Characterization Of Linear Operators Research Articles

The inner kernel theorem for a certain Segal algebra

The Segal algebra {mathbf{S}}_{0}(G) is well defined for arbitrary locally compact Abelian Hausdorff (LCA) groups G. It is a Banach space that exhibits a kernel theorem similar to the well-known Schwartz kernel theorem. Specifically, we call this characterization of the continuous linear operators from {mathbf{S}}_{0}(G_1) to {mathbf{S}}'_{0}(G_2) by generalized functions in {mathbf{S}}'_{0}(G_1 times G_2) the “outer kernel theorem”. The main subject of this paper is to formulate what we call the “inner kernel theorem”. This is the characterization of those linear operators that have kernels in {mathbf{S}}_{0}(G_1 times G_2). Such operators are regularizing—in the sense that they map {mathbf{S}}'_{0}(G_1) into {mathbf{S}}_{0}(G_2) in a w^{*} to norm continuous manner. A detailed functional analytic treatment of these operators is given and applied to the case of general LCA groups. This is done without the use of Wilson bases, which have previously been employed for the case of elementary LCA groups. We apply our approach to describe natural laws of composition for operators that imitate those of linear mappings via matrix multiplications. Furthermore, we detail how these operators approximate general operators (in a weak form). As a concrete example, we derive the widespread statement of engineers and physicists that pure frequencies “integrate” to a Dirac delta distribution in a mathematically justifiable way.

Linear Operators That Preserve Arctic Ranks of Boolean Matrices

We study some properties of arctic rank of Boolean matrices. We compare the arctic rank with Boolean rank and term rank of a given Boolean matrix. Furthermore, we obtain some characterizations of linear operators that preserve arctic rank on Boolean matrix space.

Linear Maps that Preserve Any Two Term Ranks on Matrix Spaces over Anti-Negative Semirings

There are many characterizations of linear operators from various matrix spaces into themselves which preserve term rank. In this research, we characterize the linear maps which preserve any two term ranks between different matrix spaces over anti-negative semirings, which extends the previous results on characterizations of linear operators from some matrix spaces into themselves. That is, a linear map T from p × q matrix spaces into m × n matrix spaces preserves any two term ranks if and only if T preserves all term ranks if and only if T is a ( P , Q , B )-block map.

Characterizations of Zero-Term Rank Preservers of Matrices over Semirings

Let M(S) denote the set of all m£n matrices over a semiring S. For A 2 M(S), zero-term rank of A is the minimal number of lines (rows or columns) needed to cover all zero entries in A. In (5), the authors obtained that a linear operator on M(S) preserves zero-term rank if and only if it preserves zero-term ranks 0 and 1. In this paper, we obtain new characterizations of linear operators on M(S) that preserve zero-term rank. Conse- quently we obtain that a linear operator on M(S) preserves zero-term rank if and only if it preserves two consecutive zero-term ranks k and k + 1, where 0 • kminfm;ng i 1 if and only if it strongly preserves zero-term rank h, where 1 • hminfm;ng.

Term rank preservers of Boolean matrices

We obtain some characterizations of linear operators that preserve the term rank of Boolean matrices. That is, a linear operator over Boolean matrices preserves the term rank if and only if it preserves the term ranks 1 and k(≠1) if and only if it preserves the term ranks 2 and l(≠2). Other characterizations of term rank preservers are given.

On a characterization of linear operators

On a characterization of linear operators