Abstract
By the help of power series , we can naturally construct another power series that has as coefficients the absolute values of the coefficients of f, namely, . Utilizing these functions, we show among others that
and
where denotes the numerical radius of the bounded linear operator T on a complex Hilbert space, while is its norm.
MSC: 47A63, 47A99.
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1 Introduction
The numerical radius of an operator T on H is given by [[1], p.8]
Obviously, by (1.1), for any , one has
It is well known that is a norm on the Banach algebra of all bounded linear operators , i.e.,
(i) for any and if and only if ;
(ii) for any and ;
(iii) for any .
This norm is equivalent with the operator norm. In fact, the following more precise result holds [[1], p.9].
Theorem 1 (Equivalent norm)
For any , one has
Some improvements of (1.3) are as follows.
Theorem 2 (Kittaneh, 2003 [2])
For any operator , we have the following refinement of the first inequality in (1.3):
Utilizing the Cartesian decomposition for operators, Kittaneh improved the inequality (1.3) as follows.
Theorem 3 (Kittaneh, 2005 [3])
For any operator , we have
From a different perspective, we have the following result as well.
Theorem 4 (Dragomir, 2007 [4])
For any operator , we have
The following general result for the product of two operators holds [[1], p.37].
Theorem 5 (Holbrook, 1969 [5])
If A, B are two bounded linear operators on the Hilbert space , then . In the case that , then . The constant 2 is best possible here.
The following results are also well known [[1], p.38].
Theorem 6 (Holbrook, 1969 [5])
If A is a unitary operator that commutes with another operator B, then
If A is an isometry and , then (1.7) also holds true.
We say that A and B double commute if and . The following result holds [[1], p.38].
Theorem 7 (Holbrook, 1969 [5])
If the operators A and B double commute, then
As a consequence of the above, we have the following [[1], p.39].
Corollary 1 Let A be a normal operator commuting with B. Then
A related problem with the inequality (1.8) is to find the best constant c for which the inequality
holds for any two commuting operators . It is known that ; see [6, 7] and [8].
Motivated by the above results, we establish in this paper some inequalities for the numerical radius of functions of operators defined by power series, which incorporate many fundamental functions of interest such as the exponential function, some trigonometric functions, the functions , and others. Some examples of interest are also provided.
2 Some inequalities for one operator
Now, by the help of power series , we can naturally construct another power series which will have as coefficients the absolute values of the coefficients of the original series, namely, . 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 , then .
The following simple result provides some nice inequalities for operator functions defined by power series.
Theorem 8 Let be a function defined by power series with complex coefficients and convergent on the open disk , . For any , we have the inequality
provided , and the inequality
provided .
Proof Let with . Utilizing the properties of the numerical radius, we have
Since the series is convergent on ℝ, the series is convergent in , and by the continuity of the numerical radius, we have
Then, by letting in the inequality (2.3), we deduce the desired result (2.1).
Utilizing the properties of the numerical radius and the Kittaneh inequality (1.4), we also have
Since the series , are convergent on ℝ, the series is convergent in . Then, by letting in the inequality (2.4), we deduce the desired result (2.2). □
Corollary 2 Let be a function defined by power series with nonnegative coefficients and convergent on the open disk , . For any , we have the inequality
if , and the inequality
if .
From a different perspective, we have the following.
Theorem 9 Let be a function defined by power series with complex coefficients and convergent on the open disk , . For any with and with , we have the inequality
and the inequality
Proof Let with . Utilizing the properties of the numerical radius, we have
By the weighted Cauchy-Bunyakovsky-Schwarz discrete inequality, we have
Now, on writing the inequality (1.6) for the operators , we have
for any , which implies that
On making use of the inequalities (2.9)-(2.11), we get
Since the series , and are convergent on ℝ and is convergent on , then by letting in the inequality (2.12), we deduce the desired result (2.7).
Now, on making use of the Kittaneh inequality (1.5), we also have
for any , which implies
By the inequalities (2.9) and (2.10), we then get
for any with .
The proof follows now as above and we get the desired inequality (2.8). □
Corollary 3 Let be a function defined by power series with nonnegative coefficients and convergent on the open disk , . For any with and with , we have the inequality
and the inequality
3 Some inequalities for two operators
We start with the following result.
Theorem 10 Let and such that
If is a function defined by power series with complex coefficients and convergent on the open disk , and , for , , then we have the inequalities
where
and
Proof By the properties of the numerical radius and by (3.3), we have
for any .
Let with . We have, by the above inequality,
By Hölder’s weighted inequality, we have
Then, by (3.4) and by (3.5), we get
for any with .
Since the series whose partial sums are involved in (3.6) are convergent, then by taking in (3.6), we deduce the first inequality in (3.2).
Further, by utilizing the following Hölder-type inequality obtained by Dragomir and Sándor in 1990 [9] (see also [[10], Corollary 2.34]):
that holds for nonnegative numbers and complex numbers , , where , we observe that the convergence of the series and imply the convergence of the series .
Utilizing (3.7), we can state that
for any with .
This together with (3.4) provides
for any with .
Since all the series whose partial sums are involved in (3.8) are convergent, then by taking in (3.8), we deduce the second inequality in (3.2). □
Remark 1 If we take in the first inequality in (3.2), we have
provided .
Corollary 4 Let be a function defined by power series with complex coefficients and convergent on the open disk , . Then for any with , for , , we have the inequalities
where
and
If are commutative with , for , , then we have the inequalities
where
and
The proof of the inequality (3.10) follows by Theorem 5 since in this case, we can take in (3.1), while the inequality (3.11) follows by the commutative case, in which case we can take in (3.1).
The case of commuting operators can be treated in a different way as well.
Proposition 1 Let be a function defined by power series with complex coefficients and convergent on the open disk , . If are commutative with , for , , then we have the inequalities
where
and
Proof Since are commutative, then for any , the operators and are commutative and .
Applying Theorem 5 for the commutative case, we have
for any .
Let with . We have, by the above inequality,
Now, on making use of a similar approach to the one employed in the proof of Theorem 10, we deduce the desired result (3.12). □
Remark 2 If we take in the first inequality in (3.12), we get
provided .
As pointed out in the introduction, the inequality
holds for any two commuting operators and for some . It is known that ; see [6, 7] and [8].
Proposition 2 Let be a function defined by power series with complex coefficients and convergent on the open disk , . If are commutative with , for , , then we have the inequalities
where
and
Moreover, if the operators A and B double commute, then the constant c can be taken to be 1 in (3.15).
Proof Applying the inequality (3.14) for the commuting operators and with , we have
for any .
On making use of a similar argument as in the proof of Theorem 10, we deduce the desired inequality (3.15).
If the operators A and B double commute, then the operators and also double commute, and by Theorem 7 we deduce the second part of the proposition. □
From a different perspective, we have the following result as well.
Proposition 3 Let be a function defined by power series with complex coefficients and convergent on the open disk , . If such that , then
Proof We use the following two inequalities obtained by Kittaneh in [3]
for any .
Let . We have, by the above inequalities,
The proof follows now as above and the details are omitted. □
4 Examples
As some natural examples that are useful for applications, we can point out that if
then the corresponding functions constructed by the use of the absolute values of the coefficients are
Other important examples of functions as power series representations with nonnegative coefficients are:
where Γ is a gamma function.
For any operator with , by making use of the inequality (2.1), we have the simple inequalities
and
For any operator , we also have
and
Similar inequalities may be stated by employing the other results obtained for one or two operators. However, the details are left to the interested reader.
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Dragomir, S.S. Some numerical radius inequalities for power series of operators in Hilbert spaces. J Inequal Appl 2013, 298 (2013). https://doi.org/10.1186/1029-242X-2013-298
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DOI: https://doi.org/10.1186/1029-242X-2013-298