Abstract
For an unbounded operator S on a Banach space the existence of invariant subspaces corresponding to its spectrum in the left and right half-plane is proved. The general assumption on S is the uniform boundedness of the resolvent along the imaginary axis. The projections associated with the invariant subspaces are bounded if S is strictly dichotomous, but may be unbounded in general. Explicit formulas for these projections in terms of resolvent integrals are derived and used to obtain perturbation theorems for dichotomy. All results apply, with certain simplifications, to bisectorial operators.
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References
Arendt W., Zamboni A.: Decomposing and twisting bisectorial operators. Studia Math. 197(3), 205–227 (2010)
Bart, H., Gohberg, I., Kaashoek, M.A.: Wiener-Hopf equations with symbols analytic in a strip. In: Constructive methods of Wiener-Hopf factorization. Oper. Theory Adv. Appl., vol. 21, pp. 39–74. Birkhäuser, Basel (1986)
Bart H., Gohberg I., Kaashoek M.A.: Wiener-Hopf factorization, inverse Fourier transforms and exponentially dichotomous operators. J. Funct. Anal. 68(1), 1–42 (1986)
Conway, J.B.: Functions of one complex variable. In: Graduate Texts in Mathematics, vol. 11, 2nd edn. Springer, New York (1978)
Dore G., Venni A.: Separation of two (possibly unbounded) components of the spectrum of a linear operator. Integral Equ. Oper. Theory 12(4), 470–485 (1989)
Haase, M. The functional calculus for sectorial operators. In: Operator Theory: Advances and Applications, vol. 169. Birkhäuser Verlag, Basel (2006)
Kaashoek M.A., Verduyn Lunel S.M.: An integrability condition on the resolvent for hyperbolicity of the semigroup. J. Differ. Equ. 112(2), 374–406 (1994)
Lions, J.-L., Magenes, E.: Non-homogeneous boundary value problems and applications, vol. I. Springer, New York (1972). Translated from the French by P. Kenneth, Die Grundlehren der mathematischen Wissenschaften, Band 181
Langer, H., Ran, A.C.M., van de Rotten, B.A.: Invariant subspaces of infinite dimensional Hamiltonians and solutions of the corresponding Riccati equations. In: Linear operators and matrices. Oper. Theory Adv. Appl., vol. 130, pp. 235–254. Birkhäuser, Basel (2002)
Langer, H., Tretter, C.: Diagonalization of certain block operator matrices and applications to Dirac operators. In: Operator theory and analysis (Amsterdam, 1997). Oper. Theory Adv. Appl., vol. 122, pp. 331–358. Birkhäuser, Basel (2001)
McIntosh, A., Yagi, A.: Operators of type \({\omega}\) without a bounded \({H_\infty}\) functional calculus. In: Miniconference on Operators in Analysis (Sydney, 1989). Proc. Centre Math. Anal. Austral. Nat. Univ., vol. 24, pp. 159–172. Austral. Nat. Univ., Canberra (1990)
Periago F., Straub B.: A functional calculus for almost sectorial operators and applications to abstract evolution equations. J. Evol. Equ. 2(1), 41–68 (2002)
Ran A.C.M., van der Mee C.: Perturbation results for exponentially dichotomous operators on general Banach spaces. J. Funct. Anal. 210(1), 193–213 (2004)
Sil′chenko Yu. T., Sobolevskiĭ P.E.: Solvability of the Cauchy problem for an evolution equation in a Banach space with a non-densely given operator coefficient which generates a semigroup with a singularity. Sibirsk. Mat. Zh. 27(4), 93–104, 214 (1986)
Tucsnak, M., Weiss, G.: Observation and control for operator semigroups. In: Birkhäuser Advanced Texts Birkhäuser Verlag, Basel (2009)
Tretter C., Wyss C.: Dichotomous Hamiltonians with unbounded entries and solutions of Riccati equations. J. Evol. Equ. 14(1), 121–153 (2014)
van der Mee, C.: Exponentially dichotomous operators and applications. In: Operator Theory: Advances and Applications, vol. 182. Linear Operators and Linear Systems. Birkhäuser Verlag, Basel (2008)
Wyss C., Jacob B., Zwart H.J.: Hamiltonians and Riccati equations for linear systems with unbounded control and observation operators. SIAM J. Control Optim. 50(3), 1518–1547 (2012)
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This work was partially supported by a research grant of the “Fachgruppe Mathematik und Informatik” at the University of Wuppertal and FAPA No. PI160322022 of the Facultad de las Ciencias of the Universidad de Los Andes.
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Winklmeier, M., Wyss, C. On the Spectral Decomposition of Dichotomous and Bisectorial Operators. Integr. Equ. Oper. Theory 82, 119–150 (2015). https://doi.org/10.1007/s00020-015-2218-5
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DOI: https://doi.org/10.1007/s00020-015-2218-5