Norm estimations, continuity, and compactness for Khatri-Rao products of Hilbert Space operators
DOI:
https://doi.org/10.11113/mjfas.v14n4.881Keywords:
Khatri-Rao product, compact operator, Schatten -class operator, trace-class operator, Hilbert-Schmidt class operatorAbstract
We provide estimations for the operator norm, the trace norm, and the Hilbert-Schmidt norm for Khatri-Rao products of Hilbert space operators. It follows that the Khatri-Rao product is continuous on norm ideals of compact operators equipped with the topologies induced by such norms. Moreover, if two operators are represented by block matrices in which each block is nonzero, then their Khatri-Rao product is compact if and only if both operators are compact. The Khatri-Rao product of two operators are trace-class (Hilbert-Schmidt class) if and only if each factor is trace-class (Hilbert-Schmidt class, respectively).
References
Khatri, C.G. and Rao, C.R., Solutions to some functional equations and their applications to characterization of probability distributions, Sankhya, 30(2), 167-180, 1968.
Al Zhour, Z.A. and Kilicman, A., Extension and generalization inequalities involving the Khatri-Rao product of several positive matrices, J. Inequal. Appl., Article ID 80878, 21 pages, 2006.
Civciv, H. and Taurkmen, R., On the bounds for norms of Khatri-Rao and Tracy-Singh products of Cauchy-Toeplitz matrices, Seluk J. Appl. Math., 6(2), 43-52, 2005.
Liu, S., Contributions to Matrix Calculus and Applications in Econometrics, PhD dissertation, Institute of Actuarial Science and Econometrics, University of Amsterdam, Amsterdam, 1995.
Rao, C.R. and Rao, M.B., Matrix Algebra and Its Applications to Statistics and Econometrics, World Scientific, 1998.
Van Loan, C.F., The ubiquitous Kronecker product, J. Comp. Appl. Math., 123, 85-100, 2000.
Kubrusly, C.S. and Vieira P.C.M., Convergence and decomposition for tensor products of Hilbert space operators, Oper. Matrices, 2(3), 407-416, 2008.
Schechter, M., On the spectra of operators on tensor products, J. Funct. Anal., 4(1), 95-99, 1969.
Zanni, J. and Kubrsly, C.S., A note on compactness of tensor product, Acta Math. Univ. Comenian., 84(1), 59-62, 2015.
Ploymukda, A. Chansangiam, P. and W. Lewkeeratiyutkul, Algebraic and order properties of Tracy-Singh products for operator matrices, J. Comput. Anal. Appl., 24(4), 656-664, 2018.
Ploymukda, A. Chansangiam, P. and W. Lewkeeratiyutkul, Analytic properties of Tracy-Singh products for operator matrices, Journal of Computational Analysis and Applications, 24(4), 665-674, 2018.
Ploymukda, A. and Chansangiam, P., Khatri-Rao products for operator matrices acting on the direct sum of Hilbert spaces, J. Math., 2016 (Article ID 8301709), 7 pages, 2016.
Ploymukda, A. and Chansangiam, P., Operator Inequalities Involving Khatri-Rao Sums and Moore-Penrose Inverses, Mal. J. Fund. Appl. Sci., 13(4), 742-746, 2017.
Bhatia, R. and Kittaneh, F., Norm inequalities for partitioned operators and an application, Math. Ann., 287(1), 719-726, 1990.
