Operator Subadditivity of the 𝒟-Logarithmic Integral Transform for Positive Operators in Hilbert Spaces

For a continuous and positive function w (λ), λ> 0 and µ a positive measure on [0, ∞) we consider the following 𝒟-logarithmic integral transform 𝒟 ℒ o g ( w , μ ) ( T ) : = ∫ 0 ∞ w ( λ ) 1 n ( λ + T λ ) d μ ( λ ) , \mathcal{D}\mathcal{L}og\left( {w,\mu } \right)\left( T \right): = \int_0^\infty {w\left( \lambda \right)1{\rm{n}}\left( {{{\lambda + T} \over \lambda }} \right)d\mu \left( \lambda \right),} where the integral is assumed to exist for T a positive operator on a complex Hilbert space H. We show among others that, if A, B > 0 with BA + AB ≥ 0, then 𝒟 ℒ o g ( w , μ ) ( A ) + 𝒟 ℒ o g ( w , μ ) ( B ) ≥ 𝒟 ℒ o g ( w , μ ) ( A + B ) . \mathcal{D}\mathcal{L}og\left( {w,\mu } \right)\left( A \right) + \mathcal{D}\mathcal{L}og\left( {w,\mu } \right)\left( B \right) \ge \mathcal{D}\mathcal{L}og\left( {w,\mu } \right)\left( {A + B} \right). In particular we have 1 6 π 2 + di log ( A + B ) ≥ di log ( A ) + di log ( B ) , {1 \over 6}{\pi ^2} + {\rm{di}}\log \left( {A + B} \right) \ge {\rm{di}}\log \left( A \right) + {\rm{di}}\log \left( B \right), where the dilogarithmic function dilog : [0, ∞) → ℝ is defined by di log ( t ) : = ∫ 1 t 1 n s 1 - s d s , t ≥ 0. {\rm{di}}\log \left( t \right): = \int_1^t {{{1ns} \over {1 - s}}ds,} \,\,\,\,t \ge 0. Some examples for integral transform 𝒟Log (·, ·) related to the operator monotone functions are also provided.

On weighted means and their inequalities

In (Pal et al. in Linear Multilinear Algebra 64(12):2463–2473, 2016), Pal et al. introduced some weighted means and gave some related inequalities by using an approach for operator monotone functions. This paper discusses the construction of these weighted means in a simple and nice setting that immediately leads to the inequalities established there. The related operator version is here immediately deduced as well. According to our constructions of the means, we study all cases of the weighted means from three weighted arithmetic/geometric/harmonic means by the use of the concept such as stable and stabilizable means. Finally, the power symmetric means are studied and new weighted power means are given.

Reverse and improved inequalities for operator monotone functions

In this paper we provide several refinements and reverse operator inequalities for operator monotone functions in Hilbert spaces. We also obtain refinements and a reverse of Lowner-Heinz celebrated inequality that holds in the case of power function.

Lifshitz tail for continuous Anderson models driven by Lévy operators

We investigate the behavior near zero of the integrated density of states for random Schrödinger operators [Formula: see text] in [Formula: see text], [Formula: see text], where [Formula: see text] is a complete Bernstein function such that for some [Formula: see text], one has [Formula: see text], [Formula: see text], and [Formula: see text] is a random nonnegative alloy-type potential with compactly supported single site potential [Formula: see text]. We prove that there are constants [Formula: see text] such that [Formula: see text] where [Formula: see text] is the common cumulative distribution function of the lattice random variables [Formula: see text]. For typical examples of [Formula: see text] the constants [Formula: see text] and [Formula: see text] can be eliminated from the statement above. We combine probabilistic and analytic methods which allow to treat, in a unified manner, the large class of operator monotone functions of the Laplacian. This class includes both local and nonlocal kinetic terms such as the Laplace operator, its fractional powers, the quasi-relativistic Hamiltonians and many others.