On certain spectral features inherent to scalar type spectral operators
DOI:
Keywords:
Spectral gap, scalar type spectral operatorAbstract
Important spectral features such as the emptiness of the residual spectrum, countability of the point spectrum, provided the space is separable, and a characterization of spectral gap at 0, known to hold for bounded scalar type spectral operators, are shown to naturally transfer to the unbounded case.References
William G. Bade, Unbounded spectral operators, Pacific J. Math. 4 (1954), 373-392. MathSciNet
J. M. Ball, Strongly continuous semigroups, weak solutions, and the variation of constants formula, Proc. Amer. Math. Soc. 63 (1977), no. 2, 370-373. MathSciNet
Ju. L. Daleckii and M. G. Krein, Stability of solutions of differential equations in Banach space, American Mathematical Society, Providence, R.I., 1974. MathSciNet
Nelson Dunford, Spectral operators, Pacific J. Math. 4 (1954), 321-354. MathSciNet
Nelson Dunford, A survey of the theory of spectral operators, Bull. Amer. Math. Soc. 64 (1958), 217-274. MathSciNet
Nelson Dunford and Jacob T. Schwartz, Linear operators. Part I, Wiley Classics Library, John Wiley & Sons, Inc., New York, 1988. MathSciNet
Nelson Dunford and Jacob T. Schwartz, Linear operators. Part II: Spectral theory. Self adjoint operators in Hilbert space, Interscience Publishers John Wiley & Sons, New York-London, 1963. MathSciNet
Nelson Dunford and Jacob T. Schwartz, Linear operators. Part III: Spectral operators, Interscience Publishers, John Wiley & Sons, Inc., New York-London-Sydney, 1971. MathSciNet
Klaus-Jochen Engel and Rainer Nagel, One-parameter semigroups for linear evolution equations, Graduate Texts in Mathematics, vol. 194, Springer-Verlag, New York, 2000. MathSciNet
S. R. Foguel, The relations between a spectral operator and its scalar part, Pacific J. Math. 8 (1958), 51-65. MathSciNet
Einar Hille and Ralph S. Phillips, Functional analysis and semi-groups, American Mathematical Society Colloquium Publications, vol. 31, American Mathematical Society, Providence, R. I., 1957. MathSciNet
Tosio Kato, Perturbation theory for linear operators, Classics in Mathematics, Springer-Verlag, Berlin, 1995. MathSciNet
V. S. Koroljuk and A. F. Turbin, Mathematical Foundations of Phase Consolidations of Complex Systems, Naukova Dumka, Kiev, 1978 (in Russian). MathSciNet
M. V. Markin, On the mean ergodicity of weak solutions of an abstract evolution equation, to appear
M. V. Markin, Behavior at infinity of bounded solutions of differential equations in a Banach space, Boundary value problems for operator-differential equations, Akad. Nauk Ukrainy, Inst. Mat., Kiev, 1991, pp. 56-63 (in {R}ussian). MathSciNet
M. V. Markin, Ergodicity of weak solutions of a first-order operator-differential equation, Akad. Nauk Ukrainy Inst. Mat. Preprint (1994), no. 10, 1-44. MathSciNet
M. V. Markin, A note on one decomposition of Banach spaces, Methods Funct. Anal. Topology 12 (2006), no. 3, 254-257. MathSciNet
A. I. Plesner, Spectral theory of linear operators, Nauka, Moscow, 1965 (in Russian). MathSciNet
John Wermer, Commuting spectral measures on Hilbert space, Pacific J. Math. 4 (1954), 355-361. MathSciNet