functions of selfadjoint operators

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Some Reverses of the Jensen Inequality for Functions of Selfadjoint Operators in Hilbert Spaces

Some Reverses of the Jensen Inequality for Functions of Selfadjoint Operators in Hilbert Spaces

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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Reverse Jensen’s type Trace Inequalities for Convex Functions of Selfadjoint Operators in Hilbert Spaces

Reverse Jensen’s type Trace Inequalities for Convex Functions of Selfadjoint Operators in Hilbert Spaces

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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Some Inequalities for Convex Functions of Selfadjoint Operators in Hilbert Spaces

Some Inequalities for Convex Functions of Selfadjoint Operators in Hilbert Spaces

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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Some New Grüss' Type Inequalities for Functions of Selfadjoint Operators in Hilbert Spaces

Some New Grüss' Type Inequalities for Functions of Selfadjoint Operators in Hilbert Spaces

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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Čebyšev's Type Inequalities for Functions of Selfadjoint Operators in Hilbert Spaces

Čebyšev's Type Inequalities for Functions of Selfadjoint Operators in Hilbert Spaces

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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Grüss' Type Inequalities for Functions of Selfadjoint Operators in Hilbert Spaces

Grüss' Type Inequalities for Functions of Selfadjoint Operators in Hilbert Spaces

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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Some Slater's Type Inequalities for Convex Functions of Selfadjoint Operators in Hilbert Spaces

Some Slater's Type Inequalities for Convex Functions of Selfadjoint Operators in Hilbert Spaces

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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Refinements of the Cauchy-Bunyakovsky-Schwarz Inequality for Functions of Selfadjoint Operators in Hilbert Spaces

Refinements of the Cauchy-Bunyakovsky-Schwarz Inequality for Functions of Selfadjoint Operators in Hilbert Spaces

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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Inequalities of Hermite-Hadamard type for functions of selfadjoint operators and matrices

Inequalities of Hermite-Hadamard type for functions of selfadjoint operators and matrices

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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Some Reverses of the Jensen Inequality for Functions of Selfadjoint Operators in Hilbert Spaces

Some Reverses of the Jensen Inequality for Functions of Selfadjoint Operators in Hilbert Spaces

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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Inequalities for the Čebyšev Functional of Two Functions of Selfadjoint Operators in Hilbert Spaces

Inequalities for the Čebyšev Functional of Two Functions of Selfadjoint Operators in Hilbert Spaces

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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Some Inequalities for the Čebyšev Functional of Two Functions of Selfadjoint Operators in Hilbert Spaces

Some Inequalities for the Čebyšev Functional of Two Functions of Selfadjoint Operators in Hilbert Spaces

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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Some Jensen's Type Inequalities for Twice Differentiable Functions of Selfadjoint Operators in Hilbert Spaces

Some Jensen's Type Inequalities for Twice Differentiable Functions of Selfadjoint Operators in Hilbert Spaces

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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Jensen’s type trace inequalities for convex functions of selfadjoint operators in Hilbert spaces

Jensen’s type trace inequalities for convex functions of selfadjoint operators in Hilbert spaces

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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Trace inequalities of Shisha-Mond type for operators in Hilbert spaces

Trace inequalities of Shisha-Mond type for operators in Hilbert spaces

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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Cebyšev’s type inequalities for positive linear maps of selfadjoint operators in Hilbert spaces

Cebyšev’s type inequalities for positive linear maps of selfadjoint operators in Hilbert spaces

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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Structure of the antieigenvectors of a strictly accretive operator

Structure of the antieigenvectors of a strictly accretive operator

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]

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Trace inequalities of Cassels and Grüss type for operators in Hilbert spaces

Trace inequalities of Cassels and Grüss type for operators in Hilbert spaces

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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Inequalities for the Riemann-Stieltjes integral of S-dominated integrators with applications. I

Inequalities for the Riemann-Stieltjes integral of S-dominated integrators with applications. I

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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KLEIN’S TRACE INEQUALITY AND SUPERQUADRATIC TRACE FUNCTIONS

KLEIN’S TRACE INEQUALITY AND SUPERQUADRATIC TRACE FUNCTIONS

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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