2018/08/01 by Wasiur R. KhudaBukhsh, KhudaBukhsh, Wasiur R., Mark Sinzger +3
Computer Science · Mathematics · #60B05 #Advanced Topics in Algebra #FOS: Mathematics #Mathematical Analysis and Transform Methods #Matrix Theory and Algorithms #Probability (math.PR) #math.PR #msc:60B05
paper · pdf · doi:10.48550/arxiv.1808.00258
7 pages, no figures, technical note
openalex publication_date 2018/08/01 · openalex created_date 2018/08/22 · arxiv created 2018/08/31 · arxiv updated 2018/09/05 · openalex updated_date 2026/07/28
In this short technical note, we extend a recently published result [Liao2017] on the Perron root (or the spectral radius) of non-negative matrices to real-valued non-negative kernels on an arbitrary measurable space (E, E). To be precise, for any real-valued non-negative kernel K : E× E → ℝ, we prove that the spectral radius ρ(K) of K satisfies infx ∈ E ( R K \cdotp L (x) )/( R L (x) ) ≤ ρ(K) ≤ supx ∈ E ( R K\cdotp L (x) )/( R L (x) ), where L is an arbitrary Kernel on (E, E), which is integrable with respect to the left eigenmeasure of K and satisfies R L (x) >0 for all x ∈ E, and the operator R is defined by RL (x) :=∫E L(x, dy) .