SOME GENERALIZED NUMERICAL RADIUS INEQUALITIES INVOLVING KWONG FUNCTIONS
arXiv:1801.07619v1 [math.FA] 23 Jan 2018
MOJTABA BAKHERAD Abstract. We prove several numerical radius inequalities involving positive semidefinite matrices via the Hadamard product and Kwong functions. Among other inequalities, it is shown that if X is a arbitrary n × n matrix and A, B are positive
semidefinite, then
ω(Hf,g (A)) ≤ k ω(AX + XA), which is equivalent to ω Hf,g (A, B) ± Hf,g (B, A)
≤ k ′ {ω((A + B)X + X(A + B)) + ω((A − B)X − X(A − B))} ,
(t) where f and g are two continuous functions on (0, ∞) such that h(t) = fg(t) is Kwong, n o n o f (λ)g(λ) f (λ)g(λ) ′ k = max : λ ∈ σ(A) and k = max : λ ∈ σ(A) ∪ σ(B) . λ λ
1. Introduction Let Mn be the C ∗ -algebra of all n × n complex matrices and h · , · i be the standard scalar product in Cn . A capital letter means an n × n matrix in Mn . For Hermitian
matrices A and B, we write A ≥ 0 if A is positive semidefinite, A > 0 if A is positive definite, and A ≥ B if A − B ≥ 0. The numerical radius of A ∈ Mn is defined by ω(A) := sup{| hAx, xi |: x ∈ Cn , k x k= 1}.
It is well known that ω( · ) defines a norm on Mn , which is equivalent to the usual
operator norm k · k. In fact, for any A ∈ Mn , 21 kAk ≤ ω(A) ≤ kAk; see [12]. For
further information about numerical radius inequalities we refer the reader to [12, 17]
and references therein. We use the notation J for the matrix whose entries are equal to one. The Hadamard product (Schur product) of two matrices A, B ∈ Mn is the matrix
A ◦ B whose (i, j) entry is aij bij (1 ≤ i, j ≤ n). The Schur multiplier operator SA on
2010 Mathematics Subject Classification. Primary 47A12, Secondary 47A30, 47A63. Key words and phrases. Numerical radius, Hadamard product, Operator monotone, Kwong function. 1
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M. BAKHERAD
Mn is defined by SA (X) = A ◦ X (X ∈ Mn ). The induced norm of SA with respect
to the numerical radius norm will be denoted by
ω(SA (X)) ω(A ◦ X) = sup . ω(X) ω(X) X6=0 X6=0
kSA kω = sup
A continuous real valued function f on an interval (a, b) ⊆ R is called operator monotone if A ≤ B implies f (A) ≤ f (B) for all Hermitian matrices A, B ∈ Mn with spectra
(a, b) in (a, b). Following [3], a continuous real-valued function f defined on an interval f (λi )+f (λj ) with a > 0 is called a Kwong function if the matrix Kf = is λi +λj i,j=1,2,··· ,n
positive semidefinite for any (distinct) λ1 , · · · , λn in (a, b). It is easy to see that if f is a nonzero Kwong function, then f is positive and
1 f
is Kwong. Kwong [14] showed that
the set of all Kwong functions on (0, ∞) is a closed cone and includes all non-negative operator monotone functions on (0, ∞). Also, Audenaert [3] gave a characterization
of Kwong functions by showing that, for given 0 ≤ a < b, a function f on an interval √ √ (a, b) is Kwong if and only if the function g(x) = xf ( x) is operator monotone on (a2 , b2 ). The Heinz means are defined as Hν (a, b) =
a1−ν bν +aν b1−ν 2
for a, b > 0 and 0 ≤ ν ≤ 1.
These interesting means interpolate between the geometric and arithmetic means. In √ , where a, b > 0 and fact, the Heinz inequalities assert that ab ≤ Hν (a, b) ≤ a+b 2 0 ≤ ν ≤ 1. There have been obtained several Heinz type inequalities for Hilbert space
operators and matrices; see [?] and references therein.
For two continuous functions f and g on (0, ∞) we denote Hf,g (A, B) = f (A)Xg(B) + g(A)Xf (B) and Hf,g (A) = f (A)Xg(A) + g(A)Xf (A), where A, B, X ∈ Mn such that A, B are positive semidefinite. In particular, f (t) = tα
and g(t) = t1−α (α ∈ [0, 1])
Hα (A, B) = Aα XB 1−α + A1−α XB α
and Hα (A) = Aα XA1−α + A1−α XAα .
A norm ||| · ||| on Mn is called unitarily invariant if |||UAV ||| = |||A||| for all A ∈ Mn
and all unitary matrices U, V ∈ Mn . Let A, B, X ∈ Mn such that A and B are positive
semidefinite. In [15] it was conjectured a general norm inequality of the Heinz inequality
|||Hf,g (A, B)||| ≤ |||AX + XB|||, where f and g are two continuous functions on (0, ∞) such that f (t)g(t) ≤ t and the function h(t) =
f (t) g(t)
is Kwong. In particular, if f (t) = tα
and g(t) = t1−α (α ∈ [0, 1]), then we state a Heinz type inequality |||Hα(A, B)||| ≤
|||AX + XB|||, where A, B, X ∈ Mn such that A, B are positive semidefinite. For further information, we refer the reader to [5, 7] and references therein.
SOME GENERALIZED NUMERICAL RADIUS INEQUALITIES
3
The numerical radius ω( · ) is a weakly unitarily invariant norm on Mn , that is
ω(U ∗ AU) = ω(A) for every A ∈ Mn and every unitary U ∈ Mn . In [1], the authors proved a Heinz type inequality for the numerical radius as follows ω(Hα(A)) ≤ ω(AX + XA),
(1.1)
in which A, X ∈ Mn such that A is positive semidefinite. They also showed that the inequality ω(Hα(A, B)) ≤ ω(AX + XB) is not true in general.
Our research aim is to show some numerical radius inequalities via the Hadamard
product and Kwong functions. By using some ideas of [10, 11] and [15], we obtain some extensions and generalizations of inequality (1.1), which are generalizations of a Hienz type inequality for the numerical radius. For instance, we prove if A, X ∈ Mn such that A is positive semidefinite, then
ω(Hf,g (A)) ≤ k ω(AX + XA), where f and g are two continuous functions on (0, ∞) such that o n : λ ∈ σ(A) . k = max f (λ)g(λ) λ
f (t) g(t)
is Kwong and
2. main results For our purpose we need the following lemmas. Lemma 2.1. [19, Theorem 3.4] (Spectral Decomposition) Let A ∈ Mn with eigenvalues
λ1 , · · · , λn . Then A is normal if and only if there exists a unitary matrix U such that U ∗ AU = diag(λ1 , · · · , λn ). In particular, A is positive definite if and only if the λj (1 ≤ j ≤ n) are positive. Lemma 2.2. [2, Corollary 4] Let A = [aij ] ∈ Mn be positive semidefinite. Then kSA kω = max aii . i
Lemma " 2.3. [13] #!Let X, Y ∈ Mn . Then X 0 (i) ω = max{ω(X), ω(Y )}; 0 Y " #! 0 X max(ω(X+Y ),ω(X−Y )) ≤ω ≤ (ii) 2 Y 0
ω(X+Y )+ω(X−Y ) . 2
Now, we are in position to demonstrate the first result of this section by using some ideas of [10, 11, 15].
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M. BAKHERAD
Theorem 2.4. Let A, B ∈ Mn be positive semidefinite, X ∈ Mn , and let f , g be two continuous functions on (0, ∞) such that h(t) =
f (t) g(t)
is Kwong. Then
ω(Hf,g (A)) ≤ k ω(AX + XA), n o where k = maxλ∈σ(A) f (λ)g(λ) . λ
(2.1)
Moreover, inequality (2.1) is equivalent to the inequality ω Hf,g (A, B) ± Hf,g (B, A)
≤ k ′ {ω((A + B)X + X(A + B)) + ω((A − B)X − X(A − B))} , (2.2) n o f (λ)g(λ) ′ where k = maxλ∈σ(A)∪σ(B) . λ Proof. Assume that A is positive definite. Since the numerical radius is weakly uni-
tarily invariant, we may assume that A is diagonal matrix with positive eigenvalues λ1 , · · · , λn . It follows from
f g
is a Kwong function that f (λi )g −1 (λj ) + f (λj )g −1 (λi ) Λ Z = [zij ] = Λ λi + λj (i,j=1,··· ,n)
is positive semidefinite, where Λ = diag (g(λ1), · · · , g(λn )). It follows from Lemma 2.2
that
kSZ kω = max zii = max i
or equivalently,
ω(Z◦X) ω(X)
i
f (λi )g(λi ) ≤k λi
≤ k (0 6= X ∈ Mn ).
1 If we put E = [ λi +λ ] and F = j
[f (λi )g(λj ) + f (λj )g(λi )] ∈ Mn , then
ω(E ◦ F ◦ X) = ω(Z ◦ X) ≤ k ω(X)
(X ∈ Mn ).
Let the matrix C be the entrywise inverse of E, i.e., C ◦ E = J. Thus ω(F ◦ X) ≤ k ω(C ◦ X)
(X ∈ Mn )
or equivalently ω(Hf,g (A)) = ω(f (A)Xg(A) + g(A)Xf (A)) ≤ k ω(AX + XA). (2.3) " # A1 0 Now, if A is positive semidefinite, we may assume that A = , where A1 ∈ 0 0 " # X1 X2 Mk (k < n) is a positive definite matrix. Let X = , where X1 ∈ Mk and X3 X4
SOME GENERALIZED NUMERICAL RADIUS INEQUALITIES
5
X4 ∈ Mn−k . Then we have "
ω(Hf,g (A)) = ω
f (A1 )X1 g(A1) + g(A1 )X1 f (A1 ) 0 0
"
≤ kω
0
#!
(by Lemma 2.3(i)) #! A1 X1 + X1 A1 0 (by (2.3)) 0 0
= k ω(A1 X1 + X1 A1 ) ≤ k ω(AX + XA)
(by Lemma 2.3(i)) (by [8, Lemma 2.1]).
Hence, we reach inequality (2.1). Moreover, if we replace A and X by ! 0 X in inequality (2.1), respectively, then X 0 " #! 0 Hf,g (A, B) ω Hf,g (B, A) 0 " #! 0 AX + XB ′ ≤k ω , XA + BX 0
A 0 0 B
(2.4) ! and
whence n o max ω Hf,g (A, B) ± Hf,g (B, A) " #! 0 f (A)Xg(B) + g(A)Xf (B) ≤ 2ω g(B)Xf (A) + f (B)Xg(A) 0
≤ 2k ′ ω
"
0
(by Lemma 2.3(ii)) #! AX + XB
XA + BX
0
(by (2.4))
≤ k ′ (ω(AX + XB + XA + BX) + ω(AX + XB − XA − BX)) (by Lemma 2.3(ii)). Thus, we have inequality (2.2). Also, if we put B = A in inequality (2.2), then we reach inequality (2.1).
If we take f (t) = tα and g(t) = t1−α in Theorem 2.4 for each 0 ≤ α ≤ 1, then we get
the next result.
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M. BAKHERAD
Corollary 2.5. [1, Theorem 2.4] Let A, B ∈ Mn be positive semidefinite, X ∈ Mn ,
and let 0 ≤ α ≤ 1. Then
ω(Hα(A)) ≤ ω(AX + XA).
(2.5)
Moreover, inequality (2.5) is equivalent to the inequality ω Hα (A, B) ± Hα (B, A)
≤ ω((A + B)X + X(A + B)) + ω((A − B)X − X(A − B)).
Corollary 2.6. Let A, B ∈ Mn be positive semidefinite, X ∈ Mn , and let f be a non-
negative operator monotone function on [0, ∞) such that f ′ (0) = limx→0+ f ′ (x) < ∞ and f (0) = 0. Then
ω(f (A)X + Xf (A)) ≤ f ′ (0) ω(AX + XA).
(2.6)
Moreover, inequality (2.6) is equivalent to the inequality ω(X(f (A) + f (B)) + (f (A) + f (B))X) ′ ≤ f (0) ω((A + B)X + X(A + B)) + ω((A − B)X − X(A − B)) .
Proof. A function g is non-negative operator increasing on [0, ∞) if and only if
non-negative operator increasing on [0, ∞); see [9]. Hence
is Then f (t) t f (t) have t ≤ ′ ′
decreasing. If 0 ≤ x ≤ t, then
f (t) t
≤
f (x) . x
t f (t)
t g(t)
is
is operator increasing.
Now, by taking x → 0+ we
f ′ (0). If we put g(t) = 1 (t ∈ [0, ∞)) in Theorem 2.4, it follows from
k = k ≤ f (0) that we get the required result.
We first cite the following lemma due to Fujii et al. [10], which will be needed in the next theorem. Lemma 2.7. [10, Lemma 3.1] Let λ1 , · · · , λn be any positive real numbers and −2 < t ≤ 2. If f and g are two continuous functions on (0, ∞) such that
the n × n matrix
Y =
f (λi )g −1 (λj ) + f (λj )g −1 (λi ) λ2i + tλi λj + λ2j
f (t) g(t)
is Kwong, then
i,j=1,··· ,n
is positive semidefinite. Theorem 2.8. Let A, B ∈ Mn be positive semidefinite, X ∈ Mn , f , g be two continuous functions on (0, ∞) such that
f (t) g(t)
is Kwong, and let −2 < t ≤ 2. Then
1 1 2k ω A2 X + tAXA + XA2 , ω A 2 Hf,g (A) A 2 ≤ t+2
(2.7)
SOME GENERALIZED NUMERICAL RADIUS INEQUALITIES
where k = maxλ∈σ(A)
n
f (λ)g(λ) λ
o
7
.
Moreover, inequality (2.7) is equivalent to the inequality 1 1 4k ′ ω A 2 Hf,g (A, B) B 2 ≤ ω A2 X + tAXB + XB 2 , t+2 o n . where k ′ = maxλ∈σ(A)∪σ(B) f (λ)g(λ) λ
(2.8)
Proof. First, we show inequality (2.7). It is enough to show the inequality in the case A is positive definite. Since the numerical radius is weakly unitarily invariant, we may assume that A is diagonal 1 matrix with positive eigenvalues λ1 , · · · , λn . Let 1 Σ = diag λ12 g(λ1 ), · · · , λn2 g(λn ) . It follows from Lemma 2.7 that (t + 2) (f (λ )g −1(λ ) + f (λ )g −1(λ )) i j j j Z = [zij ] = Σ Σ 2 2 2(λi + tλi λj + λj ) i,j=1,··· ,n
is positive semidefinite for −2 < t ≤ 2. In addition, all diagonal entries of Z are no
more than k. Therefore,
f (λi )g(λi ) ≤ k, i i λi i h ω(Z◦X) 1 and P = whence ω(X) ≤ k (0 6= X ∈ Mn ). Now, let M = λ2 +tλi λj +λ2 i j i,j=1,··· ,n h 1 1i t+2 2 λi f (λi )g(λj ) + f (λj )g(λi )λj2 . Then 2 kSZ kω = max zii = max
i,j=1,··· ,n
ω(M ◦ P ◦ X) = ω(Z ◦ X) ≤ k ω(X)
(0 6= X ∈ Mn ).
Let the matrix N be the entrywise inverse of M, i.e., M ◦ N = J. Hence ω(P ◦ X) ≤ k ω(N ◦ X)
(0 6= X ∈ Mn )
or equivalently 2k ω(A2 X + tAXA + XA2 ), t+2 o n : λ ∈ σ(A) . Hence we have where X ∈ Mn , −2 < t ≤ 2 and k = max f (λ)g(λ) λ 1
1
ω(A 2 (Hf,g (A)) A 2 ) ≤
inequality (2.7).
A 0
Now, if we replace A and X by
0 B
!
and
0 X 0
0
!
inequality (2.7), respectively,
then ω
"
1
1
0 A 2 (Hf,g (A, B)) B 2 0
0
#!
2k ′ ≤ ω t+2
"
0 A2 X + tAXB + XB 2 0
0
#!
.
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M. BAKHERAD
Hence "
1 1 1 ω A 2 Hf,g (A, B) B 2 ≤ ω 2
1 #! 1 0 A 2 Hf,g (A, B) B 2
0
0
(by Lemma 2.3)
2k ′ ≤ ω t+2
"
0 A2 X + tAXB + XB 2 0
0
#!
2k ′ ω A2 X + tAXB + XB 2 (by Lemma 2.3). t+2 Thus, we reach inequality (2.8). Also, if we put B = A in inequality (2.7), then we get ≤
inequality (2.8).
Corollary 2.9. Let A ∈ Mn be positive semidefinite. If f is a positive operator
monotone function on (0, ∞), then
3 3 1 1 ω A 2 f (A)Xf (A)−1A 2 + A 2 f (A)−1 Xf (A)A 2 ≤
where X ∈ Mn and −2 < t ≤ 2.
4 ω A2 X + tAXA + XA2 , t+2
Proof. Since f positive operator monotone on (0, ∞), then g(t) = monotone on (0, ∞) and also
f (t) g(t)
t f (t)
is operator
= tf 2 (t) is Kwong function [15]. So f and g satisfy
the conditions of Theorem 2.8. Hence we have the desired inequality.
Example 2.10. The function f (t) = log(1 + t) is operator monotone on (0, ∞); see [9]. If we put g(t) = 1, then
f (t) g(t)
= log(1 + t) is Kwong [14]. Using Theorem 2.4 we
have 2 ω A2 X + tAXA + XA2 , t+2 where A, X ∈ Mn such that A is positive semidefinite and −2 < t ≤ 2. 1 1 ω A 2 log(I + A)X + X log(I + A) A 2 ≤
Now, we infer the following lemma due to Zhan [18], which will be needed in the next theorem. Lemma 2.11. [18, Lemma 5] Let λ1 , · · · , λn be any positive real numbers, r ∈ [−1, 1]
and −2 < t ≤ 2. Then the n × n matrix λri + λrj L= λ2i + tλi λj + λ2j i,j=1,··· ,n is positive semidefinite. Now, we shall show the following result related to [10].
SOME GENERALIZED NUMERICAL RADIUS INEQUALITIES
9
Proposition 2.12. Let A, X ∈ Mn such that A is positive semidefinite, β > 0 and
1 ≤ 2r ≤ 3. Then
ω Ar XA2−r +A2−r XAr 4β(1 − r0 ) 2 2 ≤ ω 2(1 − 2β + 2βr0 )AXA + (A X + tAXA + XA ) , t+2 where −2 < t ≤ 2β − 2 and r0 = min{ 21 + |1 − r|, 1 − |1 − r|}. Proof. Since the numerical radius is weakly unitarily invariant, we may assume that A is diagonal matrix with positive eigenvalues λ1 , · · · , λn . Since 1 ≤ 2r ≤ 3, then 1 2 1 4
1−2β+2βr0 (t + 2) + t. It follows from −2 < t ≤ 2β − 2 2β(1−r0 ) t+2 t 1 > 0 and −2 < t0 ≤ 2, where t0 = 2β(1−r + β(1−r 4β(1−r0 ) 0) 0)
≤ r0 ≤ 43 . Let t0 =
≤ 1 − r0 ≤
1 , 4
that
Hence, by using Lemma 2.11, the n × n matrix W = [wij ] = is positive semidefinite for
1 2
and
− 2.
λ2−2r + λ2−2r t+2 i j Λr 2 Λr 4β(1 − r0 ) λi + t0 λi λj + λ2j i,j=1,··· ,n ≤ r ≤ 23 , where Λ = diag (λ1 , · · · , λn ). Therefore,
(t + 2)λri (2λ2−2r )λri i =1 i i 4β(1 − r0 )(t0 + 2)λ2i ≤ 1 (0 6= X ∈ Mn ). Now, let O = λ2i + t0 λi λj + λ2j i,j=1,··· ,n and 1 . Then 4β(1−r0 ) 2 2 kSW kω = max wii = max
whence M=
ω(W ◦X) ω(X)
2(1−2β+2βr0 )λi λj +
t+2
(λi X+tλi λj +λj )
i,j=1,··· ,n
ω(O ◦ M ◦ X) = ω(W ◦ X) ≤ ω(X)
(0 6= X ∈ Mn ).
Let the matrix N be the entrywise inverse of M, i.e., M ◦ N = J. Hence ω(O ◦ X) ≤ ω(N ◦ X)
(0 6= X ∈ Mn )
or equivalently ω Ar XA2−r +A2−r XAr 4β(1 − r0 ) 2 2 ≤ ω 2(1 − 2β + 2βr0 )AXA + (A X + tAXA + XA ) , t+2 where −2 < t ≤ 2β − 2 and r0 = min{ 21 + |1 − r|, 1 − |1 − r|}.
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M. BAKHERAD
3. Acknowledgement. The author would like to sincerely thank the referee for some useful comments and suggestions.
References 1. Aghamollaei, G. and Sheikh Hosseini, A. Some numerical radius inequalities with positive definite functions. Bull. Iranian Math. Soc. 41 (4), 889-900, 2015. 2. Ando, T. and Okubo, K.Induced norms of the Schur multiplication operator, Linear Multilinear Algebra Appl. 147, 181-199, 1991. 3. Audenaert, K.M.R. A characterization of anti-Lowner function, Proc. Amer. Math. Soc. 139 (12), 4217-4223, 2011. 4. Bakherad, M. and Kittaneh, F. Numerical Radius Inequalities Involving Commutators of G1 Operators, Complex Anal. Oper. Theory 2017. https://doi.org/10.1007/s11785-017-0726-9. 5. Bakherad, M. and Moslehian, M.S. Reverses and variations of Heinz inequality, Linear Multilinear Algebra 63 (10), 1972-1980, 2015. 6. Bakherad, M., Krnic, M. and Moslehian, M.S. Reverses of the Young inequality for matrices and operators, Rocky Mountain J. Math. 46 (2016), no. 4, 1089-1105. 7. Bhatia, R. and Davis, Ch. More matrix forms of the arithmetic-geometric mean inequality. SIAM J. Matrix Anal. Appl. 14 (1), 132-136, 1993. 8. Erfanian Omidvar, M., Moslehian, M.S. and Niknam, A. Some numerical radius inequalities for Hilbert space operators, Involve 2 (4), 469-476, 2009. 9. Fujii, M., Mi´ci´c Hot, J., Peˇcari´c, J. and Y. Seo, Recent developments of Mond–Peˇcari´c method in operator inequalities. Inequalities for bounded selfadjoint operators on a Hilbert space. II., Monographs in Inequalities 4. Zagreb: Element, 2012. 10. Fujii, M., Seo, Y. and Zuo, H. Zhan’s inequality on A-G mean inequalities. Linear Algebra Appl. 470, 241-251, 2015. 11. Fujii, J., Fujii, M., Seo, Y. and Zuo, H. Recent developments of matrix versions of the arithmeticgeometric mean inequality. Ann. Funct. Anal. 7 (1), 102-117, 2016. 12. Gustafson, K.E. and Rao, D.K.M. Numerical Range, The Field of Values of Linear Operators and Matrices, Springer, New York, 1997. 13. Hirzallah, O., Kittaneh, F. and Shebrawi, Kh. Numerical radius inequalities for certain 2 × 2 operator matrices. Integral Equations Operator Theory 71 (1), 129-147, 2011.
14. Kwong, M.K. Some results on matrix monotone functions, Linear Algebra Appl. 118, 129-153, 1989. 15. Najafi, H. Some results on Kwong functions and related inequalities, Linear Algebra Appl. 439 (9), 2634-2641, 2013. 16. Ramesh, G. On the numerical radius of a quaternionic normal operator, 2 (1), 78-86, 2017. 17. Yamazaki, T. On upper and lower bounds of the numerical radius and an equality condition, Studia Math. 178, 83-89, 2007.
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18. Zhan, X. Inequalities for unitarily invariant norms, SIAM J. Matrix Anal. Appl. 20 (2), 466-470, 1999. 19. Zhang, F. Matrix Theory, Second edition, Springer, New York, 2011. Department of Mathematics, Faculty of Mathematics, University of Sistan and Baluchestan, Zahedan, I.R.Iran. E-mail address:
[email protected];
[email protected]