Some results around quadratic maps

. This paper dedicated to study quadratic maps. We present some new operator equalities and inequalities by using quadratic map in the frame-work of B ( H ) . Applications for particular case of interest are also provided.


Introduction and preliminaries
As customary, we reserve for scalars and other capital letters denote general elements of the C -algebra B (H ) of all bounded linear operator acting on Hilbert space (H ; h ; i). The absolute value of operator A is denoted by jAj = (A A) More information on such maps can be found in [10, p. 18]. The study of linear maps on an algebra of bounded linear operators on a Hilbert space has been developed by many authors (see for instance [3,5,7,13,14]). Also, for a host of positive linear map inequalities, and for diverse applications of these inequalities, we refer to [8,11,15], and references therein. As is known to all, the linear property plays an important role to obtain this inequalities.
The motivation of this paper is to present some results concerning equalities and inequalities for maps without linear property on complex Hilbert spaces. In order to prove our main results, we need the following essential de…nitions. A map ' : B (H ) B (H ) ! B (H ) is a sesquilinear, if satisfying the following conditions:  , is called the quadratic map associated with '. It can be easily veri…ed that the de…nition of quadratic map is di¤erent from positive linear map. In fact, by using a sesquilinear map we create a quadratic map, that is not necessarily linear and positive.
The paper is organized in the following way: After this Introduction, in Section 2 we deduce some equalities. The parallelogram law is recovered (see Theorem 2.1 and 2.2) and some other interesting operator equalities are established. Afterward, in Section 3, we get an extension of some well known inequalities such as, triangle (Theorem 3.1) inequality. Especially, Bohr's inequality is generalized to the context of quadratic map (see Theorem 3.4). Some results concerning this inequality are surveyed (see Corollary 3.5 and 3.6). In Section 4 before closing the paper, we give an application of our results in the previous sections. We show that our results are a generalization of some well known works due to Fujii [9] and Hirzallah [12].

Some equalities for quadratic maps
Here and throughout, stands for the quadratic map. Our …rst main result in this section reads as follows. The following generalization of the parallelogram law holds.
Proof. We observe that which proves the theorem. Similarly, if either t 1 or t < 0, then The following result can be regarded as an extension of the well-known Apollonius's identity (see, e.g., [2, Lemma 2.12]). Proof. By Theorem 2.1, we have which is clearly equivalent to (2.5).
The following result concerning the quadratic maps may be stated. Therefore we obtain the desired equality (2.6). Hence The results in the following proposition is derived from the Theorem 2.6.

Some inequalities for quadratic maps
The following simple result is of interest in itself as well: It is worth to mention that the right side of inequality (3.1) is an extension of the triangle inequality.  This completes the proof of Theorem 3.3.
Several authors discussed operator version of Bohr inequality (see for instance [4]). In the following, we give a uni…ed version of Bohr inequality.
where the last inequality follows from the fact that p q and so the proof is complete.
The following corollary is a natural consequence of the above result.  jA + Bj 2 + jA Bj 2 = 2 jAj 2 + jBj 2 : We have from (2.5), the following well known equality jA Bj 2 = 2jC Aj 2 + 2jC Bj The following generalization of parallelogram law is derived from inequality (2.4), which is obtained in [9, Theorem 4.1].