Transfer functions and operator theory

Abstract

The notion of a transfer function from systems theory, serves as an important tool in dealing with certain problems in operator and matrix theory. This is illustrated with a variety of material taken from existing literature. A striking feature of the results is their high degree of explicitness.

Inner functions of matrix argument and conjugacy classes in unitary groups

“…But the term 'characteristic function' is overburdened (it has two other common meanings: indicator functions and Fourier transforms of measures in probability). There is another term transfer function (see, e.g., [5]) for (1.4), which came from system theory. ⊠ Remark.…”

Factorization of Matrix and Operator Functions: The State Space Method

“…However, the problem of simultaneous reduction of two matrices into complementary triangular forms, which is a core element in constructing such factorizations, appears in [14] only implicitly. This second problem was introduced and studied in [19]; see also the survey paper [10]. Section 10.1 is based on [19].…”

“…Sections 10.2 and 10.3 originate from Sections 1.1, 1.3 and 3.2 of [14]; see also [39], the references therein, and [104] for earlier material in this direction. Theorem 10.2, which appears implicitly in the proof of Theorem 1.6 in [14], is taken from Section 7.2 in the survey paper [10] (see also Section 1 in [19], where an alternative proof using lower-upper factorization is given). For an extension of the first part of Theorem 10.2 to commuting families of matrices, see Theorem 1.4 in [19].…”

Upper triangularization of matrices by lower triangular similarities

“…In 26 particular A is diagonable, which implies that W(X) does admit complete factorization (see [2, Theorems 1.6 and 3.4]; cf. also [5], [13] and [1]). The following example shows that in such a factorization the order of the poles cannot always be chosen at will.…”

“…1 and note that A-BC (being lower triangular and non-diagonable) does not admit upper triangularization by a lower triangular similarity [1. …”