For a recent monograph devoted to various inequalities for **functions** of **selfadjoint** **operators**, see [4] and the references therein. For other results, see [11], [5] and [7]. The following result that provides an operator version for the Jensen inequality is due to Mond & Peˇ cari´ c [9] (see also [4, p. 5]):

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It is known, see for instance [45, p. 356-358], that if A and B are two commut- ing bounded **selfadjoint** **operators** on the complex Hilbert space H, then there exists a bounded **selfadjoint** operator T on H and two bounded **functions** ϕ and ψ such that A = ϕ (T ) and B = ψ (T ) . Moreover, if {E λ } is the spec-

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For a recent monograph devoted to various inequalities for **functions** of **selfadjoint** **operators**, see [6] and the references therein. For other results, see [13], [7] and [9]. The following result that provides an operator version for the Jensen inequality is due to Mond & Peˇ cari´ c [11] (see also [6, p. 5]):

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Motivated by the above results we investigate in this paper other Gr¨ uss’ type inequalities for **selfadjoint** **operators** in Hilbert spaces. Some of the obtained results improve the inequalities (2.3) and (2.4) derived from the Mond-Peˇ cari´ c inequality. Others provide different operator versions for the celebrated Gr¨ uss’ inequality men- tioned above. Examples for power **functions** and the logarithmic function are given as well.

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If either r > 0, s < 0 or r < 0, s > 0, then the reverse inequality holds in (2.8). Remark 1. We observe, from the proof of the above theorem that, if A and B are **selfadjoint** **operators** and Sp (A) , Sp (B) ⊆ [m, M ] , then for any continuous synchronous (asynchronous) **functions** f, g : [m, M ] −→ R we have the more general result

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Motivated by the above results we investigate in this paper other Gr¨ uss’ type inequalities for **selfadjoint** **operators** in Hilbert spaces. Some of the obtained results improve the inequalities (2.3) and (2.4) derived from the Mond-Peˇ cari´ c inequality. Others provide different operator versions for the celebrated Gr¨ uss’ inequality men- tioned above. Examples for power **functions** and the logarithmic function are given as well.

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For a recent monograph devoted to various inequalities for **functions** of **selfadjoint** **operators**, see [7] and the references therein. For other results, see [14], [8] and [10]. The following result that provides an operator version for the Jensen inequality is due to Mond & Peˇ cari´ c [12] (see also [7, p. 5]):

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Theorem 4 . Let (ϕ, ψ) be a (DEC)-pair of continuous **functions** on [0, ∞) × [0, ∞) . If A, B are **selfadjoint** **operators** on the Hilbert space (H ; h., .i) with Sp (A) , Sp (B) ⊆ [m, M ] for some scalars m < M and if f and g are continuous on [m, M ] and with values in [0, ∞) , then we have the inequality

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Let A be a **selfadjoint** linear operator on a complex Hilbert space (H; .,.) . The Gelfand map establishes a ∗ -isometrically isomorphism Φ between the set C (Sp (A)) of all continuous **functions** defined on the spectrum of A, denoted Sp (A) , and the C ∗ - algebra C ∗ ( A ) generated by A and the identity operator 1 H on H as follows (see for

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Let A be a **selfadjoint** linear operator on a complex Hilbert space H; ·, ·. The Gelfand map establishes a ∗-isometrically isomorphism Φ between the set CSpA of all continuous **functions** defined on the spectrum of A, denoted SpA, and the C ∗ -algebra C ∗ A generated by A and the identity operator 1 H on H as follows see e.g., 1, page 3:

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The following corollary of the above Theorem 4 can be useful for applications: Corollary 1. Let A and B be **selfadjoint** **operators** with Sp (A) ; Sp (B) [m; M ] for some real numbers m < M: If f : [m; M ] ! R is absolutely continuous then we have the Ostrowski type inequality for **selfadjoint** **operators**:

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Let A be a **selfadjoint** linear operator on a complex Hilbert space (H; h :; : i ) : The Gelfand map establishes a -isometrically isomorphism between the set C (Sp (A)) of all continuous **functions** de…ned on the spectrum of A; denoted Sp (A) ; an the C -algebra C (A) generated by A and the identity operator 1 H on H as

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Let A be a **selfadjoint** linear operator on a complex Hilbert space (H; h., .i) . The Gelfand map establishes a ∗-isometrically isomorphism Φ between the set C (Sp (A)) of all continuous **functions** defined on the spectrum of A, denoted Sp (A) , an the C ∗ -algebra C ∗ (A) generated by A and the identity operator 1 H on H as

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The following result provides reverses for the inequalities (2.2) and (2.3) above: Theorem 2.2. Let A be a **selfadjoint** operator on the Hilbert space H and assume that Sp (A) ⊆ [m, M ] for some scalars m, M with m < M. If f is a continuously differentiable convex function on [m, M ] and B ∈ B 2 (H ) \ {0} , then we have

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Dragomir, Reverse Jensen’s type trace inequalities for convex functions of selfadjoint operators in Hilbert spaces.. Fink, A treatise on Grüss’ inequality, Analytic and Geometric Inequal[r]

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Let A be a **selfadjoint** operator with Sp (A) ⊆ [m, M] for some real num- bers m < M. If f, g : [m, M ] −→ R are continuous, synchronous and one is convex while the other is concave on [m, M] , then by Jensen’s inequality for convex (concave) **functions** and by (6) we have

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STATIONARY VALUF OF ^I’ FOR SELFADJOINT OPERATORS For a selfadjoint operator A we can obtain the structure of the stationary vectors of RAf which is obviously equal to ^f in this case, i[r]

Proposition 1 Let A be a **selfadjoint** operator on the Hilbert space H and assume that Sp (A) ⊆ [m, M] for some scalars m, M with 0 < m < M. If f is a continuously differentiable convex function on [m, M] with f 0 (m) > 0 and P ∈ B 1 (H) \ { 0 } , P ≥ 0, then we have

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Proposition 2. Let A be a bounded **selfadjoint** operator on the Hilbert space H. Denote m := min {λ |λ ∈ Sp (A) } = min Sp (A) and M := := max {λ |λ ∈ Sp (A) } = max Sp (A) . If f : R → C is a continuous function on [m, M ] , then we have the inequality

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Prima facie, superquadraticity looks to be stronger than convexity, but if f takes neg- ative values then it may be considered weaker. On the other hand, non-negative sub- quadratic **functions** does not need to be concave. In other words, there exist subquadratic function which are convex. This fact helps us first to improve some results for convex **functions** and second to present some counterpart results concerning convex **functions**. Some known examples of superquadratic **functions** are power **functions**. For every p ≥ 2, the function f (t) = t p is superquadratic as well as convex. If 1 ≤ p ≤ 2, then f (t) = −t p

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