Some Inequalities of the Grüss Type for the Numerical Radius of Bounded Linear Operators in Hilbert Spaces
© The Author(s). 2008
Received: 27 May 2008
Accepted: 4 August 2008
Published: 12 August 2008
Some inequalities of the Grüss type for the numerical radius of bounded linear operators in Hilbert spaces are established.
It is well known that is a norm on the Banach algebra of all bounded linear operators This norm is equivalent to the operator norm. In fact, the following more precise result holds [1, page 9].
Theorem 1.1 (equivalent norm).
For other results on numerical radius (see [2, Chapter 11]).
The following general result for the product of two operators holds [1, page 37].
The following results are also well known [1, page 38].
The following result holds [1, page 38].
Theorem 1.4 (double commute).
As a consequence of the above, one has [1, page 39] the following.
In the recent survey paper , we provided other inequalities for the numerical radius of the product of two operators. We list here some of the results.
If more information regarding one of the operators is available, then the following results may be stated as well.
are also given.
2. Numerical Radius Inequalities of Grüss Type
In order to give examples of operators and numbers such that the transform is accretive, it suffices to select a bounded linear operator and the complex numbers with the property that , and by choosing and we observe that satisfies (2.4), that is, is accretive.
The following result may be stated as well.
The first inequality has been established in  (see [6, page 62]) while the second one can be obtained in a canonical manner from the reverse of the Schwarz inequality given in . The details are omitted.
A similar argument to that in the proof of Theorem 2.3 yields the desired inequalities (2.16). The details are omitted.
respectively. These two inequalities were obtained earlier by the author using a different approach (see ).
Find general examples of bounded linear operators realizing the equality case in each of inequalities (2.6), (2.16), respectively.
3. Some Particular Cases of Interest
Finally, we also have the following proposition.
Now, on making use of a similar argument to the one in the proof of Proposition 3.1, we deduce the desired results (3.7). The details are omitted.
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