Characterizing spectra of closed operators through existence of slowly growing solutions of their Cauchy problems

Sen Huang

Studia Mathematica (1995)

  • Volume: 116, Issue: 1, page 23-41
  • ISSN: 0039-3223

Abstract

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Let A be a closed linear operator in a Banach space E. In the study of the nth-order abstract Cauchy problem u ( n ) ( t ) = A u ( t ) , t ∈ ℝ, one is led to considering the linear Volterra equation (AVE) u ( t ) = p ( t ) + A ʃ 0 t a ( t - s ) u ( s ) d s , t ∈ ℝ, where a ( · ) ∈ L l o c 1 ( ℝ ) and p(·) is a vector-valued polynomial of the form p ( t ) = ∑ j = 0 n 1 / ( j ! ) x j t j for some elements x j ∈ E . We describe the spectral properties of the operator A through the existence of slowly growing solutions of the (AVE). The main tool is the notion of Carleman spectrum of a vector-valued function. Moreover, an extension of a theorem of Pólya in complex analysis is obtained and applied to the individual “Ax = 0” and “Tx = x” problem.

How to cite

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Huang, Sen. "Characterizing spectra of closed operators through existence of slowly growing solutions of their Cauchy problems." Studia Mathematica 116.1 (1995): 23-41. <http://eudml.org/doc/216217>.

@article{Huang1995,
abstract = {Let A be a closed linear operator in a Banach space E. In the study of the nth-order abstract Cauchy problem $u^\{(n)\}(t) = Au(t)$, t ∈ ℝ, one is led to considering the linear Volterra equation (AVE) $u(t) = p(t) + A ʃ_\{0\}^\{t\} a(t-s)u(s)ds$, t ∈ ℝ, where $a(·) ∈ L_\{loc\}^\{1\}(ℝ)$ and p(·) is a vector-valued polynomial of the form $p(t) = ∑_\{j=0\}^n 1/(j!) x_j t^j$ for some elements $x_j ∈ E$. We describe the spectral properties of the operator A through the existence of slowly growing solutions of the (AVE). The main tool is the notion of Carleman spectrum of a vector-valued function. Moreover, an extension of a theorem of Pólya in complex analysis is obtained and applied to the individual “Ax = 0” and “Tx = x” problem.},
author = {Huang, Sen},
journal = {Studia Mathematica},
keywords = {Volterra equation; Carleman transform; spectrum; $C_0$-groups; closed operators; slowly growing solutions; Cauchy problems; growth of solutions; abstract initial value problems},
language = {eng},
number = {1},
pages = {23-41},
title = {Characterizing spectra of closed operators through existence of slowly growing solutions of their Cauchy problems},
url = {http://eudml.org/doc/216217},
volume = {116},
year = {1995},
}

TY - JOUR
AU - Huang, Sen
TI - Characterizing spectra of closed operators through existence of slowly growing solutions of their Cauchy problems
JO - Studia Mathematica
PY - 1995
VL - 116
IS - 1
SP - 23
EP - 41
AB - Let A be a closed linear operator in a Banach space E. In the study of the nth-order abstract Cauchy problem $u^{(n)}(t) = Au(t)$, t ∈ ℝ, one is led to considering the linear Volterra equation (AVE) $u(t) = p(t) + A ʃ_{0}^{t} a(t-s)u(s)ds$, t ∈ ℝ, where $a(·) ∈ L_{loc}^{1}(ℝ)$ and p(·) is a vector-valued polynomial of the form $p(t) = ∑_{j=0}^n 1/(j!) x_j t^j$ for some elements $x_j ∈ E$. We describe the spectral properties of the operator A through the existence of slowly growing solutions of the (AVE). The main tool is the notion of Carleman spectrum of a vector-valued function. Moreover, an extension of a theorem of Pólya in complex analysis is obtained and applied to the individual “Ax = 0” and “Tx = x” problem.
LA - eng
KW - Volterra equation; Carleman transform; spectrum; $C_0$-groups; closed operators; slowly growing solutions; Cauchy problems; growth of solutions; abstract initial value problems
UR - http://eudml.org/doc/216217
ER -

References

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