Chansangiam, Pattrawut

Preferred name

Alternative Name

Chansangiam, P.

Main Affiliation

School of Science

pattrawut.ch@kmitl.ac.th

Now showing 1 - 7 of 7

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Weighted Lim’s geometric mean of positive invertible operators on a Hilbert space
(2020-03-01)
Ploymukda, Arnon
;
Chansangiam, Pattrawut
We generalize the weighted Lim’s geometric mean of positive definite matrices to positive invertible operators on a Hilbert space. This mean is defined via a certain bijection map and parametrized over Hermitian unitary operators. We derive an explicit formula of the weighted Lim’s geometric mean in terms of weighted metric/spectral geometric means. This kind of operator mean turns out to be a symmetric Lim-Pálfia weighted mean and satisfies the idempotency, the permutation invariance, the joint homogeneity, the self-duality, and the unitary invariance. Moreover, we obtain relations between weighted Lim geometric means and Tracy-Singh products via operator identities.
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Riccati equation and metric geometric means of positive semidefinite matrices involving semi-tensor products
(2023-01-01)
Chansangiam, Pattrawut
;
Ploymukda, Arnon
We investigate the Riccati matrix equation XA−1X = B in which the conventional matrix products are generalized to the semi-tensor products ⋉. When A and B are positive definite matrices satisfying the factor-dimension condition, this equation has a unique positive definite solution, which is defined to be the metric geometric mean of A and B. We show that this geometric mean is the maximum solution of the Riccati inequality. We then extend the notion of the metric geometric mean to positive semidefinite matrices by a continuity argument and investigate its algebraic properties, order properties and analytic properties. Moreover, we establish some equations and inequalities of metric geometric means for matrices involving cancellability, positive linear map and concavity. Our results generalize the conventional metric geometric means of matrices.
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Jensen’s type inequalities involving tracy-singh products, khatri-rao products, tracy-singh sums and khatri-rao sums
(2020-01-01)
Ploymukda, Arnon
;
Chansangiam, Pattrawut
In this paper, we establish a number of Jensen’s type inequalities for Hilbert space operators involving convex/concave functions, unital positive linear maps, and certain operator products and sums. The products and sums considered here include the Tracy-Singh product, the Khatri-Rao products, the Tracy-Singh sum, and the Khatri-Rao sum. Moreover, we generalize Jensen’s type inequalities in term of functional calculus of two-variable functions. In particular, we obtain Kantorovich-type operator inequalities involving the products and sums.
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Inequalities on weighted classical pythagorean means, Tracy-Singh products, and Khatri-Rao products for hermitian operators
(2020-03-01)
Ploymukda, Arnon
;
Chansangiam, Pattrawut
We establish a number of operator inequalities between three kinds of means, namely, weighted arithmetic/harmonic/geometric means, and two kinds of operator products, namely, Tracy-Singh products and Khatri-Rao products. In this study, we have validated the data under certain assumptions relying on (opposite) synchronization, comparability, and spectra of operators. The tensor product of operators, and Tracy-Singh/Khatri-Rao products of matrices as special cases are presented.
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Chebyshev-type integral inequalities for continuous fields of operators concerning Khatri-Rao products and synchronous properties
(2020-03-01)
Ploymukda, Arnon
;
Chansangiam, Pattrawut
We consider bounded continuous fields of self-adjoint operators which are parametrized by a locally compact Hausdorff space Ω equipped with a finite Radon measure μ. Under certain assumptions on synchronous Khatri-Rao property of the fields of operators, we obtain Chebyshev-type inequalities concerning Khatri-Rao products. We also establish Chebyshev-type inequalities involving Khatri-Rao products and weighted Pythagorean means under certain assumptions of synchronous monotone property of the fields of operators. The Pythagorean means considered here are three classical symmetric means: the geometric mean, the arithmetic mean, and the harmonic mean. Moreover, we derive the Chebyshev-Gruss integral inequality via oscillations when μ is a probability Radon measure. These integral inequalities can be reduced to discrete inequalities by setting Ω to be a finite space equipped with the counting measure. Our results provide analog results for matrices and integrable functions. Furthermore, our results include the results for tensor products of operators, and Khatri-Rao/Kronecker/Hadamard products of matrices, which have been not investigated in the literature.
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Metric geometric means with arbitrary weights of positive definite matrices involving semi-tensor products
(2023-01-01)
Ploymukda, Arnon
;
Chansangiam, Pattrawut
We extend the notion of classical metric geometric mean (MGM) for positive definite matrices of the same dimension to those of arbitrary dimensions, so that usual matrix products are replaced by semi-tensor products. When the weights are arbitrary real numbers, the weighted MGMs possess not only nice properties as in the classical case, but also affine change of parameters, exponential law, and cancellability. Moreover, when the weights belong to the unit interval, the weighted MGM has remarkable properties, namely, monotonicity and continuity from above. Then we apply a continuity argument to extend the weighted MGM to positive semidefinite matrices, here the weights belong to the unit interval. It turns out that this matrix mean posses rich algebraic, order, and analytic properties, such as, monotonicity, continuity from above, congruent invariance, permutation invariance, affine change of parameters, and exponential law. Furthermore, we investigate certain equations concerning weighted MGMs of positive definite matrices. It turns out that such equations are always uniquely solvable with explicit solutions. The notion of MGMs can be applied to solve certain symmetric word equations in two letters.
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Weighted spectral geometric means and matrix equations of positive definite matrices involving semi-tensor products
(2024-01-01)
Ploymukda, Arnon
;
Tansri, Kanjanaporn
;
Chansangiam, Pattrawut
We characterized weighted spectral geometric means (SGM) of positive definite matrices in terms of certain matrix equations involving metric geometric means (MGM) ♯ and semi-tensor products ⋉. Indeed, for each real number t and two positive definite matrices A and B of arbitrary sizes, the t-weighted SGM A ⬦<inf>t</inf> B of A and B is a unique positive solution X of the equation A−1 ♯ X = (A−1 ♯ B)t. We then established fundamental properties of the weighted SGMs based on MGMs. In addition, (A ♢<inf>1/2</inf> B)2 is positively similar to A ⋉ B and, thus, they have the same spectrum. Furthermore, we showed that certain equations concerning weighted SGMs and MGMs of positive definite matrices have a unique solution in terms of weighted SGMs. Our results included the classical weighted SGMs of matrices as a special case.

Chansangiam, Pattrawut

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