Chansangiam, Pattrawut

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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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Integral inequalities of chebyshev type for continuous fields of Hermitian operators involving Tracy-Singh products and weighted pythagorean means
(2019-10-01)
Ploymukda, Arnon
;
Chansangiam, Pattrawut
In this paper, we establish several integral inequalities of Chebyshev type for bounded continuous fields of Hermitian operators concerning Tracy-Singh products and weighted Pythagorean means. The weighted Pythagorean means considered here are parametrization versions of three symmetric means: the arithmetic mean, the geometric mean, and the harmonic mean. Every continuous field considered here is parametrized by a locally compact Hausdorff space equipped with a finite Radon measure. Tracy-Singh product versions of the Chebyshev-Grüss inequality via oscillations are also obtained. Such integral inequalities reduce to discrete inequalities when the space is a finite space equipped with the counting measure. Moreover, our results include Chebyshev-type inequalities for tensor product of operators and Tracy-Singh/Kronecker products of matrices.
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Analytic properties of tracy-singh products for operator matrices
(2018-01-01)
Ploymukda, Arnon
;
Chansangiam, Pattrawut
;
Lewkeeratiyutkul, Wicharn
We show that the Tracy-Singh product of Hilbert space operators is continuous with respect to the operator-norm topology. The Tracy-Singh product of two nonzero operators is compact if and only if both factors are compact. We provide upper and lower bounds for certain Schatten p-norms of the Tracy-Singh product of operators. It turns out that this product is continuous with respect to the topologies on norm ideals of compact operators, trace class operators, and Hilbert-Schmidt class operators. Thus the Tracy-Singh product preserves such classes of operators.
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Khatri-Rao products and selection operators
(2019-08-01)
Ploymukda, Arnon
;
Chansangiam, Pattrawut
We develop further theory for Khatri-Rao products of Hilbert space operators in connections with selection operators. We provide two constructions related to selection operators. Then we establish certain identities and inequalities involving Khatri-Rao and Tracy-Singh products. As consequences, we obtain some characterizations for the mixed product property concerning the Khatri-Rao product of operators.
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Concavity and convexity of several maps involving Tracy-Singh products, Khatri-Rao products, and operator-monotone functions of positive operators
(2019-01-01)
Ploymukda, Arnon
;
Chansangiam, Pattrawut
We establish concavity and convexity theorems for a number of operator-valued maps involving Tracy-Singh products and Khatri-Rao products of positive operators on a Hilbert space. Operator means serve as useful tools for some convexity results. We also investigate certain maps dealing with positive operator-monotone functions. In this case, the concavity and the convexity of such maps are examined through suitable integral representations of the operator-monotone functions on the unit interval with respect to finite Borel measures.
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Algebraic and order properties of tracy-singh products for operator matrices
(2018-01-01)
Ploymukda, Arnon
;
Chansangiam, Pattrawut
;
Lewkeeratiyutkul, Wicharn
We generalize the tensor product for operators to the Tracy-Singh product for operator matrices acting on the direct sum of Hilbert spaces. This kind of operator product is compatible with algebraic operations and order relations for operators. It follows that this product preserves many structure properties of operators.
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Monotonicity of certain maps and Callebaut-type integral inequalities for continuous fields of operators
(2019-01-01)
Ploymukda, Arnon
;
Chansangiam, Pattrawut
In this paper, we investigate the monotonicity and the convexity of certain maps involving continuous fields of positive invertible operators, the Tracy-Singh product and the Khatri-Rao product. We obtain a number of integral inequalities of Callebaut-type for bounded continuous fields of operators involving certain kinds of operator products and weighted geometric means. Our results extend Callebaut-type inequalities for real numbers, matrices and operators.
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Algebraic, order, and analytic properties of Tracy-Singh sums for Hilbert space operators
(2019-07-01)
Ploymukda, Arnon
;
Chansangiam, Pattrawut
We introduce the Tracy-Singh sum for operators on a Hilbert space, generalizing both the Tracy-Singh sum for matrices and the tensor sum for operators. The Tracy-Singh sum is shown to be compatible with algebraic operations and order relations. Then we establish a binomial theorem involving Tracy-Singh sums, and its consequences. We also investigate continuity, convergence, and norm bounds for Tracy-Singh sums. Moreover, we derive operator identities involving Tracy-Singh sums and certain operator functions defined by power series.

Chansangiam, Pattrawut

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