Characteristic matrices and spectral functions of first order symmetric systems with maximal deficiency index of the minimal relation
DOI:
Keywords:
First-order symmetric system, characteristic matrix, spectral function, pseudospectral function, Fourier transformAbstract
Let $H$ be a finite dimensional Hilbert space and let $[H]$ be the set of all li ear operators in $H$. We consider first-order symmetric system $J y'-B(t)y=\Lambda(t) f(t)$ with $[H]$-valued coefficients defined on an interval $[a,b) $ with the regular endpoint $a$. It is assumed that the corresponding minimal relation $T_{\rm min}$ has maximally possible deficiency index $n_+(T_{\rm min})=\dim H$. The main result is a parametrization of all characteristic matrices and pseudospectral (spectral) functions of a given system by means of a Nevanlinna type boundary parameter $\tau$. Similar parametrization for regular systems has earlier been obtained by Langer and Textorius. We also show that the coefficients of the parametrization form the matrix $W(\lambda)$ with the properties similar to those of the resolvent matrix in the extension theory of symmetric operators.References
N. I. Akhiezer, The Classical Moment Problem, Oliver and Boyd, Edinburgh-London, 1965.
Sergio Albeverio, Mark Malamud, Vadim Mogilevskii, On Titchmarsh-Weyl functions and eigenfunction expansions of first-order symmetric systems, Integral Equations Operator Theory 77 (2013), no. 3, 303-354. MathSciNet CrossRef
Damir Z. Arov, Harry Dym, Bitangential direct and inverse problems for systems of integral and differential equations, Cambridge University Press, Cambridge, 2012. MathSciNet CrossRef
F. V. Atkinson, Discrete and continuous boundary problems, Academic Press, New York-London, 1964. MathSciNet
V. M. Bruk, Linear relations in a space of vector-valued functions, Mat. Zametki 24 (1978), no. 4, 499-511, 590. MathSciNet
V. A. Derkach, S. Hassi, M. M. Malamud, H. S. V. Snoo de, Generalized resolvents of symmetric operators and admissibility, Methods Funct. Anal. Topology 6 (2000), no. 3, 24-55. MathSciNet
V. A. Derkach, M. M. Malamud, Generalized resolvents and the boundary value problems for Hermitian operators with gaps, J. Funct. Anal. 95 (1991), no. 1, 1-95. MathSciNet CrossRef
Aad Dijksma, Heinz Langer, Henk Snoo de, Eigenvalues and pole functions of Hamiltonian systems with eigenvalue depending boundary conditions, Math. Nachr. 161 (1993), 107-154. MathSciNet CrossRef
Nelson Dunford, Jacob T. Schwartz, Linear operators. Part II, John Wiley & Sons, Inc., New York, 1988. MathSciNet
Charles T. Fulton, Parametrizations of Titchmarshs $m(lambda )$-functions in the limit circle case, Trans. Amer. Math. Soc. 229 (1977), 51-63. MathSciNet
I. C. Gohberg, M. G. Krein, Theory and applications of Volterra operators in Hilbert space, American Mathematical Society, Providence, R.I., 1970. MathSciNet
M. L. Gorbacuk, On spectral functions of a second order differential equation with operator coefficients, Ukrain. Mat. v Z. 18 (1966), no. 2, 3-21. MathSciNet
Granichnye zadachi dlya differentsialno-operatornykh uravnenii, Akad. Nauk Ukrainy, Inst. Mat., Kiev, 1991. MathSciNet
D. B. Hinton, J. K. Shaw, Parameterization of the $M(lambda )$ function for a Hamiltonian system of limit circle type, Proc. Roy. Soc. Edinburgh Sect. A 93 (1982/83), no. 3-4, 349-360. MathSciNet CrossRef
I. S. Kats, Linear relations generated by a canonical differential equation of dimension 2, and eigenfunction expansions, Algebra i Analiz 14 (2002), no. 3, 86-120. MathSciNet
F. Atkinson, Diskretnye i nepreryvnye granichnye zadachi, Izdat. ``Mir'', Moscow, 1968. MathSciNet
A. M. Khol′kin, Description of selfadjoint extensions of differential operators of arbitrary order on an infinite interval in the absolutely indeterminate case, Teor. Funktsiui Funktsional. Anal. i Prilozhen. (1985), no. 44, 112-122. MathSciNet CrossRef
V. I. Kogan, F. S. Rofe-Beketov, On square-integrable solutions of symmetric systems of differential equations of arbitrary order, Proc. Roy. Soc. Edinburgh Sect. A 74 (1974/75), 5-40 (1976). MathSciNet
M. G. Krein, Sh. N. Saakyan, Some new results in the theory of resolvents of Hermitian operators, Soviet Math. Dokl. 7 (1966), 1086-1089.
H. Langer, B. Textorius, $L$-resolvent matrices of symmetric linear relations with equal defect numbers; applications to canonical differential relations, Integral Equations Operator Theory 5 (1982), no. 2, 208-243. MathSciNet CrossRef
H. Langer, B. Textorius, Spectral functions of a symmetric linear relation with a directing mapping. I, Proc. Roy. Soc. Edinburgh Sect. A 97 (1984), 165-176. MathSciNet CrossRef
H. Langer, B. Textorius, Spectral functions of a symmetric linear relation with a directing mapping. II, Proc. Roy. Soc. Edinburgh Sect. A 101 (1985), no. 1-2, 111-124. MathSciNet CrossRef
Matthias Lesch, Mark Malamud, On the deficiency indices and self-adjointness of symmetric Hamiltonian systems, J. Differential Equations 189 (2003), no. 2, 556-615. MathSciNet CrossRef
M. M. Malamud, On a formula for the generalized resolvents of a non-densely defined Hermitian operator, Ukrain. Mat. Zh. 44 (1992), no. 12, 1658-1688. MathSciNet CrossRef
Vadim Mogilevskii, Boundary triplets and Krein type resolvent formula for symmetric operators with unequal defect numbers, Methods Funct. Anal. Topology 12 (2006), no. 3, 258-280. MathSciNet
V. I. Mogilevskii, On exit space extensions of symmetric operators with applications to first order symmetric systems, Methods Funct. Anal. Topology 19 (2013), no. 3, 268-292. MathSciNet
Tim Mogilevskii, On generalized resolvents and characteristic matrices of first-order symmetric systems, Methods Funct. Anal. Topology 20 (2014), no. 4, 328-348. MathSciNet
V. I. Mogilevskii, On spectral and pseudospectral functions of first-order symmetric systems; arXiv:1407.5398v1 [math.FA] 21 Jul 2014.
M. A. Naimark, Lineinye differentsialnye operatory, Izdat. ``Nauka'', Moscow, 1969. MathSciNet
Bruce Call Orcutt, CANONICAL DIFFERENTIAL EQUATIONS, ProQuest LLC, Ann Arbor, MI, 1969. MathSciNet
A. L. Sakhnovich, Spectral functions of a second-order canonical system, Mat. Sb. 181 (1990), no. 11, 1510-1524. MathSciNet
A. V. Straus, On generalized resolvents and spectral functions of differential operators of even order, Izv. Akad. Nauk SSSR. Ser. Mat. 21 (1957), 785-808. MathSciNet
