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Product of weighted Hankel and weighted Toeplitz operators

Datt G., Porwal D. K.

Demonstratio Mathematica

2013

Vol. 46, nr 3

571--583

In this paper, we discuss some properties of the weighted Hankel operator H(...) and describe the conditions on which the weighted Hankel operator H(...) and weighted Toeplitz operator T(...), with (…) on the space H(...) being a sequence of positive numbers with (…), commute. It is also proved that if a non-zero weighted Hankel operator H(...) commutes with T(...), which is not a multiple of the identity, then H(...), for some (…).

Iterative Model Reduction of Large State-Space Systems

Huhtanen M.

International Journal of Applied Mathematics and Computer Science

1999

Vol. 9, no 2

245-263

There exist criteria for reducing the order of a large state-space model based on the accuracy of the approximate solutions to the Lyapunov matrix equations and the Hankel operator. Iterative solution techniques for the Lyapunov equations with the Arnoldi method have been proposed in a number of papers. In this paper we derive error bounds for approximations to the solutions to the Lyapunov equations as well as for the Hankel operator that indicate how to precondition while solving these equations iteratively.These bounds show that the error depends on three terms: First, on the amount of invariance of the constructed subspace for A, second, on the eigenvalues of A at least in proportion to 1/|Re l|, and third, under a certain condition on projectors P_l=W_lW_l* ,on the factor min_{X in C^{l x p}}|| B-( l I-A)W_lX|| for l on a path G surrounding the spectrum of A. Consequently, in order to compensate for those parts of the spectrum where 1/|Re l| is not small, preconditioning or an inverse iteration is needed to keep the sizes of the matrices used in construction of a reduced-order model moderate.