Some inequalities of Furuta’s type for functions of operators defined by power
series
Sever S. Dragomir
Mathematics, College of Engineering & Science Victoria University, PO Box 14428 Melbourne City, MC 8001, Australia School of Computational & Applied Mathematics,
University of the Witwatersrand, Private Bag 3, Johannesburg 2050, South Africa
email:[email protected]
Abstract. Generalizations of Kato and Furuta inequalities for power series of bounded linear operators in Hilbert spaces are given. Appli- cations for normal operators and some functions of interest such as the exponential, hyperbolic and trigonometric functions are provided as well.
1 Introduction
In the following we denote by B(H) theBanach algebra of all bounded linear operators on a complex Hilbert space(H;h·,·i).
If P is a positive selfadjoint operator on H, i.e. hPx, xi ≥ 0 for any x ∈H, then the following inequality is a generalization of the Schwarz inequality in H
|hPx, yi|2≤ hPx, xi hPy, yi, (1) for any x, y∈H.
The following inequality concerning the norm of a positive operator is of interest as well, see [13, p. 221].
2010 Mathematics Subject Classification:47A63, 47A99
Key words and phrases:Bounded linear operators, operator inequalities, Kato’s inequal- ity, functions of normal operators, Euclidian norm and numerical radius
162
10.1515/ausm-2015-0004
Let P be a positive selfadjoint operator on H. Then
kPxk2≤ kPk hPx, xi (2) for any x∈H.
The “square root” of a positive selfadjoint operator onH can be defined as follows, see for instance [13, p. 240]: If the operator A∈ B(H) is selfadjoint and positive, then there exists a unique positive selfadjoint operator B:=√
A∈ B(H) such that B2=A. If Ais invertible, then so is B.
If A∈ B(H),then the operatorA∗Ais selfadjoint and positive. Define the
“absolute value” operator by|A|:=√ A∗A.
In 1952, Kato [14] proved the following celebrated generalization of Schwarz inequality for any bounded linear operatorT onH:
|hTx, yi|2 ≤D
|T|2αx, x
E D|T∗|2(1−α)y, y E
(K) for any x, y∈Hand α∈[0, 1].
In order to generalize this result, in 1994 Furuta [12] obtained the following result:
D
T|T|α+β−1x, yE
2 ≤D
|T|2αx, xE D
|T∗|2βy, yE
(F) for any x, y∈Hand α, β∈[0, 1]withα+β≥1.
If one analyses the proof from [12], that one realizes that the conditionα, β
∈[0, 1]is taken only to fit with the result from theHeinz-Kato inequality
|hTx, yi|≤ kAαxk B1−αy
(HK)
for anyx, y∈Handα∈[0, 1]whereAand Bare positive operators such that kTxk ≤ kAxk and kT∗yk ≤ kByk for all x, y∈H.
Therefore, one can state the more general result:
Theorem 1 (Furuta Inequality, 1994, [12]) Let T ∈ B(H) and α, β ≥ 0 with α+β≥1.Then for any x, y∈H we have the inequality (F).
If we take β=α, then we get
D
T|T|2α−1x, y E
2≤D
|T|2αx, x
E D|T∗|2αy, y E
(3) for any x, y∈Hand α≥ 12.In particular, forα=1we get
|hT|T|x, yi|2 ≤D
|T|2x, x
E D|T∗|2y, y E
(4)
for any x, y∈H.
If we take T = Na normal operator, i.e., we recall that NN∗ =N∗N, then we get from (F) the following inequality for normal operators
D
N|N|α+β−1x, y E
2≤D
|N|2αx, x
E D|N|2βy, y E
(5) for any x, y∈Hand α, β≥0 withα+β≥1.
This implies the inequalities
D
N|N|2α−1x, yE
2
≤D
|N|2αx, xE D
|N|2αy, yE
(6) for any x, y∈Hand α≥ 12 and, in particular,
|hN|N|x, yi|2≤D
|N|2x, xE D
|N|2y, yE
(7) for any x, y∈H.
Makingy=xin (6) produces
D
N|N|2α−1x, xE ≤D
|N|2αx, xE
for any x∈Hand α≥ 12 and, in particular,
|hN|N|x, xi|≤D
|N|2x, x E
for any x∈H.
If we take β=1−α withα∈[0, 1]in (5), then we get
|hNx, yi|2≤D
|N|2αx, x
E D|N|2(1−α)y, y E
(8) for any x, y∈H.
We can state the following corollary of Furuta’s inequality for the numerical radius w of an operator V ∈ B(H), namely w(V) = supkxk=1|hVx, xi|, which satisfies the following basic inequalities
1
2kVk ≤w(V)≤ kVk.
Corollary 1 Let T ∈B(H) and α, β ≥0 with α+β≥1.Then we have w
T|T|α+β−1
≤ 1 2
|T|2α+|T∗|2β
. (9)
In particular, we also have w
T|T|2α−1
≤ 1 2
|T|2α+|T∗|2α
, (10) for any α≥ 12 and, as a special case,
w(T|T|)≤ 1 2
|T|2+|T∗|2
. (11) Proof.We have from (F) for any x∈H that
D
T|T|α+β−1x, x E
≤ D
|T|2αx, x E1/2D
|T∗|2βx, x E1/2
(12)
≤ 1 2
Dh|T|2α+|T∗|2βi x, x
E
whereα, β ≥0 withα+β≥1.
Utilising the inequality in (12) and taking the supremum overx∈H,kxk=1 we get
w
T|T|α+β−1
= sup
kxk=1
D
T|T|α+β−1x, x E
≤ 1 2 sup
kxk=1
Dh|T|2α+|T∗|2βi x, x
E
= 1 2
|T|2α+|T∗|2β .
For various interesting generalizations, extension of Kato and Furuta in- equalities, see the papers [3]-[12], [17]-[21] and [23].
Motivated by the above results, we establish in this paper some generaliza- tions of Kato and Furuta inequalities for functions of operators that can be expresses as power series with real coefficients. Applications for some functions of interest such as the exponential, hyperbolic and trigonometric functions are provided as well.
2 Functional inequalities
Now, by the help of power seriesf(z) =P∞
n=0anznwe can naturally construct another power series which will have as coefficients the absolute values of the coefficient of the original series, namely, fA(z) := P∞
n=0|an|zn. It is obvious that this new power series will have the same radius of convergence as the original series. We also notice that if all coefficients an≥0,then fA=f.
Theorem 2 Let f(z) = P∞
n=0anzn and be g(z) = P∞
n=0bnzn be two func- tions defined by power series with real coefficients and both of them convergent on the open disk D(0, R)⊂C, R > 0. If T is a bounded linear operator on the Hilbert space Hand z, u∈C with the property that
|z|2,|u|2,kTk2 < R, (13) then we have the inequality
|hTf(z|T|)g(u|T|)x, yi|2 (14)
≤fA |z|2
gA
|u|2 D fA
|T|2 x, x
E D|T∗|2gA
|T∗|2 y, y
E
for any x, y∈H.
Proof. From Furuta’s inequality (F) we have for any natural numbers n≥0 and m≥1the following power inequality
D
T|T|n+m−1x, y E
≤D
|T|2nx, x E1/2D
|T∗|2my, y E1/2
, (15)
wherex, y∈H.
If we multiply this inequality with the positive quantities|an| |z|nand|bm−1|
|u|m−1,use the triangle inequality and the Cauchy-Bunyakowsky-Schwarz dis- crete inequality we have successively:
Xk
n=0
Xl
m=1
anznbm−1um−1 D
T|T|n+m−1x, y E
(16)
≤ Xk
n=0
Xl
m=1
|an| |z|n|bm−1| |u|m−1 D
T|T|n+m−1x, y E
≤ Xk
n=0
|an| |z|nD
|T|2nx, x
E1/2Xl m=1
|bm−1| |u|m−1D
|T∗|2my, y E1/2
≤ Xk
n=0
|an| |z|2n
!1/2* k X
n=0
|an| |T|2nx, x +1/2
× Xl m=1
|bm−1| |u|2(m−1)
!1/2*Xl
m=1
|bm−1| |T∗|2my, y +1/2
for any x, y∈Hand k≥0, l≥1.
Observe also that Xk n=0
Xl m=1
anznbm−1um−1D
T|T|n+m−1x, yE
(17)
=
* T
Xk n=0
anzn|T|n
! Xl
m=1
bm−1um−1|T|m−1
! x, y
+
for any x, y∈Hand k≥0, l≥1.
Making use of (16) and (17) we get
* T
Xk n=0
anzn|T|n
! Xl
m=1
bm−1um−1|T|m−1
! x, y
+
(18)
≤ Xk n=0
|an| |z|2n
!1/2*Xk
n=0
|an| |T|2nx, x +1/2
× Xl m=1
|bm−1| |u|2(m−1)
!1/2*
|T∗|2 Xl m=1
|bm−1| |T∗|2(m−1)y, y +1/2
for any x, y∈Hand k≥0, l≥1.
Due to the assumption (13) in the theorem, we have that the seriesP∞
n=0anzn
|T|n,P∞
m=0bmum|T|m,P∞
n=0|an| |T|2nandP∞
m=0|bm| |T∗|2mare convergent in B(H) and the series P∞
n=0|an| |z|2n and P∞
m=0|bm| |u|2m are convergent in R and then, by taking the limit over k→ ∞ and l→ ∞ in (18), we deduce the
desired result (14).
Remark 1 The above inequality (14) can provide various particular instances of interest.
For instance, if we take g=f in Theorem 2 then we get
D
Tf2(z|T|)x, y E
(19)
≤fA
|z|2 D fA
|T|2 x, x
E1/2D
|T∗|2fA
|T∗|2 y, y
E1/2
for any x, y∈H.
Also if we take g(z) =1 in (14), then we get
|hTf(z|T|)x, yi|2 ≤fA
|z|2 D fA
|T|2 x, x
E D|T∗|2y, y E
(20) for any x, y∈H.
Corollary 2 With the assumptions of Theorem2we have the norm inequality kTf(z|T|)g(u|T|)k2 (21)
≤fA |z|2
gA
|u|2 fA
|T|2
|T∗|2gA
|T∗|2
and the numerical radius inequality
w(Tf(z|T|)g(u|T|)) (22)
≤ 1 2
h fA
|z|2 gA
|u|2i1/2 fA
|T|2
+|T∗|2gA
|T∗|2 .
Proof. The inequality (21) follows from (14) by taking the supremum over x, y∈Hwithkxk=kyk=1.
From (14) we also have the inequality
|hTf(z|T|)g(u|T|)x, xi|
≤h fA
|z|2 gA
|u|2i1/2D
fA
|T|2
x, xE1/2D
|T∗|2gA
|T∗|2
x, xE1/2
≤ 1 2 h
fA |z|2
gA
|u|2i1/2Dh fA
|T|2
+|T∗|2gA
|T∗|2i x, x
E1/2
for any x ∈ H, which, by taking the supremum over kxk = 1 produces the
desired result (22).
The following result also holds:
Theorem 3 Letf(z) =P∞
n=0anznbe a function defined by power series with real coefficients and convergent on the open disk D(0, R)⊂C,R > 0. If T is a bounded linear operator on the Hilbert spaceHwith the property thatkTk2< R, then we have the inequality
D
T|T|f |T|2
x, y E
2 ≤D
|T|2fA |T|2
x, x
E D|T∗|2fA
|T∗|2 y, y
E
(23) for any x, y∈H.
Proof. From Furuta’s inequality (F) we have for any natural numbers n≥1 the power inequality
D
T|T|2n−1x, yE ≤D
|T|2nx, xE1/2D
|T∗|2ny, yE1/2
(24) wherex, y∈H.
If we multiply this inequality with the positive quantities |an−1|, use the triangle inequality and the Cauchy-Bunyakowsky-Schwarz discrete inequality we have successively
* k X
n=1
an−1T|T|2n−1x, y +
(25)
≤ Xk n=1
|an−1|
D
T|T|2n−1x, yE
≤ Xk n=1
|an−1|D
|T|2nx, xE1/2D
|T∗|2ny, yE1/2
≤
* k X
n=1
|an−1| |T|2nx, x
+1/2* k X
n=1
|an−1| |T∗|2ny, y +1/2
for any x, y∈Hand k≥1.
Observe also that Xk
n=1
an−1T|T|2n−1=T|T| Xk
n=1
an−1|T|2(n−1),
Xk n=1
|an−1| |T|2n=|T|2 Xk n=1
|an−1| |T|2(n−1) and
Xk n=1
|an−1| |T∗|2n=|T∗|2 Xk n=1
|an−1| |T∗|2(n−1) for any k≥1.
Therefore, by (25) we have the inequality
* T|T|
Xk
n=1
an−1|T|2(n−1)x, y +2
(26)
≤
*
|T|2 Xk
n=1
|an−1| |T|2(n−1)x, x + *
|T∗|2 Xk
n=1
|an−1| |T∗|2(n−1)y, y +
for any x, y∈Hand k≥1.
Due to the assumption kTk2 < R, we have that the series P∞
n=0an|T|2n, P∞
n=0|an| |T|2n and P∞
n=0|an| |T∗|2n are convergent in B(H) and taking the limit overk→ ∞ in (26) we deduce the desired result from (23).
Corollary 3 With the assumptions of Theorem3we have the norm inequality
T|T|f
|T|2
2≤ |T|2fA
|T|2 |T∗|2fA
|T∗|2
and the numerical radius inequality w
T|T|f
|T|2
≤ 1 2
|T|2fA
|T|2
+|T∗|2fA
|T∗|2 . The following result for functions of normal operators holds.
Theorem 4 Letf(z) =P∞
n=0anznbe a function defined by power series with real coefficients and convergent on the open disk D(0, R) ⊂C, R > 0. If N is a normal operator on the Hilbert space H and α, β ≥ 0 with α+β≥ 1 with the property that kNk2α,kNk2β< R, then we have the inequality
D
f
N|N|(α+β−1) x, y
E
2≤D fA
|N|2α x, x
E D fA
|N|2β y, y
E
(27) for any x, y∈H.
Proof.Utilising Furuta’s inequality written for Nn we have
D
Nn|Nn|α+β−1x, yE
2≤D
|Nn|2αx, xE D
|(Nn)∗|2βy, yE
(28) for any x, y∈H.
Since Nis normal, then
|Nn|2 = (Nn)∗Nn=N∗...N∗N...N
= N∗...NN∗...N=...
= (N∗N)...(N∗N) =|N|2n for any natural number n, and, similarly,
|(Nn)∗|2=|(N∗)n|2=|N∗|2n=|N|2n for any n∈N.
These imply that |Nn|2α = |N|2αn, |(Nn)∗|2β = |N|2βn and |Nn|α+β−1 =
|N|(α+β−1)n for any α, β≥0 and for any n∈N.
Utilising the spectral representation for Borel functions of normal operators on Hilbert spaces, see for instance [1, p. 67], we have for anyα, β≥0 and for any n∈Nthat
Nn|N|(α+β−1)n = Z
σ(N)
zn|z|(α+β−1)ndP(z)
= Z
σ(N)
h
z|z|(α+β−1)in
dP(z)
= h
N|N|(α+β−1)in
,
whereP is the spectral measure associated to the operatorNand σ(N) is its spectrum.
Therefore, the inequality (28) can be written as
Dh
N|N|(α+β−1)in
x, yE ≤Dh
|N|2αin
x, xE1/2Dh
|N|2βin
y, yE1/2
(29) for any x, y∈Hand for any n∈N.
If we multiply the inequality (29) by |an| ≥ 0, sum over n from 0 to k ≥ 1 and utilize the Cauchy-Bunyakowsky-Schwarz discrete inequality, we have successively
*Xk
n=0
an h
N|N|(α+β−1)in
x, y +
(30)
≤ Xk n=0
|an|
Dh
N|N|(α+β−1)in
x, y E
≤ Xk n=0
|an|Dh
|N|2αin
x, x E1/2Dh
|N|2βin
y, y E1/2
≤
*Xk
n=0
|an|h
|N|2αin
x, x
+1/2*Xk
n=0
|an|h
|N|2βin
y, y +1/2
for any x, y∈Hand for any k≥1.
Since kNk2α,kNk2β < Rthen
N|N|(α+β−1)
< Rand the series X∞
n=0
|an|h
|N|2αin
, X∞ n=0
|an|h
|N|2βin
and
X∞ n=0
anh
N|N|(α+β−1)in
are convergent in the Banach algebraB(H).
Taking the limit over k → ∞ in the inequality (30) we deduce the desired
result from (27).
Corollary 4 With the assumptions of Theorem 4, we have the inequality
f
N|N|(α+β−1)
2≤ fA
|N|2α fA
|N|2β
. (31) Remark 2 If we take β = 1−α with α ∈ [0, 1] in (27), then we get the following generalization of Kato’s inequality for normal operators (8)
|hf(N)x, yi|2 ≤D fA
|N|2α x, x
E D fA
|N|2(1−α) y, y
E
(32) where x, y∈H andkNk2α,kNk2(1−α)< R.
3 Applications
As some natural examples that are useful for applications, we can point out that, if
f(z) = X∞ n=1
(−1)n
n! zn=ln 1
1+z, z∈D(0, 1) ; (33) g(z) =
X∞ n=0
(−1)n
(2n) !z2n=cosz, z∈C; h(z) =
X∞ n=0
(−1)n
(2n+1) !z2n+1 =sinz, z∈C; l(z) =
X∞ n=0
(−1)nzn = 1
1+z, z∈D(0, 1) ;
then the corresponding functions constructed by the use of the absolute values of the coefficients are
fA(z) = X∞ n=1
1
n!zn=ln 1
1−z, z∈D(0, 1) ; (34) gA(z) =
X∞ n=0
1
(2n) !z2n=coshz, z∈C; hA(z) =
X∞ n=0
1
(2n+1) !z2n+1 =sinhz, z∈C; lA(z) =
X∞ n=0
zn= 1
1−z, z∈D(0, 1).
Other important examples of functions as power series representations with nonnegative coefficients are:
exp(z) = X∞ n=0
1
n!zn z∈C, (35)
1 2ln
1+z 1−z
= X∞ n=1
1
2n−1z2n−1, z∈D(0, 1) ; sin−1(z) =
X∞ n=0
Γ n+ 12
√π(2n+1)n!z2n+1, z∈D(0, 1) ;
tanh−1(z) = X∞ n=1
1
2n−1z2n−1, z∈D(0, 1)
2F1(α, β, γ, z) = X∞ n=0
Γ(n+α)Γ(n+β)Γ(γ)
n!Γ(α)Γ(β)Γ(n+γ) zn, α, β, γ > 0, z∈D(0, 1) ;
whereΓ is theGamma function.
Example 1 Let x, y∈H.
a) If we take f(z) =sinzand g(z) =coszin (14), then we get
|hTsin(z|T|)cos(u|T|)x, yi|2 (36)
≤sinh
|z|2 cosh
|u|2
×D sinh
|T|2 x, x
E D|T∗|2cosh
|T∗|2 y, y
E
for any z∈C and T ∈B(H).
b) If we take f(z) =ln1+z1 and g(z) =ln1−z1 in (14), then we get
D
Tln(1H+z|T|)−1ln(1H−z|T|)−1x, yE
2
(37)
≤ ln 1 1−|z|2
!2
×
ln
1H−|T|2−1
x, x |T∗|2ln
1H−|T∗|2−1
y, y
for any z∈C and T ∈B(H) with |z|< 1 and kTk< 1.
c) If we take f(z) =exp(z) and g(z) =exp(z) in (14), then we get
|hTexp[(z+u)|T|]x, yi|2 (38)
≤exp |z|2
exp |u|2
×D exp
|T|2 x, xE D
|T∗|2exp
|T∗|2 y, yE for any z, u∈C and T ∈B(H).
d) By the inequality (20) we have
D
Tsin−1(z|T|)x, y E
2≤sin−1
|z|2 D sin−1
|T|2 x, x
E D|T∗|2y, y E
(39) and
D
Ttanh−1(z|T|)x, y E
2
(40)
≤tanh−1
|z|2 D
tanh−1
|T|2 x, xE D
|T∗|2y, yE for any z∈C and T ∈B(H) with |z|< 1 and kTk< 1.
Example 2 Let x, y∈H.
a) If we take f(z) = 1±z1 in (23), then we get
T|T|
1H±|T|2−1
x, y
2
(41)
≤
|T|2
1H−|T|2−1
x, x |T∗|2
1H−|T∗|2−1
y, y
for any T ∈B(H) with kTk< 1.
b) If we take f(z) =ln1±z1 in (23), then we get
T|T|ln
1H±|T|2−1
x, y
2
(42)
≤
|T|2ln
1H−|T|2−1
x, x |T∗|2ln
1H−|T∗|2−1
y, y
for any T ∈B(H) with kTk< 1.
c) If we take f(z) =exp(z) in (23), then we get
D
T|T|exp
|T|2 x, yE
2
(43)
≤D
|T|2exp
|T|2 x, x
E D|T∗|2exp
|T∗|2 y, y
E
for any T ∈B(H).
Example 3 Let Nbe a normal operator on the Hilbert spaceH,α, β≥0with α+β≥1 and x, y∈H.
a) If we take f(z) = 1±z1 in (27), then we get
1H±N|N|(α+β−1)−1
x, y
2
(44)
≤
1H−|N|2α−1
x, x
1H−|N|2β−1
y, y
provided kNk< 1.
In particular, we have
D
(1H±N)−1x, yE
2
(45)
≤
1H−|N|2α−1
x, x
1H−|N|2(1−α)−1
y, y
, for α∈[0, 1].
b) If we take f(z) =exp(z) in (27), then we get
D
exp
N|N|(α+β−1) x, y
E
2≤D exp
|N|2α x, x
E D exp
|N|2β y, y
E . (46) As a special case, we have
|hexp(N)x, yi|2≤D exp
|N|2α
x, xE D exp
|N|2(1−α) y, yE
, (47) for α∈[0, 1].
References
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Received: 13 June 2014
