inequality by t − m ≥ 0, then we deduce
′
(t − m) f (t) − (t − m) f (M ) ≥ f−
(M ) (t − M ) (t − m)
(2.17)
for any t ∈ [m, M ]. Similarly, we get
′
(M − t) f (t) − (M − t) f (m) ≥ f+
(m) (M − t) (t − m)
for any t ∈ [m, M ].
(2.18)
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Vector and Operator Trapezoidal Type Inequalities
Summing the above inequalities and dividing by M − m we deduce the inequality
Φf (t) ≤
′
1
(M − t) (t − m) ′
′
′
(M ) − f+
f− (M ) − f+
(m)
(m) ≤ (M − m) f−
M −m
4
for any t ∈ [m, M ]. By the convexity of f , we also have that
1
[(M − t) f (m) + (t − m) f (M )] ≥ f
M −m
(M − t) m + (t − m) M
M −m
= f (t)
(2.19)
giving that
Φf (t) ≥ 0 for any t ∈ [m, M ] .
(2.20)
Utilizing (2.3) for the representation (2.2) we deduce from (2) and (2.20) the
desired result (2.16).
We observe that a similar result holds for concave functions as well.
3. Inequalities in the operator order. Before we state the next results, we
recall that if A is a bounded linear operator, then the operator A√∗ A is selfadjoint and
positive and the “absolute value” operator is defined by |A| := A∗ A.
In what follows, we use the fact that if A is selfadjoint and f ∈ C (Sp (A)), then
|f (A)| = (|f |) (A).
The following result providing some inequalities in the operator order may be
stated:
Theorem 3.1. Let A be a selfadjoint operator in the Hilbert space H with the
spectrum Sp (A) ⊆ [m, M ] for some real numbers m < M .
1. If f : [m, M ] → R is continuous on [m, M ], then
f (m) (M 1H − A) + f (M ) (A − m1H )
− f (A)
M −m
≤ max f (t) − min f (t) 1H .
t∈[m,M ]
t∈[m,M ]
(3.1)
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2. If f : [m, M ] → C is continuous and of bounded variation on [m, M ], then
f (m) (M 1H − A) + f (M ) (A − m1H )
−
f
(A)
(3.2)
M −m
A
M
M 1H − A _
A − m1H _
(f ) +
(f )
M −m m
M −m
A
"
# M
_
1 A − m+M
2 1H
(f ) ,
+
≤
2
M −m
m
≤
where
A
_
(f ) denotes the operator generated by the scalar function
m
[m, M ] ∋ t 7−→
The same notation applies for
M
_
t
_
m
(f ) ∈ R.
(f ).
A
3. If f : [m, M ] → C is Lipschitzian with the constant L > 0 on [m, M ], then
f (m) (M 1H − A) + f (M ) (A − m1H )
− f (A)
(3.3)
M −m
A − m1H
M 1H − A
|f (A) − f (m) 1H | +
|f (M ) 1H − f (A)|
≤
M −m
M −m
1
≤ (M − m) L1H .
2
4. If f : [m, M ] → R is continuous convex on [m, M ] with finite lateral derivatives
′
′
(m), then we have the inequalities
(M ) and f+
f−
f (m) (M 1H − A) + f (M ) (A − m1H )
− f (A)
M −m
(M 1H − A) (A − m1H ) ′
′
≤
f− (M ) − f+
(m)
M −m
′
1
′
(M ) − f+
(m) 1H .
≤ (M − m) f−
4
0≤
(3.4)
Proof. the proof follows by applying the property (P) to the scalar inequalities
(2.4), (2.8), (2.11), (2) and (2.20). The details are omitted.
The following particular case is perhaps more useful for applications:
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Corollary 3.2. Let A be a selfadjoint operator in the Hilbert space H with the
spectrum Sp (A) ⊆ [m, M ] for some real numbers m < M . If l, L ∈ R with L > l and
f : [m, M ] → R is (l, L) −Lipschitzian on [m, M ], then we have
f (m) (M 1H − A) + f (M ) (A − m1H )
1
− f (A) ≤ (M − m) (L − l) 1H . (3.5)
M −m
4
4. More inequalities for differentiable functions. The following result holds:
Theorem 4.1. Let A be a selfadjoint operator in the Hilbert space H with the
spectrum Sp (A) ⊆ [m, M ] for some real numbers m < M . Assume that the function
f : [m, M ] → C is continuously differentiable on [m, M ].
1. If the derivative f ′ is of bounded variation on [m, M ], then we have the inequalities
f (m) (M 1H − A) + f (M ) (A − m1H )
x,
y
−
hf
(A)
x,
yi
(4.1)
M −m
≤
M
M
_
_
1
E(·) x, y
(M − m) (f ′ )
4
m
m
M
≤
_
1
(M − m) (f ′ ) kxk kyk
4
m
for any x, y ∈ H.
2. If the derivative f ′ is Lipschitzian with the constant K > 0 on [m, M ], then
we have the inequalities
f (m) (M 1H − A) + f (M ) (A − m1H )
x,
y
−
hf
(A)
x,
yi
(4.2)
M −m
≤
≤
M
_
1
2
E(·) x, y
(M − m) K
8
m
1
2
(M − m) K kxk kyk
8
for any x, y ∈ H.
Proof. First of all we notice that if f : [m, M ] → C is continuously differentiable
on [m, M ], then we have the following representation in terms of the Riemann-Stieltjes
integral:
Z M
1
Φf (t) =
K (t, s) df ′ (s), t ∈ [m, M ] ,
(4.3)
M −m m
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2
where the kernel K : [m, M ] → R is given by
(
(M − t) (s − m) if m ≤ s ≤ t
K (t, s) :=
(t − m) (M − s) if t < s ≤ M.
(4.4)
Indeed, since f ′ is continuously differentiable on [m, M ], then the Riemann-Stieltjes
Rt
RM
integrals m (s − m) df ′ (s) and t (M − s) df ′ (s) exist for each t ∈ [m, M ]. Now,
integrating by parts in the Riemann-Stieltjes integral yields
Z M
Z t
Z M
(M − s) df ′ (s)
(s − m) df ′ (s) + (t − m)
K(t, s)df ′ (s) = (M − t)
t
m
m
Z t
t
′
′
f (s)ds
= (M − t) (s − m) f (s) m −
m
#
"
Z M
M
′
′
f (s)ds
+ (t − m) (M − s) f (s) +
t
t
= (M − t) [(t − m) f ′ (t) − (f (t) − f (m))]
+ (t − m) [− (M − t) f ′ (t) + f (M ) − f (t)]
= (t − m) [f (M ) − f (t)] − (M − t) [f (t) − f (m)]
= (M − m) Φf (t)
for any t ∈ [m, M ], which provides the desired representation (4.3).
Now, utilizing the representation (4.3) and the property (2.3), we have
|Φf (t)|
=
≤
≤
=
(4.5)
Z M
Z t
1
(M − s) df ′ (s)
(s − m) df ′ (s) + (t − m)
(M − t)
M −m
t
m
Z
#
"
Z t
M
1
(M − s) df ′ (s)
(s − m) df ′ (s) + (t − m)
(M − t)
M −m
t
m
#
"
M
t
_
_
1
′
′
(M − t) (f ) sup (s − m) + (t − m) (f ) sup (M − s)
M −m
s∈[t,M ]
s∈[m,t]
t
m
" t
#
M
_
(t − m) (M − t) _ ′
(f ) + (f ′ )
M −m
m
t
M
=
M
_
(t − m) (M − t) _ ′
1
(f ) ≤ (M − m) (f ′ )
M −m
4
m
m
for any t ∈ [m, M ].
Using the representation (2.2), we deduce the desired result (4.1).
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Further, we utilize the fact that for an L−Lipschitz continuous function, v :
[α, β] → C and a Riemann integrable function p : [α, β] → C, the Riemann-Stieltjes
Rβ
integral α p (s) dv(s) exists and
Z
Z β
β
|p(s)| ds.
p(s)dv(s)
≤
L
α
α
Then, by utilizing (4.5) we have
|Φf (t)|
(4.6)
Z
#
Z t
M
1
≤
(M − s) df ′ (s)
(s − m) df ′ (s) + (t − m)
(M − t)
M −m
t
m
#
"
Z M
Z t
K
(M − s) ds
(s − m) ds + (t − m)
(M − t)
≤
M −m
t
m
"
#
2
2
(M − t) (t − m)
(t − m) (M − t)
K
+
=
M −m
2
2
=
"
1
1
2
(M − m) (t − m) (M − t) K ≤ (M − m) K
2
8
for any t ∈ [m, M ].
Using the representation (2.2), we deduce the desired result (4.2).
The following inequalities in the operator order are of interest as well:
Theorem 4.2. Let A be a selfadjoint operator in the Hilbert space H with the
spectrum Sp (A) ⊆ [m, M ] for some real numbers m < M . Assume that the function
f : I → C with [m, M ] ⊂ ˚
I (the interior of I) is differentiable on ˚
I.
1. If the derivative f ′ is continuous and of bounded variation on [m, M ], then we
have the inequalities
f (m) (M 1H − A) + f (M ) (A − m1H )
− f (A)
M −m
(4.7)
M
(A − m1H ) (M 1H − A) _ ′
(f )
≤
M −m
m
M
≤
_
1
(M − m) (f ′ ) 1H .
4
m
2. If the derivative f ′ is Lipschitzian with the constant K > 0 on [m, M ], then
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S.S. Dragomir
we have the inequalities
f (m) (M 1H − A) + f (M ) (A − m1H )
−
f
(A)
M −m
1
≤ (M − m) (A − m1H ) (M 1H − A) K
2
1
2
≤ (M − m) K1H .
8
(4.8)
Proof. The proof follows by the property (P) applied for the scalar inequalities
(4.5) and (4.6).
5. Applications for particular functions. It is obvious that the above results
can be applied for various particular functions. However, we will restrict here only to
the power and logarithmic functions.
1. Consider now the power function f : (0, ∞) → R, f (t) = tp with p 6= 0.
Applying Theorem 2.8, we can state the following proposition:
Proposition 5.1. Let A be a selfadjoint operator in the Hilbert space H with
the spectrum Sp (A) ⊆ [m, M ] for some real numbers 0 < m < M . Then for any
x, y ∈ H, we have the inequality
p
m (M 1H − A) + M p (A − m1H )
p
x, y − hA x, yi
(5.1)
M −m
1
≤ (M − m) ∆p kxk kyk
4
where
∆p = p ×
In particular,
p−1
− mp−1
M
mp−1 − M p−1
if p ∈ (−∞, 0) ∪ [1, ∞)
if 0 < p < 1.
M (M 1H − A) + m (A − m1H )
−1
x, y − A x, y
mM (M − m)
(5.2)
2
≤
1 (M − m) (M + m)
kxk kyk
4
m2 M 2
for any x, y ∈ H.
The following inequalities in the operator order also hold:
Proposition 5.2. Let A be a selfadjoint operator in the Hilbert space H with
the spectrum Sp (A) ⊆ [m, M ] for some real numbers 0 < m < M .
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If p ∈ (−∞, 0) ∪ [1, ∞), then
mp (M 1H − A) + M p (A − m1H )
− Ap
M −m
(M 1H − A) (A − m1H )
≤p
M p−1 − mp−1
M −m
1
≤ p (M − m) M p−1 − mp−1 1H .
4
0≤
(5.3)
If p ∈ (0, 1), then
mp (M 1H − A) + M p (A − m1H )
M −m
(M 1H − A) (A − m1H )
≤p
mp−1 − M p−1
M −m
1
≤ p (M − m) mp−1 − M p−1 1H .
4
0 ≤ Ap −
(5.4)
In particular, we have the inequalities
M (M 1H − A) + m (A − m1H )
− A−1
mM (M − m)
(M 1H − A) (A − m1H ) M 2 − m2
≤
·
M −m
m2 M 2
2
1 (M − m) (M + m)
1H .
≤
4
m2 M 2
0≤
(5.5)
Proof. The proof follows from (3.4) and the details are omitted.
2. The case of logarithmic function is as follows:
Proposition 5.3. Let A be a selfadjoint operator in the Hilbert space H with
the spectrum Sp (A) ⊆ [m, M ] for some real numbers 0 < m < M . Then for any
x, y ∈ H, we have the inequality
(M 1H − A) ln m + (A − m1H ) ln M
x, y − hln Ax, yi
(5.6)
M −m
2
1 (M − m)
kxk kyk .
4 mM
We also have the following inequality in the operator order
≤
(M 1H − A) ln m + (A − m1H ) ln M
M −m
2
1 (M − m)
(M 1H − A) (A − m1H )
≤
≤
1H .
Mm
4 mM
0 ≤ ln A −
(5.7)
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S.S. Dragomir
Remark 5.4. Similar results can be obtained if ones uses the inequalities from
Theorem 4.1 and 4.2. However the details are left to the interested reader.
Acknowledgment. The author would like to thank the anonymous referee for
valuable suggestions that have been implemented in the final version of this paper.
REFERENCES
[1] P. Cerone and S.S. Dragomir. Trapezoidal-type rules from an inequalities point of view. Handbook of Analytic-Computational Methods in Applied Mathematics, G.A. Anastassiou (editor), Chapman & Hall/CRC Press, New York, 2000, 65–134.
[2] P. Cerone, S.S. Dragomir, and C.E.M. Pearce. A generalised trapezoid inequality for functions
of bounded variation. Turkish J. Math., 24(2):147–163, 2000.
[3] S.S. Dragomir. On the trapezoid quadrature formula for Lipschitzian mappings and applications. Tamkang J. Math., 30(2):133–138, 1999.
[4] S.S. Dragomir. An inequality improving the second Hermite-Hadamard inequality for convex
functions defined on linear spaces and applications for semi-inner products. J. Inequal.
Pure Appl. Math., 3, 2002, Article 3.
ˇ
[5] S.S. Dragomir. Cebyˇ
sev’s type inequalities for functions of selfadjoint operators in Hilbert
spaces. Preprint, RGMIA Res. Rep. Coll., 11(e), 2008, Article 9. Availiable at
http://rgmia.org/issues.php.
[6] S.S. Dragomir. Gr¨
uss’ type inequalities for functions of selfadjoint operators in Hilbert
spaces. Preprint, RGMIA Res. Rep. Coll., 11(e), 2008, Article 11. Availiable at
http://rgmia.org/issues.php.
ˇ
[7] S.S. Dragomir. Inequalities for the Cebyˇ
sev functional of two functions of selfadjoint operators
in Hilbert spaces. Preprint, RGMIA Res. Rep. Coll., 11(e), 2008, Article 17. Availiable at
http://rgmia.org/issues.php.
[8] S.S. Dragomir. Some trapezoidal vector inequalities for continuous functions of selfadjoint
operators in Hilbert spaces. Preprint, RGMIA Res. Rep. Coll., 13(2), 2010, Article 14.
Availiable at http://rgmia.org/issues.php.
[9] S.S. Dragomir. Some generalized trapezoidal vector inequalities for continuous functions of
selfadjoint operators in Hilbert spaces. Preprint, RGMIA Res. Rep. Coll., 13(e), 2010,
Article 14. Availiable at http://rgmia.org/issues.php.
[10] T. Furuta, J. Mi´
ci´
c Hot, J. Peˇ
cari´
c, and Y. Seo. Mond-Peˇ
cari´
c Method in Operator Inequalities.
Inequalities for Bounded Selfadjoint Operators on a Hilbert Space. Element, Zagreb, 2005.
[11] Z. Liu. Refinement of an inequality of Gr¨
uss type for Riemann-Stieltjes integral. Soochow J.
Math., 30(4):483–489, 2004.
[12] A. Matkovi´
c, J. Peˇ
cari´
c, and I. Peri´
c. A variant of Jensen’s inequality of Mercer’s type for
operators with applications. Linear Algebra Appl., 418:551–564, 2006.
[13] B. Mond and J. Peˇ
cari´
c. Convex inequalities in Hilbert spaces. Houston J. Math., 19:405–420,
1993.
[14] B. Mond and J. Peˇ
cari´
c. Classical inequalities for matrix functions. Utilitas Math., 46:155–166,
1994.
[15] J. Peˇ
cari´
c, J. Mi´
ci´
c, and Y. Seo. Inequalities between operator means based on the MondPeˇ
cari´
c method. Houston J. Math., 30(1):191–207, 2004.
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VECTOR AND OPERATOR TRAPEZOIDAL TYPE INEQUALITIES
FOR CONTINUOUS FUNCTIONS OF SELFADJOINT OPERATORS
IN HILBERT SPACES∗
SEVER S. DRAGOMIR†
Abstract. Some vector and operator generalized trapezoidal inequalities for continuous functions of selfadjoint operators in Hilbert spaces are given. Applications for power and logarithmic
functions of operators are provided as well.
Key words. Selfadjoint operators, Functions of selfadjoint operators, Spectral representation,
Inequalities for selfadjoint operators.
AMS subject classifications. 47A63, 47A99.
1. Introduction. Let A be a selfadjoint linear operator on a complex Hilbert
space (H; h·, ·i). The Gelfand map establishes an isometric ∗-isomorphism Φ between
the C ∗ -algebra C (Sp (A)) of all continuous functions defined on the spectrum of
A, denoted by Sp (A), and the C ∗ -algebra C ∗ (A) generated by A and the identity
operator 1H on H (see for instance [10, p. 3]) with the property that Φ (f0 ) = 1H
and Φ (f1 ) = A, where f0 (t) = 1 and f1 (t) = t for t ∈ Sp (A).
With this notation, we define
f (A) := Φ (f ) for all f ∈ C (Sp (A))
and we call it the continuous functional calculus for a selfadjoint operator A.
If A is a selfadjoint operator and f is a real valued continuous function on Sp (A),
then f (t) ≥ 0 for any t ∈ Sp (A) implies that f (A) ≥ 0, i.e., f (A) is a positive
operator on H. Moreover, if both f and g are real valued functions on Sp (A), then
the following important property holds:
f (t) ≥ g (t) for any t ∈ Sp (A) implies that f (A) ≥ g (A)
(P)
in the operator order of B (H).
∗ Received
by the editors on November 29, 2010. Accepted for publication on February 17, 2011.
Handling Editor: Moshe Goldberg.
† School of Engineering & Science, Victoria University, PO Box 14481, Melbourne City, MC
8001, Australia ([email protected]), and School of Computational & Applied Mathematics,
University of the Witwatersrand, Private Bag-3, Wits-2050, Johannesburg, South Africa.
161
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S.S. Dragomir
For a recent monograph devoted to various inequalities for continuous functions
of selfadjoint operators, see [10] and the references therein.
For other recent results, see [5]–[7], [12]–[14] and [15].
Let U be a selfadjoint operator on the complex Hilbert space (H, h·, ·i) with the
spectrum Sp (U ) 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 :
hf (U ) x, yi =
Z
M
f (λ) d (hEλ x, yi) ,
(1.1)
m−0
for any x, y ∈ H. The function gx,y (λ) := hEλ x, yi is of bounded variation on the
interval [m, M ], and
gx,y (m − 0) = 0 and gx,y (M ) = hx, yi
for any x, y ∈ H. It is also well known that gx (λ) := hEλ x, xi is monotonic nondecreasing and right continuous on [m, M ].
With the notations introduced above, we have considered in the recent paper [8]
the problem of bounding the error
f (M ) + f (m)
· hx, yi − hf (A) x, yi
2
(m)
·hx, yi, where
in approximating hf (A) x, yi by the trapezoidal type formula f (M )+f
2
x, y are vectors in the Hilbert space H and f is a continuous function of the selfadjoint
operator A with the spectrum in the compact interval of real numbers [m, M ].
We recall here only two such results. The first deals with the case of continuous
functions of bounded variation and is incorporated in the following theorem [8]:
Theorem 1.1. Let A be a selfadjoint operator in the Hilbert space H with the
spectrum Sp (A) ⊆ [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 ],
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Vector and Operator Trapezoidal Type Inequalities
then we have the inequalities
f (M ) + f (m)
· hx, yi − hf (A) x, yi
2
h
1
1/2
1/2
max hEλ x, xi hEλ y, yi
≤
2 λ∈[m,M ]
+ h(1H − Eλ ) x, xi
1/2
(1.2)
1/2
h(1H − Eλ ) y, yi
M
≤
M
i_
(f )
m
_
1
kxk kyk (f )
2
m
for any x, y ∈ H.
The case of Lipschitzian functions is as follows [8]:
Theorem 1.2. Let A be a selfadjoint operator in the Hilbert space H with the
spectrum Sp (A) ⊆ [m, M ] for some real numbers m < M , and let {Eλ }λ be its spectral
family. If f : [m, M ] → C is Lipschitzian with the constant L > 0 on [m, M ], then we
have the inequalities
f (M ) + f (m)
·
hx,
yi
−
hf
(A)
x,
yi
2
Z M h
1
1/2
1/2
hEλ x, xi hEλ y, yi
≤ L
2
m−0
+ h(1H − Eλ ) x, xi
≤
1/2
1
(M − m) L kxk kyk
2
h(1H − Eλ ) y, yi
(1.3)
1/2
i
dλ
for any x, y ∈ H.
In order to provide error bounds in approximating hf (A) x, yi with the quantity
1
[f (m) (M hx, yi − hAx, yi) + f (M ) (hAx, yi − m hx, yi)] ,
M −m
where x, y ∈ H, which is a generalized trapezoid formula, we obtained in the recent
paper [9] the following results:
Theorem 1.3. Let A be a selfadjoint operator in the Hilbert space H with the
spectrum Sp (A) ⊆ [m, M ] for some real numbers m < M , and let {Eλ }λ be its spectral
family.
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1. If f : [m, M ] → C is continuous and of bounded variation on [m, M ], then
f (m) (M 1H − A) + f (M ) (A − m1H )
x, y − hf (A) x, yi
(1.4)
M −m
#M
"
t
M
_
M −t _
t−m _
E(·) x, y +
E(·) x, y
(f )
≤ sup
M −m t
t∈[m,M ] M − m m
m
≤
M
_
E(·) x, y
M
_
m
m
for any x, y ∈ H.
(f ) ≤ kxk kyk
M
_
(f )
m
2. If f : [m, M ] → C is Lipschitzian with the constant L > 0 on [m, M ], then
f (m) (M 1H − A) + f (M ) (A − m1H )
x, y − hf (A) x, yi
(1.5)
M −m
#
Z M "
t
M
M −t _
t−m _
dt
E(·) x, y +
E(·) x, y
≤L
M −m t
m−0 M − m m−0
≤ L (M − m)
M
_
m−0
for any x, y ∈ H.
E(·) x, y
≤ L (M − m) kxk kyk
3. If f : [m, M ] → R is continuous and monotonic nondecreasing on [m, M ], then
f (m) (M 1H − A) + f (M ) (A − m1H )
x,
y
−
hf
(A)
x,
yi
(1.6)
M −m
#
Z M "
t
M
M −t _
t−m _
df (t)
E(·) x, y +
E(·) x, y
≤
M −m t
m−0 M − m m−0
≤
M
_
m
for any x, y ∈ H.
E(·) x, y
[f (M ) − f (m)] ≤ kxk kyk [f (M ) − f (m)]
For other results of this type, see [9]. For scalar trapezoidal inequalities, see
[1]–[4].
The main aim of this paper is to provide more results of the above type as well
as bounds in the operator order for the selfadjoint linear operator
f (m) (M 1H − A) + f (M ) (A − m1H )
,
−
f
(A)
M −m
where f is a continuous function of bounded variation, Lipschitzian or convex on
[m, M ]. Applications for certain functions of large interest such as the power and
logarithmic functions are provided as well.
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2. Some new vector inequalities. The following result for general continuous
functions holds:
Theorem 2.1. Let A be a selfadjoint operator in the Hilbert space H with the
spectrum Sp (A) ⊆ [m, M ] for some real numbers m < M , and let {Eλ }λ be its spectral
family. If f : [m, M ] → R is continuous on [m, M ], then we have the inequalities
f (m) (M 1H − A) + f (M ) (A − m1H )
(2.1)
x,
y
−
hf
(A)
x,
yi
M −m
_
M
≤ max f (t) − min f (t)
E(·) x, y
t∈[m,M ]
≤
t∈[m,M ]
m
max f (t) − min f (t) kxk kyk
t∈[m,M ]
t∈[m,M ]
for any x, y ∈ H.
Proof. We observe that, by the spectral representation (1.1), we have the equality
f (m) (M 1H − A) + f (M ) (A − m1H )
x, y − hf (A) x, yi
(2.2)
M −m
Z M
Φf (t) d (hEt x, yi)
=
m−0
for any x, y ∈ H, where Φf : [m, M ] → R is given by
Φf (t) =
1
[(M − t) f (m) + (t − m) f (M )] − f (t) .
M −m
Since Φf is continuous on [m, M ] and Φf (m) = 0, we can replace in what follows
RM
RM
Φ (t) dv (t) by m Φf (t) dv (t) for every function v : [m, M ] → C of bounded
m−0 f
variation.
It is well known that if p : [a, b] → C is a continuous function and v : [a, b] → C
Rb
is of bounded variation, then the Riemann-Stieltjes integral a p (t) dv (t) exists and
the following inequality holds:
Z
b
b
_
p (t) dv (t) ≤ sup |p (t)| (v) ,
(2.3)
a
t∈[a,b]
a
where
b
_
(v) denotes the total variation of v on [a, b].
a
Now, if we denote by γ := mint∈[m,M ] f (t) and by Γ := maxt∈[m,M ] f (t), then we
have
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γ (M − t) ≤ (M − t) f (m) ≤ Γ (M − t) ,
γ (t − m) ≤ (t − m) f (M ) ≤ Γ (t − m)
and
− (M − m) Γ ≤ − (M − m) f (t) ≤ −γ (M − m)
for any t ∈ [m, M ]. If we add these three inequalities, then we get
− (M − m) (Γ − γ) ≤ (M − m) Φf (t) ≤ (M − m) (Γ − γ)
for any t ∈ [m, M ], which shows that
|Φf (t)| ≤ Γ − γ for any t ∈ [m, M ] .
(2.4)
Applying the inequality (2.3) for the representation (2.2), we have from (2.4) that
Z
M
M
_
Φf (t)d (hEt x, yi) ≤ (Γ − γ)
E(·) x, y
m
m
for any x, y ∈ H, which proves the first part of (2.1).
If P is a nonnegative operator on H, i.e., hP x, xi ≥ 0 for any x ∈ H, then the
following inequality is a generalization of the Schwarz inequality in H:
2
|hP x, yi| ≤ hP x, xi hP y, yi
(2.5)
for any x, y ∈ H.
Now, if d : m = t0 < t1 < · · · < tn−1 < tn = M is an arbitrary partition of the
interval [m, M ], then we have by Schwarz’s inequality for nonnegative operators that
M
_
m
E(·) x, y
(n−1
)
X
Et − Et x, y
= sup
i
i+1
d
≤ sup
d
i=0
(n−1
X h
i=0
Eti+1 − Eti
)
1/2
1/2 i
x, x
Eti+1 − Eti y, y
:= I.
By the Cauchy-Bunyakovsky-Schwarz inequality for sequences of real numbers, we
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Vector and Operator Trapezoidal Type Inequalities
also have that
I ≤ sup
d
≤ sup
d
=
"
n−1
X
i=0
"
n−1
X
i=0
"M
_
m
Eti+1 − Eti
x, x
Eti+1 − Eti
x, x
E(·) x, x
#1/2 " M
_
#1/2 "n−1
X
i=0
#1/2
E(·) y, y
m
sup
d
#1/2
Eti+1 − Eti
"
n−1
X
i=0
#1/2
y, y
Eti+1 − Eti
= kxk kyk
#1/2
y, y
for any x, y ∈ H. These prove the last part of (2.1).
When the generating function is of bounded variation, we have the following result
that complements Theorem 1.3:
Theorem 2.2. Let A be a selfadjoint operator in the Hilbert space H with the
spectrum Sp (A) ⊆ [m, M ] for some real numbers m < M , and let {Eλ }λ be its spectral
family. If f : [m, M ] → C is continuous and of bounded variation on [m, M ], then we
have the inequalities
f (m) (M 1H − A) + f (M ) (A − m1H )
x, y − hf (A) x, yi
M −m
#M
"
t
M
_
t−m _
M −t _
(f ) +
(f )
E(·) x, y
≤ max
M −m t
t∈[m,M ] M − m
m
m
≤
M
_
m
E(·) x, y
M
_
m
(f ) ≤
M
_
m
(2.6)
(f ) kxk kyk
for any x, y ∈ H.
Proof. First of all, observe that
(M − m) Φf (t) = (t − M ) [f (t) − f (m)] + (t − m) [f (M ) − f (t)]
for any t ∈ [m, M ], where here Φf : [m, M ] → C.
(2.7)
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S.S. Dragomir
Therefore,
t−m
M −t
|f (t) − f (m)| +
|f (M ) − f (t)|
M −m
M −m
t
M
M −t _
t−m _
≤
(f ) +
(f )
M −m m
M −m t
#
"_
t
M
_
M −t t−m
,
(f ) +
(f )
≤ max
M −m M −m
m
t
"
# M
_
1 t − m+M
2
(f )
=
+
2
M −m
m
|Φf (t)| ≤
for any t ∈ [m, M ], which implies that
"
#
t
M
M −t _
t−m _
max |Φf (t)| ≤ max
(f ) +
(f )
M −m t
t∈[m,M ]
t∈[m,M ] M − m
m
"
# M
M
_
_
1 t − m+M
2
≤ max
(f ) =
(f ) .
+
M −m
t∈[m,M ] 2
m
m
(2.8)
(2.9)
Applying the inequality (2.3) for the representation (2.2), we have from (2.9) that
Z
"
#M
t
M
M
_
M −t _
t−m _
Φf (t) d (hEt x, yi) ≤ max
(f ) +
(f )
E(·) x, y
t∈[m,M ] M − m m
m
M −m t
m
≤
M
_
m
(f )
M
_
m
E(·) x, y
for any x, y ∈ H, which implies the desired result (2.6).
The case of Lipschitzian functions is as follows:
Theorem 2.3. Let A be a selfadjoint operator in the Hilbert space H with the
spectrum Sp (A) ⊆ [m, M ] for some real numbers m < M , and let {Eλ }λ be its spectral
family. If f : [m, M ] → C is Lipschitzian with the constant L > 0 on [m, M ], then we
have the inequalities
f (m) (M 1H − A) + f (M ) (A − m1H )
x, y − hf (A) x, yi
(2.10)
M −m
M
_
M −t
t−m
|f (t) − f (m)| +
|f (M ) − f (t)|
E(·) x, y
max
≤
M −m
t∈[m,M ] M − m
m
≤
M
_
1
1
E(·) x, y ≤ (M − m) L kxk kyk
(M − m) L
2
2
m
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for any x, y ∈ H.
Proof. We have from the first part of the equality (2.7) that
M −t
t−m
|f (t) − f (m)| +
|f (M ) − f (t)|
M −m
M −m
2L
1
≤
(M − t) (t − m) ≤ (M − m) L
M −m
2
|Φf (t)| ≤
(2.11)
for any t ∈ [m, M ], which, by arguments similar to the arguments of the proof of
Theorem 2.2, yields the desired result (2.10). The details are omitted.
The following lemma may be stated.
Lemma 2.4. Let u : [a, b] → R and ϕ, Φ ∈ R be such that Φ > ϕ. The following
statements are equivalent:
(i) The function u− ϕ+Φ
2 ·e, where e (t) = t, t ∈ [a, b], is
(ii) It holds that
ϕ≤
1
2
(Φ − ϕ) −Lipschitzian;
u (t) − u (s)
≤ Φ for each t, s ∈ [a, b] with t 6= s;
t−s
(2.12)
(iii) It holds that
ϕ (t − s) ≤ u (t) − u (s) ≤ Φ (t − s) for each t, s ∈ [a, b] with t > s. (2.13)
Following [11], we can introduce the concept:
Definition 2.5. A function u : [a, b] → R which satisfies one of the equivalent
conditions (i)–(iii) is said to be (ϕ, Φ) −Lipschitzian on [a, b].
Notice that in [11], the definition was introduced on utilizing the statement (iii)
and only the equivalence (i) ⇔ (iii) was considered.
Utilizing Lagrange’s mean value theorem, we can state the following result that
provides practical examples of (ϕ, Φ) −Lipschitzian functions.
Proposition 2.6. Let u : [a, b] → R be continuous on [a, b] and differentiable on
(a, b). If
−∞ < γ := inf u′ (t) and
t∈(a,b)
sup u′ (t) =: Γ < ∞,
(2.14)
t∈(a,b)
then u is (γ, Γ) −Lipschitzian on [a, b].
The following corollary holds:
Corollary 2.7. Let A be a selfadjoint operator in the Hilbert space H with the
spectrum Sp (A) ⊆ [m, M ] for some real numbers m < M , and let {Eλ }λ be its spectral
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S.S. Dragomir
family. If l, L ∈ R are such that L > l and f : [m, M ] → R is (l, L) −Lipschitzian on
[m, M ], then we have the inequality
f (m) (M 1H − A) + f (M ) (A − m1H )
(2.15)
x, y − hf (A) x, yi
M −m
≤
M
_
1
1
E(·) x, y ≤ (M − m) (L − l) kxk kyk
(M − m) (L − l)
4
4
m
for any x, y ∈ H.
Proof. The proof follows by applying the inequality (2.10) to the 21 (L − l)–
Lipschitzian function f − 12 (l + L) e, where e (t) = t, t ∈ [m, M ]. The details are
omitted.
When the generating function is continuous convex, we can state the following
result as well:
Theorem 2.8. Let A be a selfadjoint operator in the Hilbert space H with the
spectrum Sp (A) ⊆ [m, M ] for some real numbers m < M , and let {Eλ }λ be its
spectral family. If f : [m, M ] → R is continuous convex on [m, M ] with finite lateral
′
′
(m), then we have the inequalities
(M ) and f+
derivatives f−
f (m) (M 1H − A) + f (M ) (A − m1H )
x, y − hf (A) x, yi
M −m
(2.16)
M
′
_
1
′
E(·) x, y
(M − m) f−
(M ) − f+
(m)
4
m
′
1
′
≤ (M − m) f− (M ) − f+ (m) kxk kyk
4
≤
for any x, y ∈ H.
Proof. By the convexity of f on [m, M ], we have
′
f (t) − f (M ) ≥ f−
(M ) (t − M )
for any t ∈ [m, M ]. If we multiply this inequality by t − m ≥ 0, then we deduce
′
(t − m) f (t) − (t − m) f (M ) ≥ f−
(M ) (t − M ) (t − m)
(2.17)
for any t ∈ [m, M ]. Similarly, we get
′
(M − t) f (t) − (M − t) f (m) ≥ f+
(m) (M − t) (t − m)
for any t ∈ [m, M ].
(2.18)
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Summing the above inequalities and dividing by M − m we deduce the inequality
Φf (t) ≤
′
1
(M − t) (t − m) ′
′
′
(M ) − f+
f− (M ) − f+
(m)
(m) ≤ (M − m) f−
M −m
4
for any t ∈ [m, M ]. By the convexity of f , we also have that
1
[(M − t) f (m) + (t − m) f (M )] ≥ f
M −m
(M − t) m + (t − m) M
M −m
= f (t)
(2.19)
giving that
Φf (t) ≥ 0 for any t ∈ [m, M ] .
(2.20)
Utilizing (2.3) for the representation (2.2) we deduce from (2) and (2.20) the
desired result (2.16).
We observe that a similar result holds for concave functions as well.
3. Inequalities in the operator order. Before we state the next results, we
recall that if A is a bounded linear operator, then the operator A√∗ A is selfadjoint and
positive and the “absolute value” operator is defined by |A| := A∗ A.
In what follows, we use the fact that if A is selfadjoint and f ∈ C (Sp (A)), then
|f (A)| = (|f |) (A).
The following result providing some inequalities in the operator order may be
stated:
Theorem 3.1. Let A be a selfadjoint operator in the Hilbert space H with the
spectrum Sp (A) ⊆ [m, M ] for some real numbers m < M .
1. If f : [m, M ] → R is continuous on [m, M ], then
f (m) (M 1H − A) + f (M ) (A − m1H )
− f (A)
M −m
≤ max f (t) − min f (t) 1H .
t∈[m,M ]
t∈[m,M ]
(3.1)
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2. If f : [m, M ] → C is continuous and of bounded variation on [m, M ], then
f (m) (M 1H − A) + f (M ) (A − m1H )
−
f
(A)
(3.2)
M −m
A
M
M 1H − A _
A − m1H _
(f ) +
(f )
M −m m
M −m
A
"
# M
_
1 A − m+M
2 1H
(f ) ,
+
≤
2
M −m
m
≤
where
A
_
(f ) denotes the operator generated by the scalar function
m
[m, M ] ∋ t 7−→
The same notation applies for
M
_
t
_
m
(f ) ∈ R.
(f ).
A
3. If f : [m, M ] → C is Lipschitzian with the constant L > 0 on [m, M ], then
f (m) (M 1H − A) + f (M ) (A − m1H )
− f (A)
(3.3)
M −m
A − m1H
M 1H − A
|f (A) − f (m) 1H | +
|f (M ) 1H − f (A)|
≤
M −m
M −m
1
≤ (M − m) L1H .
2
4. If f : [m, M ] → R is continuous convex on [m, M ] with finite lateral derivatives
′
′
(m), then we have the inequalities
(M ) and f+
f−
f (m) (M 1H − A) + f (M ) (A − m1H )
− f (A)
M −m
(M 1H − A) (A − m1H ) ′
′
≤
f− (M ) − f+
(m)
M −m
′
1
′
(M ) − f+
(m) 1H .
≤ (M − m) f−
4
0≤
(3.4)
Proof. the proof follows by applying the property (P) to the scalar inequalities
(2.4), (2.8), (2.11), (2) and (2.20). The details are omitted.
The following particular case is perhaps more useful for applications:
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Corollary 3.2. Let A be a selfadjoint operator in the Hilbert space H with the
spectrum Sp (A) ⊆ [m, M ] for some real numbers m < M . If l, L ∈ R with L > l and
f : [m, M ] → R is (l, L) −Lipschitzian on [m, M ], then we have
f (m) (M 1H − A) + f (M ) (A − m1H )
1
− f (A) ≤ (M − m) (L − l) 1H . (3.5)
M −m
4
4. More inequalities for differentiable functions. The following result holds:
Theorem 4.1. Let A be a selfadjoint operator in the Hilbert space H with the
spectrum Sp (A) ⊆ [m, M ] for some real numbers m < M . Assume that the function
f : [m, M ] → C is continuously differentiable on [m, M ].
1. If the derivative f ′ is of bounded variation on [m, M ], then we have the inequalities
f (m) (M 1H − A) + f (M ) (A − m1H )
x,
y
−
hf
(A)
x,
yi
(4.1)
M −m
≤
M
M
_
_
1
E(·) x, y
(M − m) (f ′ )
4
m
m
M
≤
_
1
(M − m) (f ′ ) kxk kyk
4
m
for any x, y ∈ H.
2. If the derivative f ′ is Lipschitzian with the constant K > 0 on [m, M ], then
we have the inequalities
f (m) (M 1H − A) + f (M ) (A − m1H )
x,
y
−
hf
(A)
x,
yi
(4.2)
M −m
≤
≤
M
_
1
2
E(·) x, y
(M − m) K
8
m
1
2
(M − m) K kxk kyk
8
for any x, y ∈ H.
Proof. First of all we notice that if f : [m, M ] → C is continuously differentiable
on [m, M ], then we have the following representation in terms of the Riemann-Stieltjes
integral:
Z M
1
Φf (t) =
K (t, s) df ′ (s), t ∈ [m, M ] ,
(4.3)
M −m m
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2
where the kernel K : [m, M ] → R is given by
(
(M − t) (s − m) if m ≤ s ≤ t
K (t, s) :=
(t − m) (M − s) if t < s ≤ M.
(4.4)
Indeed, since f ′ is continuously differentiable on [m, M ], then the Riemann-Stieltjes
Rt
RM
integrals m (s − m) df ′ (s) and t (M − s) df ′ (s) exist for each t ∈ [m, M ]. Now,
integrating by parts in the Riemann-Stieltjes integral yields
Z M
Z t
Z M
(M − s) df ′ (s)
(s − m) df ′ (s) + (t − m)
K(t, s)df ′ (s) = (M − t)
t
m
m
Z t
t
′
′
f (s)ds
= (M − t) (s − m) f (s) m −
m
#
"
Z M
M
′
′
f (s)ds
+ (t − m) (M − s) f (s) +
t
t
= (M − t) [(t − m) f ′ (t) − (f (t) − f (m))]
+ (t − m) [− (M − t) f ′ (t) + f (M ) − f (t)]
= (t − m) [f (M ) − f (t)] − (M − t) [f (t) − f (m)]
= (M − m) Φf (t)
for any t ∈ [m, M ], which provides the desired representation (4.3).
Now, utilizing the representation (4.3) and the property (2.3), we have
|Φf (t)|
=
≤
≤
=
(4.5)
Z M
Z t
1
(M − s) df ′ (s)
(s − m) df ′ (s) + (t − m)
(M − t)
M −m
t
m
Z
#
"
Z t
M
1
(M − s) df ′ (s)
(s − m) df ′ (s) + (t − m)
(M − t)
M −m
t
m
#
"
M
t
_
_
1
′
′
(M − t) (f ) sup (s − m) + (t − m) (f ) sup (M − s)
M −m
s∈[t,M ]
s∈[m,t]
t
m
" t
#
M
_
(t − m) (M − t) _ ′
(f ) + (f ′ )
M −m
m
t
M
=
M
_
(t − m) (M − t) _ ′
1
(f ) ≤ (M − m) (f ′ )
M −m
4
m
m
for any t ∈ [m, M ].
Using the representation (2.2), we deduce the desired result (4.1).
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Vector and Operator Trapezoidal Type Inequalities
175
Further, we utilize the fact that for an L−Lipschitz continuous function, v :
[α, β] → C and a Riemann integrable function p : [α, β] → C, the Riemann-Stieltjes
Rβ
integral α p (s) dv(s) exists and
Z
Z β
β
|p(s)| ds.
p(s)dv(s)
≤
L
α
α
Then, by utilizing (4.5) we have
|Φf (t)|
(4.6)
Z
#
Z t
M
1
≤
(M − s) df ′ (s)
(s − m) df ′ (s) + (t − m)
(M − t)
M −m
t
m
#
"
Z M
Z t
K
(M − s) ds
(s − m) ds + (t − m)
(M − t)
≤
M −m
t
m
"
#
2
2
(M − t) (t − m)
(t − m) (M − t)
K
+
=
M −m
2
2
=
"
1
1
2
(M − m) (t − m) (M − t) K ≤ (M − m) K
2
8
for any t ∈ [m, M ].
Using the representation (2.2), we deduce the desired result (4.2).
The following inequalities in the operator order are of interest as well:
Theorem 4.2. Let A be a selfadjoint operator in the Hilbert space H with the
spectrum Sp (A) ⊆ [m, M ] for some real numbers m < M . Assume that the function
f : I → C with [m, M ] ⊂ ˚
I (the interior of I) is differentiable on ˚
I.
1. If the derivative f ′ is continuous and of bounded variation on [m, M ], then we
have the inequalities
f (m) (M 1H − A) + f (M ) (A − m1H )
− f (A)
M −m
(4.7)
M
(A − m1H ) (M 1H − A) _ ′
(f )
≤
M −m
m
M
≤
_
1
(M − m) (f ′ ) 1H .
4
m
2. If the derivative f ′ is Lipschitzian with the constant K > 0 on [m, M ], then
Electronic Journal of Linear Algebra ISSN 1081-3810
A publication of the International Linear Algebra Society
Volume 22, pp. 161-178, March 2011
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176
S.S. Dragomir
we have the inequalities
f (m) (M 1H − A) + f (M ) (A − m1H )
−
f
(A)
M −m
1
≤ (M − m) (A − m1H ) (M 1H − A) K
2
1
2
≤ (M − m) K1H .
8
(4.8)
Proof. The proof follows by the property (P) applied for the scalar inequalities
(4.5) and (4.6).
5. Applications for particular functions. It is obvious that the above results
can be applied for various particular functions. However, we will restrict here only to
the power and logarithmic functions.
1. Consider now the power function f : (0, ∞) → R, f (t) = tp with p 6= 0.
Applying Theorem 2.8, we can state the following proposition:
Proposition 5.1. Let A be a selfadjoint operator in the Hilbert space H with
the spectrum Sp (A) ⊆ [m, M ] for some real numbers 0 < m < M . Then for any
x, y ∈ H, we have the inequality
p
m (M 1H − A) + M p (A − m1H )
p
x, y − hA x, yi
(5.1)
M −m
1
≤ (M − m) ∆p kxk kyk
4
where
∆p = p ×
In particular,
p−1
− mp−1
M
mp−1 − M p−1
if p ∈ (−∞, 0) ∪ [1, ∞)
if 0 < p < 1.
M (M 1H − A) + m (A − m1H )
−1
x, y − A x, y
mM (M − m)
(5.2)
2
≤
1 (M − m) (M + m)
kxk kyk
4
m2 M 2
for any x, y ∈ H.
The following inequalities in the operator order also hold:
Proposition 5.2. Let A be a selfadjoint operator in the Hilbert space H with
the spectrum Sp (A) ⊆ [m, M ] for some real numbers 0 < m < M .
Electronic Journal of Linear Algebra ISSN 1081-3810
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Vector and Operator Trapezoidal Type Inequalities
177
If p ∈ (−∞, 0) ∪ [1, ∞), then
mp (M 1H − A) + M p (A − m1H )
− Ap
M −m
(M 1H − A) (A − m1H )
≤p
M p−1 − mp−1
M −m
1
≤ p (M − m) M p−1 − mp−1 1H .
4
0≤
(5.3)
If p ∈ (0, 1), then
mp (M 1H − A) + M p (A − m1H )
M −m
(M 1H − A) (A − m1H )
≤p
mp−1 − M p−1
M −m
1
≤ p (M − m) mp−1 − M p−1 1H .
4
0 ≤ Ap −
(5.4)
In particular, we have the inequalities
M (M 1H − A) + m (A − m1H )
− A−1
mM (M − m)
(M 1H − A) (A − m1H ) M 2 − m2
≤
·
M −m
m2 M 2
2
1 (M − m) (M + m)
1H .
≤
4
m2 M 2
0≤
(5.5)
Proof. The proof follows from (3.4) and the details are omitted.
2. The case of logarithmic function is as follows:
Proposition 5.3. Let A be a selfadjoint operator in the Hilbert space H with
the spectrum Sp (A) ⊆ [m, M ] for some real numbers 0 < m < M . Then for any
x, y ∈ H, we have the inequality
(M 1H − A) ln m + (A − m1H ) ln M
x, y − hln Ax, yi
(5.6)
M −m
2
1 (M − m)
kxk kyk .
4 mM
We also have the following inequality in the operator order
≤
(M 1H − A) ln m + (A − m1H ) ln M
M −m
2
1 (M − m)
(M 1H − A) (A − m1H )
≤
≤
1H .
Mm
4 mM
0 ≤ ln A −
(5.7)
Electronic Journal of Linear Algebra ISSN 1081-3810
A publication of the International Linear Algebra Society
Volume 22, pp. 161-178, March 2011
ELA
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178
S.S. Dragomir
Remark 5.4. Similar results can be obtained if ones uses the inequalities from
Theorem 4.1 and 4.2. However the details are left to the interested reader.
Acknowledgment. The author would like to thank the anonymous referee for
valuable suggestions that have been implemented in the final version of this paper.
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