On the spectrum of the operator which is a composition of integration and substitution

Ignat Domanov. "On the spectrum of the operator which is a composition of integration and substitution." Studia Mathematica 185.1 (2008): 49-65. <http://eudml.org/doc/285287>.

@article{IgnatDomanov2008,
abstract = {Let ϕ: [0,1] → [0,1] be a nondecreasing continuous function such that ϕ(x) > x for all x ∈ (0,1). Let the operator $V_\{ϕ\} : f(x) ↦ ∫_\{0\}^\{ϕ(x)\} f(t)dt$ be defined on L₂[0,1]. We prove that $V_\{ϕ\}$ has a finite number of nonzero eigenvalues if and only if ϕ(0) > 0 and ϕ(1-ε) = 1 for some 0 < ε < 1. Also, we show that the spectral trace of the operator $V_\{ϕ\}$ always equals 1.},
author = {Ignat Domanov},
journal = {Studia Mathematica},
keywords = {eigenvalue; integral operator; Fredholm determinant},
language = {eng},
number = {1},
pages = {49-65},
title = {On the spectrum of the operator which is a composition of integration and substitution},
url = {http://eudml.org/doc/285287},
volume = {185},
year = {2008},
}

TY - JOUR
AU - Ignat Domanov
TI - On the spectrum of the operator which is a composition of integration and substitution
JO - Studia Mathematica
PY - 2008
VL - 185
IS - 1
SP - 49
EP - 65
AB - Let ϕ: [0,1] → [0,1] be a nondecreasing continuous function such that ϕ(x) > x for all x ∈ (0,1). Let the operator $V_{ϕ} : f(x) ↦ ∫_{0}^{ϕ(x)} f(t)dt$ be defined on L₂[0,1]. We prove that $V_{ϕ}$ has a finite number of nonzero eigenvalues if and only if ϕ(0) > 0 and ϕ(1-ε) = 1 for some 0 < ε < 1. Also, we show that the spectral trace of the operator $V_{ϕ}$ always equals 1.
LA - eng
KW - eigenvalue; integral operator; Fredholm determinant
UR - http://eudml.org/doc/285287
ER -