Specht Ratio Research Articles

Estimations of Karcher mean by Hadamard product

In this paper, we estimate the difference between the Hadamard product and the Karcher mean of n positive invertible operators on the Hilbert space in terms of the Specht ratio and the Kantorovich constant. Also, we improve the obtained inequalities in the case of n=2. Moreover, we give ratio inequalities of the operator power means by the Hadamard product.

Specht's ratio and logarithmic mean in time scale dynamic inequalities and their retrospective variants

In this research article, we investigate reverse Radon's inequality, reverse Bergström's inequality, the reverse weighted power mean inequality, reverse Schlömilch's inequality, reverse Bernoulli's inequality and reverse Lyapunov's inequality with Specht's ratio on time scales. We also present reverse Rogers--Holder's inequality with logarithmic mean and Specht's ratio on time scales. The time scale dynamic inequalities unify and extend some continuous inequalities and their corresponding discrete and quantum versions.

MULTIDIMENSIONAL REVERSE HÖLDER INEQUALITY ON TIME SCALES

This paper develops the study of Hölder's inequality with weighted functions where we can establish some new multidimensional reverse Hölder inequality on time scale measure spaces. Our results will be proved by using the definition and some properties of a Specht's ratio function. We will prove these inequalities in a time scale calculus to avoid proving them twice once in the continuous case and the second in the discrete case.

Specht Oranına Göre Berezin Sayı Eşitsizlikleri

Smooth functions are associated with operators on Hilbert spaces of analytic functions through the Berezin transform. The Berezin symbol and the Berezin number of an operator A on the Hilbert functional space H(Ω) over some set Ω with the reproducing kernel are defined, respectively, by A ̃(μ)=〈A K_μ/K_μ ,K_μ/K_μ 〉,μ∈Ω and ber(A)=sup┬(μ∈Ω)⁡|A ̃(μ)|. By using this bounded function A ̃, we present some new Berezin number inequalities of Hilbert functional space operators. Some inequalities with respect to Specht's ratio are improved and generalized. Using these modifications, we also establish various new inequalities for the Berezin radius and Berezin norm of operators.

The quantum ♮α-Rényi divergence of real order

There are many proposals for defining a quantum version of the Rényi divergence on the differentiable manifold P of positive definite matrices. The geodesic connecting two points on P is a weighted matrix geometric mean, and its tangent vector is the relative operator entropy. It is a natural idea to use the relative operator entropy to describe the difference between two points on P. The quantum version of the classical results of Rényi divergence does not always hold due to its non-commutativity. Due to this difficulty, we use the upper bound and the lower one of the spectrum of positive definite matrices to estimate the order relation between quantum divergences, which is called the Mond Pečarić method in operator theory. In this paper, we show fundamental properties of the ♮α-Rényi divergence of real order α∈R, including monotonicity of order, the relation of max- and min-divergence, data processing inequality, bound estimates, and convexity. We also estimate the difference without the α–z-Rényi divergence and ♮α-Rényi divergence in terms of the generalized Kantorovich constant and the Specht ratio.

An extension of the operator Kantorovich inequality

In this paper, we present an extension of Kantorovich inequality for two operators on a Hilbert space. Also, the multiple version and a related inequality for positive linear maps are obtained. Moreover, we introduce the concept of Specht's ratio and improve some inequalities related to Specht's ratio.

VARIANTS OF ANDO–HIAI TYPE INEQUALITIES FOR DEFORMED MEANS AND APPLICATIONS

AbstractFor an n-tuple of positive invertible operators on a Hilbert space, we present some variants of Ando–Hiai type inequalities for deformed means from an n-variable operator mean by an operator mean, which is related to the information monotonicity of a certain unital positive linear map. As an application, we investigate the monotonicity of the power mean from the deformed mean in terms of the generalized Kantorovich constants under the operator order. Moreover, we improve the norm inequality for the operator power means related to the Log-Euclidean mean in terms of the Specht ratio.

Some reverse Hölder inequalities with Specht's ratio on time scales

Some reverse Hölder inequalities with Specht's ratio on time scales

Generalized Pólya–Szegö type inequalities for some non-commutative geometric means

In this paper, we shall show generalized Pólya–Szegö type inequalities of n positive invertible operators on a Hilbert space for any integer n⩾3 in terms of the following two typical non-commutative geometric means, that is, one is the higher order weighted geometric mean due to Lawson–Lim which is an extension of the Ando–Li–Mathias geometric mean, and the other is the weighted chaotic geometric mean. Among others, the Specht ratio plays an important role in our discussion, which is the upper bound of a ratio type reverse of the weighted arithmetic–geometric mean inequality.

Alternative reverse inequalities for Young's inequality

Two reverse inequalities for Young's inequality were shown by M. Tominaga, using Specht ratio. In this short paper, we show alternative reverse inequalities for Young's inequality without using Specht ratio.

Refined Young inequality with Kantorovich constant

The Specht ratio S(h) is the optimal constant in the reverse of the arithmetic-geometric mean inequality, i.e., if 0 0, μ ∈ (0,1) ,w hereh = b and r = min{μ,1 − μ}. In this paper, we improve it by virtue of the Kantorovich constant, utilizing the refined scalar Young inequality we establish a weighted arithmetic-geometric-harmonic mean inequality for two positive operators. In the remainder of this work we focus on extending the refined weighted arithmetic-harmonic mean inequality to an operator version for another type of improvement.

A sharp converse inequality of three weightedv arithmetic and geometric means of positive definite operators

In this paper we consider three-weighted arithmetic and geometric means of positive definite operators constructed by the symmetrization method that appeared in the definition of Ando-Li-Mathias's geometric mean of several positive definite operators and we establish a converse inequality of three-weighed arithmetic-geometric means via Specht ratio.

A converse inequality of higher order weighted arithmetic and geometric means of positive definite operators

In this paper we consider weighted arithmetic and geometric means of higher orders constructed by the symmetrization method appeared in Ando–Li–Mathias’s definition of multi-variable geometric means and the arithmetic–geometric mean inequality of higher order weighted version. We establish a converse inequality of higher order weighed arithmetic and geometric means via Specht ratio.

Reverse inequality to Golden–Thompson type inequalities: Comparison of [formula omitted] and [formula omitted

Let A and B be n-by- n Hermitian matrices. Then Tr e A e B ⩽ S ( α ) Tr e A + B where α is the condition number of e A and S ( t ) is the Specht ratio of the reverse arithmetic-geometric mean inequality. It is a sharp reverse result to the Golden–Thompson inequality. This can be extended to each eigenvalue. Equivalently there exists a unitary V such that e A / 2 e B e A / 2 ⩽ S ( α ) V e A + B V ∗ . We also show that there exists a unitary W such that W e A + B W ∗ ⩽ S ( α ) e A / 2 e B e A / 2 .

Kantorovich type operator inequalities for Furuta inequality

In this paper, we shall present Kantorovich type operator inequalities for Furuta inequality related to the usual order and the chaotic one in terms of a generalized Kantorovich constant, a generalized condition number and the Specht ratio, in which we use variants of the grand Furuta inequality. Mathematics subject classification (2000): 47A63.

Two reverse inequalities associated with Tsallis relative operator entropy via generalized Kantorovich constant and their applications

Recently Tsallis relative operator entropy T p ( A∣ B) and Tsallis relative entropy D p ( A∥ B) are discussed by Furuichi–Yanagi–Kuriyama. We shall show two reverse inequalities involving Tsallis relative operator entropy T p ( A∣ B) via generalized Kantorovich constant K( p). As some applications of two reverse inequalities, we shall show two trace reverse inequalities involving −Tr[ T p ( A∣ B)] and D p ( A∥ B) and also a known reverse trace inequality involving the relative operator entropy S ^ ( A | B ) by Fujii–Kamei and the Umegaki relative entropy S( A, B) is shown as a simple corollary. We show the following result: Let A and B be strictly positive operators on a Hilbert space H such that M 1 I ⩾ A ⩾ m 1 I > 0 and M 2 I ⩾ B ⩾ m 2 I > 0. Put m = m 2 M 1 , M = M 2 m 1 , h = M m = M 1 M 2 m 1 m 2 > 1 and p ∈ (0, 1]. Let Φ be normalized positive linear map on B( H). Then the following inequalities hold: (i) 1 - K ( p ) p Φ ( A ) ♯ p Φ ( B ) + Φ ( T p ( A | B ) ) ⩾ T p ( Φ ( A ) | Φ ( B ) ) ⩾ Φ ( T p ( A | B ) ) and (ii) F ( p ) Φ ( A ) + Φ ( T p ( A | B ) ) ⩾ T p ( Φ ( A ) | Φ ( B ) ) ⩾ Φ ( T p ( A | B ) ) , where K( p) is the generalized Kantorovich constant defined by K ( p ) = ( h p - h ) ( p - 1 ) ( h - 1 ) ( p - 1 ) p ( h p - 1 ) ( h p - h ) p and K( p) ∈ (0, 1] and F ( p ) = m p p h p - h h - 1 1 - K ( p ) 1 p - 1 ⩾ 0 . In addition, let A and B be strictly positive definite matrices, (iii) 1 - K ( p ) p ( Tr [ A ] ) 1 - p ( Tr [ B ] ) p + D p ( A ‖ B ) ⩾ - Tr [ T p ( A | B ) ] ⩾ D p ( A ‖ B ) and (iv) F ( p ) Tr [ A ] + D p ( A ‖ B ) ⩾ - Tr [ T p ( A | B ) ] ⩾ D p ( A ‖ B ) . In particular, both (iii) and (iv) yield the following known result: log S ( 1 ) Tr [ A ] + S ( A , B ) ⩾ - Tr [ S ^ ( A | B ) ] ⩾ S ( A , B ) , where S ( 1 ) = h 1 h - 1 e log h 1 h - 1 is said to be the Specht ratio and S(1) > 1.

Specht's ratio and logarithmic mean in the Young inequality

Specht's ratio and logarithmic mean in the Young inequality

Kantorovich type operator inequalities via the Specht ratio

As a converse of the arithmetic and geometric mean inequality, Specht gave the ratio of the arithmetic one by the geometric one in 1960. We can reap the rich harvest of the Specht ratio in operator theory. In this paper, we shall present other characterizations of the chaotic order and the usual one associated with Kantorovich type inequalities via the Specht ratio. Among others, as an application of the grand Furuta inequality, we show that if A and B are positive operators and k⩾A⩾ 1 k for some k⩾1, then A⩾ B is equivalent to S k(2(p−1)s) 2 s A p⩾B p holds for all p⩾1, s⩾1 such that p−1⩾ 1 s , where the Specht ratio S k ( r) is defined for each r>0 as S k(r)= (k r−1)k r k r−1 re logk (k>0,k≠1) and S 1(r)=1.

Operator inequalities associated with Alog A via Specht ratio

An operator means a bounded linear operator on a Hilbert space H. We obtained the basic property between Specht ratio S(1) and generalized Kantorovich constant K( p) in [Math. Inequal. Appl., in press], that is, Specht ratio S(1) can be expressed by generalized Kantorovich constant K( p): S(1)=e K ′(1) . We shall investigate several product type and difference type inequalities associated with Alog A by applying this basic property to several Kantorovich type inequalities.

Reverse inequalities on chaotically geometric mean via Specht ratio

Reverse inequalities on chaotically geometric mean via Specht ratio