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Abstract and Applied Analysis
Volume 2012, Article ID 204031, 11 pages
doi:10.1155/2012/204031
Research Article
More on α, β-Normal Operators in Hilbert Spaces
Rasoul Eskandari, Farzollah Mirzapour, and Ali Morassaei
Department of Mathematics, Faculty of Sciences, University of Zanjan, Zanjan 45195-313, Iran
Correspondence should be addressed to Ali Morassaei, morassaei@znu.ac.ir
Received 18 November 2011; Revised 3 July 2012; Accepted 20 July 2012
Academic Editor: Gerd Teschke
Copyright q 2012 Rasoul Eskandari et al. 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.
We study some properties of α, β-normal operators and we present various inequalities between
the operator norm and the numerical radius of α, β-normal operators on Banach algebra BH of
all bounded linear operators T : H → H, where H is Hilbert space.
1. Introduction
Throughout the paper, let BH denote the algebra of all bounded linear operators acting on
a complex Hilbert space H, ·, ·, Bh H denote the algebra of all self-adjoint operators in
BH, and I is the identity operator. In case of dim H n, we identify BH with the full
matrix algebra Mn C of all n × n matrices with entries in the complex field. An operator
A ∈ Bh H is called positive if Ax, x ≥ 0 is valid for any x ∈ H, and then we write A ≥ 0.
Moreover, by A > 0 we mean Ax, x > 0 for any x ∈ H. For A, B ∈ Bh H, we say A ≤ B if
B−A ≥ 0. An operator A is majorized by B, if there exists a constant λ such that Ax ≤ λBx
for all x ∈ H or equivalently A∗ A ≤ λ2 B∗ B 1.
For real numbers α and β with 0 ≤ α ≤ 1 ≤ β, an operator T acting on a Hilbert space
H is called α, β-normal 2, 3 if
α2 T ∗ T ≤ T T ∗ ≤ β2 T ∗ T.
1.1
An immediate consequence of above definition is
α2 T ∗ T x, x ≤ T T ∗ x, x ≤ β2 T ∗ T x, x,
1.2
2
Abstract and Applied Analysis
from which we obtain
αT x ≤ T ∗ x ≤ βT x,
1.3
for all x ∈ H.
Notice that, according to 1.1, if T is α, β-normal operator, then T and T ∗ majorize
each other.
In 3, Moslehian posed two problems about α, β-normal operators as follows.
For fixed α > 0 and β / 1,
i give an example of an α, β-normal operator which is neither normal nor
hyponormal;
ii is there any nice relation between norm, numerical radius, and spectral radius of an
α, β-normal operator?
Dragomir and Moslehian answered these problems in 2, as more as, they
propounded a nice example of α, β-normal operator that is neither normal nor hyponormal,
as follows.
√
3 − 5/2 and β The matrix 11 01 in BC2 is an α, β-normal with α √
3 5/2.
The numerical radius wT of an operator T on H is defined by
wT sup{|T x, x| : x 1}.
1.4
Obviously, by 1.4, for any x ∈ H we have
|T x, x| ≤ wT x2 .
1.5
It is well known that w· is a norm on the Banach algebra BH of all bounded linear
operators. Moreover, we have
wT ≤ T ≤ 2wT T ∈ BH.
1.6
For other results and historical comments on the numerical radius see 4.
The antieigenvalue of an operator T ∈ BH defined by
μ1 T : inf
Tx /
0
ReT x, x
.
T xx
1.7
The vector x ∈ H which takes μ1 T is called an antieigenvector of T . We refer more study on
this matter to 4.
In this paper, we prove some properties of α, β-normal operators and state various
inequalities between the operator norm and the numerical radius of α, β-normal operators
in Hilbert spaces.
Abstract and Applied Analysis
3
2. Some Properties of α, β-Normal Operators
In this section, we establish some properties of α, β-normal operators. It is easy to see that if
T is an α, β-normal α > 0 then T ∗ is 1/β, 1/α-normal. We find numbers z ∈ C such that
z T is α, β-normal where T is α, β-normal.
We know by the Cauchy-Schwartz inequality that −1 ≤ μ1 T ≤ 1. Also we can write
ReT x, x
.
T x
2.1
ReT x, x
.
T x
x1
2.2
μ1 T inf
x1
Tx /
0
We define
μ2 T : sup
Tx /
0
We know that if T is normal operator then z T is also normal.
Theorem 2.1. Let T be an α, β-normal operator on a Hilbert space such that 0 ≤ α < 1 < β and
z ∈ C. Then z T is α, β-normal, if provided one of the following conditions holds:
i μ1 zT ≥ 0,
ii μ1 zT < 0, |z|2 ≥ −2|z|T μ1 zT .
Proof. In both of above cases, we show that
|z|2 2 RezT x, x ≥ 0,
∀x ∈ Hwith x 1, T x /
0.
2.3
By the assumption i, μ1 zT ≥ 0, we have RezT x, x/|z|T x ≥ 0 for every x ∈ H with
x 1 and T x /
0, consequently we get RezT x, x ≥ 0, and therefore 2.3 is valid. On the
other hand, if ii holds and we set B : μ1 zT then we get B ≤ RezT x, x/|z|T x for every
x ∈ H with x 1 and T x /
0, consequently:
RezT x, x
inf{BT x : x 1, T x /
0} ≤ inf T x
: x 1, T x /
0 .
|z|T x
2.4
Since B < 0, we obtain
RezT x, x
−B inf{−T x : x 1, T x /
0} ≤ inf T x
: x 1, T x /
0 ,
|z|T x
2.5
RezT x, x
B sup{T x : x 1, T x /
0} ≤ inf T x
: x 1, T x /
0 .
|z|T x
2.6
and so
4
Abstract and Applied Analysis
Now, by using the last inequality, we have
⎛
⎞⎛
⎞
⎜
⎟⎜
⎟ inf RezT x, x ⎟
|z|2 2|z|T μ1 zT |z|2 2|z|⎜
⎠
⎝ sup T x⎠⎝x1
|z|T x
x1
Tx /0
2
≤ |z| 2|z| inf
x1
Tx /0
RezT x, x
T x
|z|T x
2.7
|z|2 2 inf {RezT x, x}.
x1
This shows that 2.3 holds for ii, too. Thus, for any x ∈ H with x 1 we have
α2 z T ∗ z T x, x α2 |z|2 x, x zT x, x zT ∗ x, x α2 T ∗ T x, x
≤ |z|2 x, x zT x, x zT ∗ x, x T T ∗ x, x
z T z T ∗ x, x
≤ β2 |z|2 x, x zT x, x zT ∗ x, x β2 T ∗ T x, x
2.8
β2 z T ∗ z T x, x
and this completes the proof.
Corollary 2.2. Let T be an α, β-normal operator. We have the following.
i If μ1 T ≥ 0 then z T is α, β-normal operator for any z > 0.
ii If μ2 T ≤ 0 then z T is α, β-normal operator for any z < 0.
Proof. i By the definition of the first antieigenvalue of T , for all z > 0 we have
μ1 zT μ1 zT μ1 T ≥ 0.
2.9
By using Theorem 2.1i we imply that z T is an α, β-normal.
ii If z < 0, then
μ1 zT −μ2 T ≥ 0.
By using Theorem 2.1i we imply that z T is an α, β-normal.
Corollary 2.3. Let T be an injective and α, β-normal operator with α > 0. Then
i RT is dense,
ii T ∗ is injective,
iii if T is surjective then T −1 is also α, β-normal.
2.10
Abstract and Applied Analysis
5
Proof. Since the inequality 1.3 is valid, we obtain NT ∗ NT , and therefore RT ⊥ NT ∗ NT 0, thus RT is a dense subspace of H and T ∗ is injective. This proves i
and ii.
To prove iii, we note that since T is surjective, we imply that T is invertible. On
the other hand we have T ∗ −1 T −1 ∗ . Also we know that if A and B are two positive and
invertible operators with 0 < A ≤ B then B−1 ≤ A−1 . Since T is α, β-normal, by taking inverse
from all sides of 1.1, we get
1 −1 ∗ −1
1
T T ≤ T ∗ −1 T −1 ≤ 2 T −1 T ∗ −1 .
β2
α
2.11
∗
This means that T −1 is 1/β, 1/α-normal, thus T −1 is α, β-normal.
Example 2.4. Consider the following matrix T in BC2 :
T
1 0
.
1 1
T is an α, β-normal operator, with parameters α 1 0
T −1 −1
1 is α, β-normal.
For T ∈ BH we call
2.12
3 −
√
√
5/2 and β 3 5/2. Then
rT sup{|λ| : λ ∈ σT }
2.13
the spectral radius of T , where σT is the spectrum of T and it is known that rT limn → ∞ T n 1/n 5, page 102.
n
Theorem 2.5. Let T be an α, β-normal operator such that T 2 is α, β-normal operator for every
n ∈ N, too. Then, we have
1
T ≤ rT ≤ T .
β
2.14
T ∗ T T 2 .
2.15
Proof. For any T ∈ BH we have
In particular, if T is a self-adjoint operator then T 2 T 2 . Thus, by the definition of α, βnormal operator, we have
1
1
∗2 2
2
T T ≥ 2 T ∗ T 2 T 4 .
β
β
2.16
6
Abstract and Applied Analysis
By induction on n, we imply that
n n
∗2 2 T T ≥
1
n1
β2n1 −2
T 2 ,
2.17
from which we obtain
n n 1/2n
rT 2 rT ∗ rT lim T ∗ 2 T 2 n→∞
n n 1/2n
≥ lim T ∗ 2 T 2 n→∞
≥ lim
n→∞
1
β
1/2n
2n1
2.18
T 2n1 −2
1
1
1
T 2 lim −2/2n 2 T 2 .
2
n
→
∞
β
β
β
Therefore, we get 1/βT ≤ rT ≤ T . This completes the proof.
Below, we give an example of α, β-normal operator such that it satisfies in
Theorem 2.5.
Example 2.6. Assume that H is a separable Hilbert space and {en : n ∈ Z} is an orthonormal
basis for H. We define the operator T ∈ BH as follows:
⎧
e ,
⎪
⎪
⎨ n−1
1
T en en−1 ,
⎪
2
⎪
⎩
2en−1 ,
n ≡ 0 mod 3,
n ≡ 1 mod 3,
2.19
n ≡ 2 mod 3,
so
⎧1
⎪
⎪
⎨ 2 en1 ,
∗
T en 2en1 ,
⎪
⎪
⎩
en1 ,
n ≡ 0 mod 3,
n ≡ 1 mod 3,
n ≡ 2 mod 3,
2.20
and by simple computation we get
⎧1
⎪
⎪
⎨ 4 en , n ≡ 0 mod 3,
∗
T T en 4en , n ≡ 1 mod 3,
⎪
⎪
⎩
en ,
n ≡ 2 mod 3,
⎧
n ≡ 0 mod 3,
e ,
⎪
⎪
⎨ n
1
T ∗ T en en , n ≡ 1 mod 3,
⎪
⎪
⎩4
4en , n ≡ 2 mod 3.
2.21
Consequently, T is 1/4, 4-normal operator and also T n is 1/4, 4-normal operator, for any
integer n ≥ 0. Thus we have T 2 and rT 1, hence 2.14 is valid.
Abstract and Applied Analysis
7
3. Inequalities Involving Norms and Numerical Radius
In this section we state some inequalities involving norms and numerical radius.
Theorem 3.1. Let T ∈ BH be an α, β-normal operator.
i For positive real numbers p and q with p ≥ 2 and 1/p 1/q 1 we have
T T ∗ p T − T ∗ p ≥ 21 αq p−1 T p .
3.1
ii If 0 ≤ p ≤ 1 or p ≥ 2, then we have
T T ∗ 2 T − T ∗ 2
p
≥ T 2p ϕ α, p ,
3.2
where ϕα, p 2p 1 αp 2 2p − 22 αp .
iii If NT 0 and for any x ∈ H with x 1 we have
Tx
T ∗x −
T ∗ x T x ≤ ρ,
3.3
ρ2
αT 2 ≤ ω T 2 βT 2 .
2
3.4
then, we obtain
Proof. i We use the following known inequality:
p−1
,
a bp a − bp ≥ 2 aq bq
3.5
which is valid for any a, b ∈ H where H is a Hilbert space.
Now, if we take a T x and b T ∗ x in 3.5, then for any x ∈ H we get
p−1
T x T ∗ xp T x − T ∗ xp ≥ 2 T xq T ∗ xq
p−1
≥ 2 T xq αq T xq
21 αq p−1 T xqp−1
3.6
21 αq p−1 T xp .
Taking the supremum in 3.6 over x ∈ H with x 1, we get the desired result 3.1.
ii We use the following inequality 6, Theorem 8, page 551:
a b2 a − b2
p
≥ 2p
ap bp
2
2p − 22 ap bp ,
where a and b are two vectors in a Hilbert space and 0 ≤ p ≤ 1 or p ≥ 2.
3.7
8
Abstract and Applied Analysis
Now, if we put a T x and b T ∗ x in 3.7, then we obtain
T x T ∗ x2 T x − T ∗ x2
≥ 2p
T xp T ∗ xp
p
2
2p − 22 T xp T ∗ xp ,
≥ 2p T x2p 1 αp 2 2p − 22 αp T x2p
3.8
2p T x2p 1 αp 2 2p − 22 αp
T x2p ϕ α, p .
Now, taking the supremum over x 1 in 3.8, we get the desired result 3.2.
iii We use the following reverse of Schwarz’s inequality:
0 ≤ab − |a, b| ≤ ab − Rea, b ≤
1 2
ρ ab,
2
3.9
which is valid for a, b ∈ H \ {0} and ρ > 0, with a/b − b/a ≤ ρ see 7. We take
a T x and b T ∗ x in 3.9 to get
1
T xT ∗ x ≤ |T x, T ∗ x| ρ2 T xT ∗ x.
2
3.10
1
αT x2 ≤ |T x, T ∗ x| ρ2 βT x2 .
2
3.11
Thus, we obtain
Now, taking the supremum over x 1 in recent inequality, we get the desired result 3.4.
Theorem 3.2. Assume that T is an α, β-normal operator. Then, we have
1
1 α2 T 2 ≤ T − T ∗ 2 ω T 2 .
2
3.12
Proof. By 2, Theorem 3.1, we have
21 αp T p ≤
1
T T ∗ p T − T ∗ p ,
2
3.13
and also
∗
T T T T ∗ p/2 1 ≤ T T ∗ p T − T ∗ p .
2
4
3.14
Abstract and Applied Analysis
9
On the other hand, it is known 8 that for A, B ∈ BH we have
A B 2 1
2 ≤2
∗
!
A A B∗ B ωB∗ A .
2
3.15
∗
!
T T TT∗ ω T2 .
2
3.16
By using this inequality we get
T T ∗ 2 1
2 ≤2
If we put p 2 in 3.14, we obtain
!
T T ∗ 2 1 1 ∗ 2
∗ 2
≤
ω
T2
−
T
T
T
T
2 2 4
"
#
2 T − T ∗ 2
T T∗ 1 2
.
ω T
2 2 2 3.17
∗
∗
2
2 ω T 2
1
T T ≤ 1T − T .
2 2 2 2 2
3.18
Thus we get
Now, we take p 2 in 3.13 to obtain
1
T − T ∗ 2 T − T ∗ 2
2
∗ 2
2
1 α2 T 2 ≤ 2 2 ω T 2 T − T ω T .
3.19
This completes the proof.
Theorem 3.3. Assume that T is an α, β-normal operator. Then for any real s with 0 ≤ s ≤ 1, we
have
1 − s
1
s
β2
1 − s s
2
1
4
2
2
∗ 2
≤
1
−
s
sβ
w
T2 .
−
T
T
T
T
β2
3.20
Proof. By 9, Theorem 2.6 see also 10, Theorem 2.4, we have
1 − sa2 sb2 1 − sb2 sa2 − |a, b|2
≤ 1 − sa2 sb2 1 − sb − ta2 stb − a2 ,
3.21
10
Abstract and Applied Analysis
where 0 ≤ s ≤ 1, t ∈ R and a, b ∈ H. By taking t 1, a T x, and b T ∗ x in 3.21, we get
1 − sT x2 sT ∗ x2 1 − sT ∗ x2 sT x2 − |T x, T ∗ x|2
≤ 1 − sT x2 sT ∗ x2 1 − sT ∗ x − T x2 sT ∗ x − T x2 ,
3.22
thus, we have
!
! $
$2
s ∗ 2
1 − s ∗ 2
$
$ 2
2
2
∗
∗
T
x
sT
x
x
x
x,
x
−
sT
−
1
T
T
$
$
β2
β2
$
$2
$
$
≤ 1 − sT x2 sT ∗ x2 1 − sT ∗ x2 sT x2 − $ T 2 x, x $
2
∗
2
≤ 1 − sT x sT x
∗
2
∗
2
1 − sT x − T x sT x − T x
3.23
≤ 1 − sT x2 sβ2 T x2 T ∗ x − T x2 .
Finally, we take supremum over x 1 from both sides of
s
1 − s
s
−
s
1
T ∗ x4
β2
β2
$
$2
$
$
≤ 1 − sT x2 sβ2 T x2 T ∗ x − T x2 $ T 2 x, x $ ,
3.24
and we use triangle inequality for supremums to complete the proof.
Corollary 3.4. Let T be an α, β-normal operator. Then, we have
1
T 2 ≤ T T − T ∗ ω T 2 .
β
3.25
Proof. By using the inequality 3.21 we get
1 − s sα2
2
1 − sα2 s T 4 ≤ 1 − s sα2 T 2 T − T ∗ 2 w T 2 .
3.26
We take s 0 in inequalities 3.20 and 3.26 to imply
max
2
1 2
4
2
∗ 2
,
α
≤
x
ω
T2 .
x
−
T
T
T
T
β2
3.27
Thus, max{1/β, α}T x2 ≤ T xT x − T ∗ x ωT 2 . Now, taking supremum overall x with
x 1, the desired inequality is obtained.
Abstract and Applied Analysis
11
Acknowledgments
The authors would like to sincerely thank the anonymous referee for several useful comments
improving the paper and also Professor Mehdi Hassani for a useful discussion.
References
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2 S. S. Dragomir and M. S. Moslehian, “Some inequalities for α, β-normal operators in Hilbert spaces,”
Facta Universitatis, vol. 23, pp. 39–47, 2008.
3 M. S. Moslehian, “On α, β-normal operators in Hilbert spaces, IMAGE 39,” Problem, pp. 39–34, 2007.
4 K. E. Gustafson and D. K. M. Rao, Numerical Range, Springer, New York, NY, USA, 1997.
5 J. Weidmann, Linear Operators in Hilbert Spaces, vol. 68, Springer, New York, NY, USA, 1980.
6 D. S. Mitrinović, J. E. Pečarić, and A. M. Fink, Classical and New Inequalities in Analysis, vol. 61, Kluwer
Academic, Dordrecht, The Netherlands, 1993.
7 S. S. Dragomir, “A potpourri of Schwarz related inequalities in inner product spaces. I,” Journal of
Inequalities in Pure and Applied Mathematics, vol. 6, no. 3, article 59, 2005.
8 S. S. Dragomir, “A survey of some recent inequalities for the norm and numerical radius of operators
in Hilbert spaces,” Banach Journal of Mathematical Analysis, vol. 1, no. 2, pp. 154–175, 2007.
9 S. S. Dragomir, “A potpourri of Schwarz related inequalities in inner product spaces. II,” Journal of
Inequalities in Pure and Applied Mathematics, vol. 7, no. 1, article 14, 2006.
10 S. S. Dragomir, “Some inequalities for normal operators in Hilbert spaces,” Acta Mathematica
Vietnamica, vol. 31, no. 3, pp. 291–300, 2006.
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