On the core of a cone-preserving map

Abstract

This is the third of a sequence of papers in an attempt to study the Perron-Frobenius theory of a nonnegative matrix and its generalizations from the cone-theoretic viewpoint. Our main object of interest here is the core of a cone-preserving map. If <italic>A</italic> is an <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="n times n"> <mml:semantics> <mml:mrow> <mml:mi>n</mml:mi> <mml:mo> × </mml:mo> <mml:mi>n</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">n \times n</mml:annotation> </mml:semantics> </mml:math> </inline-formula> real matrix which leaves invariant a proper cone <italic>K</italic> in <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-struck upper R Superscript n"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">R</mml:mi> </mml:mrow> <mml:mi>n</mml:mi> </mml:msup> </mml:mrow> <mml:annotation encoding="application/x-tex">{\mathbb {R}^n}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , then by the core of <italic>A</italic> relative to <italic>K</italic> , denoted by <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="core Subscript upper K Baseline left-parenthesis upper A right-parenthesis"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mtext>core</mml:mtext> </mml:mrow> <mml:mi>K</mml:mi> </mml:msub> <mml:mo stretchy="false">(</mml:mo> <mml:mi>A</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">{\text {core}}_K(A)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , we mean the convex cone <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="intersection Underscript i equals 1 Overscript normal infinity Endscripts upper A Superscript i Baseline upper K"> <mml:semantics> <mml:mrow> <mml:msubsup> <mml:mo movablelimits="false"> ⋂ </mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>i</mml:mi> <mml:mo>=</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> <mml:mi mathvariant="normal"> ∞ </mml:mi> </mml:msubsup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msup> <mml:mi>A</mml:mi> <mml:mi>i</mml:mi> </mml:msup> </mml:mrow> <mml:mi>K</mml:mi> </mml:mrow> </mml:mrow> <mml:annotation encoding="application/x-tex">\bigcap \nolimits _{i = 1}^\infty {{A^i}K}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . It is shown that when <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="core Subscript upper K Baseline left-parenthesis upper A right-parenthesis"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mtext>core</mml:mtext> </mml:mrow> <mml:mi>K</mml:mi> </mml:msub> <mml:mo stretchy="false">(</mml:mo> <mml:mi>A</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">{\text {core}}_K(A)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is polyhedral, which is the case whenever <italic>K</italic> is, then <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="core Subscript upper K Baseline left-parenthesis upper A right-parenthesis"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mtext>core</mml:mtext> </mml:mrow> <mml:mi>K</mml:mi> </mml:msub> <mml:mo stretchy="false">(</mml:mo> <mml:mi>A</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">{\text {core}}_K(A)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is generated by the distinguished eigenvectors of positive powers of <italic>A</italic> . The important concept of a distinguished <italic>A</italic> -invariant face is introduced, which corresponds to the concept of a distinguished class in the nonnegative matrix case. We prove a significant theorem which describes a one-to-one correspondence between the distinguished <italic>A</italic> -invariant faces of <italic>K</italic> and the cycles of the permutation induced by <italic>A</italic> on the extreme rays of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="core Subscript upper K Baseline left-parenthesis upper A right-parenthesis"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mtext>core</mml:mtext> </mml:mrow> <mml:mi>K</mml:mi> </mml:msub> <mml:mo stretchy="false">(</mml:mo> <mml:mi>A</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">{\text {core}}_K(A)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , provided that the latter cone is nonzero, simplicial. By an interplay between cone-theoretic and graph-theoretic ideas, the extreme rays of the core of a nonnegative matrix are fully described. Characterizations of <italic>K</italic> -irreducibility or <italic>A</italic> -primitivity of <italic>A</italic> are also found in terms of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="core Subscript upper K Baseline left-parenthesis upper A right-parenthesis"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mrow class="MJX-TeXAtom-ORD">