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

Studia Mathematica (1995)

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

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topHuang, 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

top- [A-P] W. Arendt and J. Prüss, Vector-valued Tauberian theorems and asymptotic behavior of linear Volterra equations, SIAM J. Math. Anal. 23 (1992), 412-448. Zbl0765.45009
- [A] A. Atzmon, Operators which are annihilated by analytic functions and invariant subspaces, Acta Math. 114 (1980), 27-63. Zbl0449.47007
- [A1] A. Atzmon, On the existence of hyperinvariant subspaces, J. Operator Theory 11 (1984), 3-40. Zbl0583.47009
- [B-M] A. Beurling and P. Malliavin, On Fourier transforms of measures with compact support, Acta Math. 107 (1962), 291-309. Zbl0127.32601
- [C] T. Carleman, L'intégrale de Fourier et Questions qui s'y Rattachent, Almqvist and Wiksel, Uppsala, 1944. Zbl0060.25504
- [D] Y. Domar, On the analytic transform of bounded linear functionals on certain Banach algebras, Studia Math. 53 (1975), 203-224. Zbl0272.46030
- [E] J. Esterle, Quasimultipliers, representations of ${H}^{\infty}$, and the closed ideal problem for commutative Banach algebras, in: Lecture Notes in Math. 975, Springer, 1983, 66-162.
- [H-P] E. Hille and R. S. Phillips, Functional Analysis and Semi-groups, Amer. Math. Soc. Colloq. Publ. 31, Amer. Math. Soc., Providence, R.I., 1958.
- [K] Y. Katznelson, An Introduction to Harmonic Analysis, Wiley, New York, 1968. Zbl0169.17902
- [L] N. Levinson, Gap and Density Theorems, Amer. Math. Soc. Colloq. Publ. 26, Amer. Math. Soc., New York, 1940. Zbl0145.08003
- [Ly] Yu. I. Lyubich, Introduction to the Theory of Banach Representations of Groups, Birkhäuser, 1988.
- [N] R. Nagel (ed.), One-Parameter Semigroups of Positive Operators, Lecture Notes in Math. 1184, Springer, 1986. Zbl0585.47030
- [N-H] R. Nagel and S. Huang, Spectral mapping theorems for ${C}_{0}$-groups satisfying non-quasianalytic growth conditions, Math. Nachr. 169 (1994), 207-218. Zbl0941.47036
- [Pr] J. Prüss, Linear Evolutionary Integral Equations in Banach Spaces and Applications, Birkhäuser, Basel 1993.
- [Sh] S. M. Shah, On the singularities of a class of functions on the unit circle, Bull. Amer. Math. Soc. 52 (1946), 1053-1056. Zbl0061.14603
- [V] A. Vretblad, Spectral analysis in weighted ${L}^{1}$ spaces on ℝ, Ark. Mat. 11 (1973), 109-138. Zbl0258.46047
- [Vu] Vũ Quôc Phóng, On the spectrum, complete trajectories and asymptotic stability of linear semi-dynamical systems, J. Differential Equations 115 (1993), 30-45.
- [Z] M. Zarrabi, Spectral synthesis and applications to ${C}_{0}$-groups, preprint, 1992.
- [Ze] J. Zemánek, On the Gelfand-Hille theorems, in: Functional Analysis and Operator Theory, J. Zemánek (ed.), Banach Center Publ. 30, Inst. Math., Polish Acad. Sci., Warszawa, 1994, 369-385. Zbl0822.47005

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