# On a generalization of Lumer-Phillips' theorem for dissipative operators in a Banach space

Studia Mathematica (1998)

- Volume: 130, Issue: 1, page 1-7
- ISSN: 0039-3223

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topDrissi, Driss. "On a generalization of Lumer-Phillips' theorem for dissipative operators in a Banach space." Studia Mathematica 130.1 (1998): 1-7. <http://eudml.org/doc/216538>.

@article{Drissi1998,

abstract = {Using [1], which is a local generalization of Gelfand's result for powerbounded operators, we first give a quantitative local extension of Lumer-Philips' result that states conditions under which a quasi-nilpotent dissipative operator vanishes. Secondly, we also improve Lumer-Phillips' theorem on strongly continuous semigroups of contraction operators.},

author = {Drissi, Driss},

journal = {Studia Mathematica},

keywords = {dissipative operators; local spectrum; semigroup of contraction operators; dissipative operator; Lumer-Phillips and Gelfand-Hille theorems},

language = {eng},

number = {1},

pages = {1-7},

title = {On a generalization of Lumer-Phillips' theorem for dissipative operators in a Banach space},

url = {http://eudml.org/doc/216538},

volume = {130},

year = {1998},

}

TY - JOUR

AU - Drissi, Driss

TI - On a generalization of Lumer-Phillips' theorem for dissipative operators in a Banach space

JO - Studia Mathematica

PY - 1998

VL - 130

IS - 1

SP - 1

EP - 7

AB - Using [1], which is a local generalization of Gelfand's result for powerbounded operators, we first give a quantitative local extension of Lumer-Philips' result that states conditions under which a quasi-nilpotent dissipative operator vanishes. Secondly, we also improve Lumer-Phillips' theorem on strongly continuous semigroups of contraction operators.

LA - eng

KW - dissipative operators; local spectrum; semigroup of contraction operators; dissipative operator; Lumer-Phillips and Gelfand-Hille theorems

UR - http://eudml.org/doc/216538

ER -

## References

top- [1] B. Aupetit and D. Drissi, Some spectral inequalities involving generalized scalar operators, Studia Math. 109 (1994), 51-66. Zbl0829.47002
- [2] B. Aupetit and D. Drissi, Local spectrum theory and subharmonicity, Proc. Edinburgh Math. Soc. 39 (1996), 571-579)
- [3] F. F. Bonsall and J. Duncan, Numerical Ranges I and II, London Math. Soc. Lecture Note Ser. 2 and 10, Cambridge Univ. Press, 1971 and 1973.
- [4] I. Gelfand, Zur Theorie der Charaktere der Abelschen topologischen Gruppen, Mat. Sb. 9 (1941), 49-50. Zbl67.0407.02
- [5] E. Hille, On the theory of characters of groups and semigroups in normed vector rings, Proc. Nat. Acad. Sci. U.S.A. 30 (1944), 58-60. Zbl0061.25305
- [6] E. Hille and R. S. Phillips, Functional Analysis and Semi-Groups, Amer. Math. Soc., Providence, 1957. Zbl0078.10004
- [7] G. Lumer and R. S. Phillips, Dissipative operators in Banach space, Pacific J. Math. 11 (1961), 679-698. Zbl0101.09503
- [8] J. Zemánek, On the Gelfand-Hille theorems, in: Banach Center Publ. 30, Inst. of Math., Polish Acad. Sci., 1994, 369-385. Zbl0822.47005

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