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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. [ABSTRACT FROM AUTHOR]

We investigate the Riccati matrix equation XA-1X = B in which the conventional matrix products are generalized to the semi-tensor products n. 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. [ABSTRACT FROM AUTHOR]

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 ge- ometric 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. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

We investigate relationship between Tracy-Singh products and certain classes of Hilbert space operators. We show that the normality, hyponor-mality, paranormality of operators are preserved by Tracy-Singh products.Operators of class-A type are also preserved under Tracy-Singh products.Moreover, we obtain necessary and suffIcient conditions for the Tracy-Singh product of two operators to be normal, quasinormal, (co)isometry,and unitary. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

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. Keywords: tensor product, Tracy-Singh product, operator matrix, compact operator, Schatten p-class operator [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

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–Grüss 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. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]