By Hille E.

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T I / x; xi D 1h j 4 j2 jh2 hT x; xi . 63). Corollary 46. 63) on taking the infimum and the supremum over x 2 H; kxk D 1; respectively. H / and ; 2 K. Remark 47. 71) for each x 2 H; kxk D 1: The following identity that links the norm with the inner product also holds. Lemma 48. H / and ; 2 K. I T / x; x D j j2 4 2 Proof. 72). 3 Some Associated Functionals 23 Corollary 49. 75) We recall that a bounded linear operator T W H ! H is called strongly c 2 accretive (with c ¤ 0) if Re hT y; yi c 2 for each y 2 H; kyk D 1: For c D 0; the operator is called accretive.

I C tT / t 1 for any t > 0; which implies, by letting t ! 108) is obtained. H / : The following inequalities may be stated as well: Theorem 67 (Dragomir [18], 2008). 112) I/ : respectively. Proof. 111) and the one from the first branch in the second. 111). 36 2 Inequalities for One Operator The second inequality may be proven in a similar way. The details are omitted. i / defined on Banach spaces are not commutative. H / : Corollary 68. 116) Proof. 117) for any t > 0: Taking the limit over t ! 115).

84). 84). 89) above in which we take the infimum over x 2 H; kxk D 1: Corollary 56. H / and ; 2 K. 91) respectively. T / under various assumptions for the operator T . In our recent paper [13] several such inequalities have been obtained. In order to establish some new results that would complement the inequalities outlined in the Introduction, we need the following lemma which provides two simple identities of interest: Lemma 57 (Dragomir [17], 2007). 92) for each x 2 H; kxk D 1: Proof. The first identity is obvious by direct calculation.