Symmetry of numerical range and semigroup generation of in nite dimensional Hamiltonian operators

This paper deals with the in nite dimensional Hamiltonian operator with unbounded entries. Using the core of its entries, we obtain the conditions under which the numerical range of such an operator is symmetric with respect to the imaginary axis. Based on the symmetry above, a necessary and sufficient condition for generating C 0 semigroups is further given.

PDF

___

[1] Azizov TY, Dijksma AA. On the boundness of Hamiltonian operators. Proc Amer Math Soc 2002; 131: 563-576.
[2] Alatancang, Huang JJ, Fan XY. Structure of the spectrum of in nite dimensional Hamiltonian operators. Science in China Series A: Mathematics 2008; 51: 915-924.
[3] Engel KJ, Nagel R. One Parameter Semigroups for Linear Evolution Equations. New York, NY, USA: Springer- Verlag, 2000.
[4] Gustafson K, Rao DKM. Numerical Range: the Field of Values of Linear Operators and Matrices. New York, NY, USA: Springer-Verlag, 1997.
[5] Jin GH, Chen A. Some basic properties of block operator matrices. arXiv: 1403.7732vl [math.FA] 30 Mar 2014.
[6] Kato T. Perturbation Theory for Linear Operators. Berlin, Germany: Springer-Verlag, 1995.
[7] Kurina GA. Invertibility of an operator appearing in the control theory for linear systems. Math Notes 2001; 70: 206-212.
[8] Kurina GA. Invertibility of nonnegatively Hamiltonian operators in a Hilbert space. Differential Equations 2001; 37: 880-882.
[9] Mugnolo D. Asymptotics of semigroups generated by operator matrices. Arab J Math 2014; 3: 419-435.
[10] Nagel R. Towards a matrix theory" for unbounded operator matrices. Math Z 1989; 201: 57-68.
[11] Tretter C. Spectral Theory of Block Operator Matrices and Applications. London, UK: Imperial College Press, 2008.
[12] Wu DY, Chen A. Spectral inclusion properties of the numerical range in a space with an inde nite metric. Linear Algebra and its Applications 2011; 435: 1131-1136.
[13] Zhong WX. A New Systematic Methodology for Theory of Elasticity. Dalian, LN, China: Dalian University of Technology Press, 1995 (in Chinese).

ISSN: 1300-0098
Yayın Aralığı: Yılda 6 Sayı
Yayıncı: TÜBİTAK

Arşiv