Abstract
The behavior of the domain, the range, the kernel and the multi-valued part of a rational function in a linear relation is analyzed, respectively. We give some basic properties of such linear relations, and we prove that the rational form of the spectral mapping theorem holds in terms of ascent, essential ascent, descent and essential descent.
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References
Arens, R.: Operational calculus of linear relations. Pac. J. Math. 11, 9–23 (1961)
Baskakov, A.G., Chernyshov, K.I.: Spectral analysis of linear relations and degenerate operator semigroups. Mat. Sb. 193(11), 3–42 (2002). (In Russian; translated in Sb. Math. 193:11--12 (2002), 1573–1610)
Chafai, E., Mnif, M.: Spectral mapping theorem for ascent, essential ascent, descent and essential descent spectrum of linear relations. Acta Math. Sci. 34(B4), 1212–1224 (2014)
Chafai, E., Mnif, M.: Descent and essential descent spectrum of linear relations. Extracta Math. 29(1–2), 117–139 (2014)
Chafai, E., Alvarez, T.: Ascent, essential ascent, descent and essential descent for a linear relation in a linear space. Filomat 31(3), 709–721 (2017)
Coddington, E.A.: Multivalued operators and boundary value problems. In: Analytic Theory of Differential Equations (Proceedings Conference, Western Michigan University, Kalamazoo, Michigan, 1970) Lecture Notes in Mathematics, vol. 183, pp. 2–8. Springer, Berlin (1971)
Cross, R.W.: Multivalued Linear Operators. Marcel Dekker, New York (1998)
Cross, R.W., Favini, A., Yakukov, Y.: Perturbation results for multivalued linear operators. Prog. Nonlinear Differ. Equ. Appl. 80, 111–130 (2011)
Fakhfakh, F., Mnif, M.: Perturbtion theory of lower semi-Browder multivalued linear operators. Publ. Math. Debr. 78(3/4), 595–606 (2011)
Favini, A., Yagi, A.: Multivalued linear operators and degenerate evolution equations. Ann. Mat. Pura. Appl. (4) 163, 353–384 (1993)
Gheorghe, D., Vasilescu, F.: Spectral theory for linear relations via linear operators. Pac. J. Math. 255(2), 349–372 (2012)
Grunenfelder, L., Omladic̆, M.: Ascent and descent for finite sequences of commuting endomorphisms. Pac. J. Math. 191, 95–121 (1999)
Harte, R.E.: Invertibility and Singularity for Bounded Linear Operators. Marcel Dekker Inc, New York (1988)
Kascic Jr., M.J.: Polynomial in linear relations. Pac. J. Math. 24(2), 291–295 (1968)
Ren, G., Shi, Y.: Defect indices and definiteness conditions for discrete linear Hamiltonian systems. Appl. Math. Comput. 218, 3414–3429 (2011)
Ren, G.: On the density of the minimal subspaces generated by discrete linear Hamiltonian systems. Appl. Math. Lett. 27, 1–5 (2014)
Riesz, F.: Uber lineare Functionalgleichungen. Acta Math. 41, 71–98 (1918)
Sandovici, A., de Snoo, H., Winkler, H.: Ascent, descent, nullity, defect, and related notions for linear relations in linear spaces. Linear Algebra Appl. 423, 456–497 (2007)
Sandovici, A.: Some basic properties of polynomials in a linear relation in linear spaces. Oper. Theory Adv. Appl. 175, 231–240 (2007)
Sandovici, A., de Snoo, H.: An index formula for the product of linear relations. Linear Alg. Appl. 431, 2160–2171 (2009)
Taylor, A.E.: Theorems on ascent, descent, nullity and defect of linear operators. Math. Ann. 163, 18–49 (1966)
Taylor, A.E., Lay, D.C.: Introduction to Functional Analysis, 2nd edn. Wiley, New York (1980)
Von Neumann, J.: Functorial Operators, II The Geometry of Orthogonal Spaces, in Annals of Mathematics Studies. Princeton University Press, Princeton, NJ (1950)
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Communicated by Fuad Kittaneh.
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Chafai, E. On a Rational Function in a Linear Relation. Bull. Malays. Math. Sci. Soc. 42, 2963–2984 (2019). https://doi.org/10.1007/s40840-018-0643-8
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DOI: https://doi.org/10.1007/s40840-018-0643-8