November  2013, 33(11&12): 5067-5088. doi: 10.3934/dcds.2013.33.5067

On a class of model Hilbert spaces

1. 

Department of Mathematics, University of Missouri, Columbia, MO 65211

2. 

Department of Mathematics, University of Alabama at Birmingham, Birmingham, AL 35294, United States

3. 

Department of Mathematics and Statistics, University of New Mexico, Albuquerque, NM 87131, United States

Received  November 2011 Published  May 2013

A detailed description of the model Hilbert space $L^2(\mathbb{R}; d\Sigma; K)$, where $K$ represents a complex, separable Hilbert space, and $\Sigma$ denotes a bounded operator-valued measure, is provided. In particular, we show that several alternative approaches to such a construction in the literature are equivalent.
    These spaces are of fundamental importance in the context of perturbation theory of self-adjoint extensions of symmetric operators, and the spectral theory of ordinary differential operators with operator-valued coefficients.
Citation: Fritz Gesztesy, Rudi Weikard, Maxim Zinchenko. On a class of model Hilbert spaces. Discrete and Continuous Dynamical Systems, 2013, 33 (11&12) : 5067-5088. doi: 10.3934/dcds.2013.33.5067
References:
[1]

V. M. Adamjan and H. Langer, Spectral properties of a class of rational operator valued functions, J. Operator Th., 33 (1995), 259-277.

[2]

G. D. Allen and F. J. Narcowich, On the representation and approximation of a class of operator-valued analytic functions, Bull. Amer. Math. Soc., 81 (1975), 410-412. doi: 10.1090/S0002-9904-1975-13761-X.

[3]

G. D. Allen and F. J. Narcowich, $R$-operators I. Representation theory and applications, Indiana Univ. Math. J., 25 (1976), 945-963. doi: 10.1512/iumj.1976.25.25075.

[4]

Yu. Arlinskii, S. Belyi and E. Tsekanovskii, "Conservative Realizations of Herglotz-Nevanlinna Functions," Operator Theory advances and Applications, 217, Birkhäuser, Springer, Basel, 2011. doi: 10.1007/978-3-7643-9996-2.

[5]

H. Baumgärtel and M. Wollenberg, "Mathematical Scattering Theory," Operator Theory: Advances and Applications, 9, Birkhäuser, Boston, 1983.

[6]

S. V. Belyi and E. R. Tsekanovskii, Classes of operator $R$-functions and their realization by conservative systems, Sov. Math. Dokl., 44 (1992), 692-696.

[7]

Ju. Berezanskiĭ, "Expansions in Eigenfunctions of Selfadjoint Operators," Transl. Math. Mongraphs, 17, Amer. Math. Soc., Providence, R.I., 1968.

[8]

Yu. Berezanskiĭ, "Selfadjoint Operators in Spaces of Functions of Infinitely Many Variables," Transl. Math. Mongraphs, 63, Amer. Math. Soc., Providence, R.I., 1986.

[9]

Y. M. Berezansky, Z. G. Sheftel and G. F. Us, "Functional Analysis Vol. II," Operator Theory: Advances and Applications, 86, Birkhäuser, Basel, 1996.

[10]

M. Š. Birman and S. B. Èntina, The stationary method in the abstract theory of scattering theory, Math. SSSR Izv., 1 (1967), 391-420. doi: 10.1070/IM1967v001n02ABEH000564.

[11]

M. S. Birman and M. Z. Solomjak, "Spectral Theory of Self-Adjoint Operators in Hilbert Space," Reidel, Dordrecht, 1987. doi: 10.1007/978-94-009-4586-9.

[12]

J. F. Brasche, M. Malamud and H. Neidhardt, Weyl function and spectral properties of self-adjoint extensions, Integr. Eq. Oper. Th., 43 (2002), 264-289. doi: 10.1007/BF01255563.

[13]

M. S. Brodskii, "Triangular and Jordan Representations of Linear Operators," Transl. Math. Mongraphs, 32, Amer. Math. Soc., Providence, RI, 1971.

[14]

R. W. Carey, A unitary invariant for pairs of self-adjoint operators, J. Reine Angew. Math., 283 (1976), 294-312.

[15]

V. A. Derkach and M. M. Malamud, Generalized resolvents and the boundary value problems for Hermitian operators with gaps, J. Funct. Anal., 95 (1991), 1-95. doi: 10.1016/0022-1236(91)90024-Y.

[16]

V. A. Derkach and M. M. Malamud, The extension theory of Hermitian operators and the moment problem, J. Math. Sci., 73 (1995), 141-242. doi: 10.1007/BF02367240.

[17]

V. A. Derkach and M. M. Malamud, On some classes of holomorphic operator functions with nonnegative imaginary part, in "Operator Algebras and Related Topics," 16th International Conference on Operator Theory, (eds. A. Gheondea, R. N. Gologan and T. Timotin), The Theta Foundation, Bucharest, (1997), 113-147.

[18]

J. Diestel and J. J. Uhl, "Vector Measures," Mathematical Surveys, 15, Amer. Math. Soc., Providence, RI, 1977.

[19]

J. Dixmier, "Les Algèbres d'Opérateurs dans l'Espace Hilbertien (Algebrès de von Neumann)," 2nd extended ed., Gauthier-Villars, Paris, 1969, Éditions Jaques Gabai, 1996.

[20]

N. Dunford and J. T. Schwartz, "Linear Operators Part II: Spectral Theory," Interscience, New York, 1988.

[21]

P. A. Fuhrmann, "Linear Systems and Operators in Hilbert Space," McGraw-Hill, New York, 1981.

[22]

I. M. Gel'fand and A. G. Kostyuchenko, Expansion in eigenfunctions of differential and other operators, Dokl. Akad. Nauk SSSR (N.S.), 103 (1955), 349-352. (Russian.)

[23]

I. M. Gel'fand and G. E. Shilov, "Generalized Functions. Volume 3: Theory of Differential Equations," Academic Press, New York, 1967.

[24]

F. Gesztesy, N. J. Kalton, K. A. Makarov and E. Tsekanovskii, Some applications of operator-valued Herglotz functions, in "Operator Theory, System Theory and Related Topics," Oper. Theory Adv. Appl., 123, Birkhäuser, Basel, (2001), 271-321.

[25]

F. Gesztesy, K. A. Makarov and E. Tsekanovskii, An Addendum to Krein's formula, J. Math. Anal. Appl., 222 (1998), 594-606. doi: 10.1006/jmaa.1998.5948.

[26]

F. Gesztesy and M. Mitrea, Robin-to-Robin maps and Krein-type resolvent formulas for Schrödinger operators on bounded Lipschitz domains, in "Modern Analysis and Applications. The Mark Krein Centenary Conference" (eds. V. Adamyan, Yu. Berezansky, I. Gohberg, M. Gorbachuk, V. Gorbachuk, A. Kochubei, H. Langer and G. Popov), Operator Theory: Advances and Applications, 191, Birkhäuser, Basel, (2009), 81-113. doi: 10.1007/978-3-7643-9921-4_6.

[27]

F. Gesztesy and M. Mitrea, Generalized Robin boundary conditions, Robin-to-Dirichlet maps, and Krein-type resolvent formulas for Schrödinger operators on bounded Lipschitz domains, in "Perspectives in Partial Differential Equations, Harmonic Analysis and Applications: A Volume in Honor of Vladimir G. Maz'ya's 70th Birthday" (eds. D. Mitrea and M. Mitrea), Proceedings of Symposia in Pure Mathematics, 79, Amer. Math. Soc., Providence, RI, (2008), 105-173.

[28]

F. Gesztesy, M. Mitrea and M. Zinchenko, Variations on a theme of Jost and Pais, J. Funct. Anal., 253 (2007), 399-448. doi: 10.1016/j.jfa.2007.05.009.

[29]

F. Gesztesy and E. Tsekanovskii, On matrix-valued Herglotz functions, Math. Nachr., 218 (2000), 61-138. doi: 10.1002/1522-2616(200010)218:1<61::AID-MANA61>3.0.CO;2-D.

[30]

F. Gesztesy, R. Weikard and M. Zinchenko, Initial value problems and Weyl-Titchmarsh theory for Schrödinger operators with operator-valued potentials, Operators and Matrices, 7 (2013), 241-283.

[31]

F. Gesztesy, R. Weikard and M. Zinchenko, On spectral theory for Schrödinger operators with operator-valued potentials,, , (). 

[32]

M. L. Gorbačuk, On spectral functions of a second order differential operator with operator coefficients, Ukrain. Math. J., 18 (1966), 3-21. (Russian.) Engl. transl. in Amer. Math. Soc. Transl. (2), , 72 (1968), 177-202.

[33]

E. Heinz, Beiträge zur Störungstheorie der Spektralzerlegung, Math. Ann., 123 (1951), 415-438. doi: 10.1007/BF02054965.

[34]

T. Kato, Notes on some operator inequalities for linear operators, Math. Ann., 125 (1952), 208-212. doi: 10.1007/BF01343117.

[35]

I. S. Kats, On Hilbert spaces generated by monotone Hermitian matrix-functions, Zap. Mat. Otd. Fiz.-Mat. Fak. i Har'kov. Mat. Obšč. (4), 22 (1950), 95-113.

[36]

I. S. Kats, Linear relations generated by a canonical differential equation of dimension 2, and eigenfunction expansions, St. Petersburg Math. J., 14 (2003), 429-452.

[37]

M. G. Krein and I. E. Ovčarenko, $Q$-functions and sc-resolvents of nondensely defined Hermitian contractions, Sib. Math. J., 18 (1977), 728-746.

[38]

M. G. Krein and I. E. Ovčarenko, Inverse problems for $Q$-functions and resolvent matrices of positive Hermitian operators, Sov. Math. Dokl., 19 (1978), 1131-1134.

[39]

M. M. Malamud and S. M. Malamud, On the spectral theory of operator measures, Funct. Anal. Appl., 36 (2002), 154-158. doi: 10.1023/A:1015655005114.

[40]

M. M. Malamud and S. M. Malamud, On the spectral theory of operator measures in Hilbert space, St. Petersburg Math. J., 15 (2004), 323-373. doi: 10.1090/S1061-0022-04-00812-X.

[41]

M. Malamud and H. Neidhardt, On the unitary equivalence of absolutely continuous parts of self-adjoint extensions, J. Funct. Anal., 260 (2011), 613-638. doi: 10.1016/j.jfa.2010.10.021.

[42]

M. Malamud and H. Neidhardt, Sturm-Liouville boundary value problems with operator potentials and unitary equivalence, J. Diff. Eq., 252 (2012), 5875-5922. doi: 10.1016/j.jde.2012.02.018.

[43]

V. Mogilevskii, Boundary triplets and Titchmarsh-Weyl functions of differential operators with arbitrary deficiency indices, Meth. Funct. Anal. Topology, 15 (2009), 280-300.

[44]

S. N. Naboko, Boundary values of analytic operator functions with a positive imaginary part, J. Soviet Math., 44 (1989), 786-795. doi: 10.1007/BF01463185.

[45]

S. N. Naboko, Nontangential boundary values of operator-valued $R$-functions in a half-plane, Leningrad Math. J., 1 (1990), 1255-1278.

[46]

S. N. Naboko, The boundary behavior of ${\textfrakS}_p$-valued functions analytic in the half-plane with nonnegative imaginary part, Functional Analysis and Operator Theory, Banach Center Publications, 30, Institute of Mathematics, Polish Academy of Sciences, Warsaw, (1994), 277-285.

[47]

F. J. Narcowich, Mathematical theory of the $R$ matrix. II. The $R$ matrix and its properties, J. Math. Phys., 15 (1974), 1635-1642. doi: 10.1063/1.1666518.

[48]

F. J. Narcowich, $R$-operators II. On the approximation of certain operator-valued analytic functions and the Hermitian moment problem, Indiana Univ. Math. J., 26 (1977), 483-513. doi: 10.1512/iumj.1977.26.26038.

[49]

F. J. Narcowich and G. D. Allen, Convergence of the diagonal operator-valued Padé approximants to the Dyson expansion, Commun. Math. Phys., 45 (1975), 153-157. doi: 10.1007/BF01629245.

[50]

F. S. Rofe-Beketov, Expansions in eigenfunctions of infinite systems of differential equations in the non-self-adjoint and self-adjoint cases, Mat. Sb., 51 (1960), 293-342. (Russian.)

[51]

M. Rosenberg, The square-integrability of matrix-valued functions with respect to a non-negative Hermitian measure, Duke Math. J., 31 (1964), 291-298. doi: 10.1215/S0012-7094-64-03128-X.

[52]

Y. Saitō, Eigenfunction expansions associated with second-order differential equations for Hilbert space-valued functions,, Publ. RIMS, 7 (): 1.  doi: 10.2977/prims/1195193780.

[53]

Yu. L. Shmul'yan, On operator $R$-functions, Siberian Math. J., 12 (1971), 315-322. doi: 10.1007/BF00969054.

[54]

I. Trooshin, Asymptotics for the spectral and Weyl functions of the operator-valued Sturm-Liouville problem, in "Inverse Problems and Related Topics" (eds. G. Nakamura, S. Saitoh, J. K. Seo and M. Yamamoto), Chapman & Hall/CRC, Boca Raton, FL, (2000), 189-208.

[55]

J. von Neumann, "Functional Operators. Vol. II: The Geometry of Orthogonal Spaces," Ann. Math. Stud., 22, 2nd printing, Princeton University Press, Princeton, 1951.

[56]

J. Weidmann, "Linear Operators in Hilbert Spaces," Graduate Texts in Mathematics, 68, Springer, New York, 1980.

[57]

A. Wilansky, "Topology for Analysis," Reprint of the 1970 edition with corrections, Krieger, Melbourne, FL, 1983.

show all references

References:
[1]

V. M. Adamjan and H. Langer, Spectral properties of a class of rational operator valued functions, J. Operator Th., 33 (1995), 259-277.

[2]

G. D. Allen and F. J. Narcowich, On the representation and approximation of a class of operator-valued analytic functions, Bull. Amer. Math. Soc., 81 (1975), 410-412. doi: 10.1090/S0002-9904-1975-13761-X.

[3]

G. D. Allen and F. J. Narcowich, $R$-operators I. Representation theory and applications, Indiana Univ. Math. J., 25 (1976), 945-963. doi: 10.1512/iumj.1976.25.25075.

[4]

Yu. Arlinskii, S. Belyi and E. Tsekanovskii, "Conservative Realizations of Herglotz-Nevanlinna Functions," Operator Theory advances and Applications, 217, Birkhäuser, Springer, Basel, 2011. doi: 10.1007/978-3-7643-9996-2.

[5]

H. Baumgärtel and M. Wollenberg, "Mathematical Scattering Theory," Operator Theory: Advances and Applications, 9, Birkhäuser, Boston, 1983.

[6]

S. V. Belyi and E. R. Tsekanovskii, Classes of operator $R$-functions and their realization by conservative systems, Sov. Math. Dokl., 44 (1992), 692-696.

[7]

Ju. Berezanskiĭ, "Expansions in Eigenfunctions of Selfadjoint Operators," Transl. Math. Mongraphs, 17, Amer. Math. Soc., Providence, R.I., 1968.

[8]

Yu. Berezanskiĭ, "Selfadjoint Operators in Spaces of Functions of Infinitely Many Variables," Transl. Math. Mongraphs, 63, Amer. Math. Soc., Providence, R.I., 1986.

[9]

Y. M. Berezansky, Z. G. Sheftel and G. F. Us, "Functional Analysis Vol. II," Operator Theory: Advances and Applications, 86, Birkhäuser, Basel, 1996.

[10]

M. Š. Birman and S. B. Èntina, The stationary method in the abstract theory of scattering theory, Math. SSSR Izv., 1 (1967), 391-420. doi: 10.1070/IM1967v001n02ABEH000564.

[11]

M. S. Birman and M. Z. Solomjak, "Spectral Theory of Self-Adjoint Operators in Hilbert Space," Reidel, Dordrecht, 1987. doi: 10.1007/978-94-009-4586-9.

[12]

J. F. Brasche, M. Malamud and H. Neidhardt, Weyl function and spectral properties of self-adjoint extensions, Integr. Eq. Oper. Th., 43 (2002), 264-289. doi: 10.1007/BF01255563.

[13]

M. S. Brodskii, "Triangular and Jordan Representations of Linear Operators," Transl. Math. Mongraphs, 32, Amer. Math. Soc., Providence, RI, 1971.

[14]

R. W. Carey, A unitary invariant for pairs of self-adjoint operators, J. Reine Angew. Math., 283 (1976), 294-312.

[15]

V. A. Derkach and M. M. Malamud, Generalized resolvents and the boundary value problems for Hermitian operators with gaps, J. Funct. Anal., 95 (1991), 1-95. doi: 10.1016/0022-1236(91)90024-Y.

[16]

V. A. Derkach and M. M. Malamud, The extension theory of Hermitian operators and the moment problem, J. Math. Sci., 73 (1995), 141-242. doi: 10.1007/BF02367240.

[17]

V. A. Derkach and M. M. Malamud, On some classes of holomorphic operator functions with nonnegative imaginary part, in "Operator Algebras and Related Topics," 16th International Conference on Operator Theory, (eds. A. Gheondea, R. N. Gologan and T. Timotin), The Theta Foundation, Bucharest, (1997), 113-147.

[18]

J. Diestel and J. J. Uhl, "Vector Measures," Mathematical Surveys, 15, Amer. Math. Soc., Providence, RI, 1977.

[19]

J. Dixmier, "Les Algèbres d'Opérateurs dans l'Espace Hilbertien (Algebrès de von Neumann)," 2nd extended ed., Gauthier-Villars, Paris, 1969, Éditions Jaques Gabai, 1996.

[20]

N. Dunford and J. T. Schwartz, "Linear Operators Part II: Spectral Theory," Interscience, New York, 1988.

[21]

P. A. Fuhrmann, "Linear Systems and Operators in Hilbert Space," McGraw-Hill, New York, 1981.

[22]

I. M. Gel'fand and A. G. Kostyuchenko, Expansion in eigenfunctions of differential and other operators, Dokl. Akad. Nauk SSSR (N.S.), 103 (1955), 349-352. (Russian.)

[23]

I. M. Gel'fand and G. E. Shilov, "Generalized Functions. Volume 3: Theory of Differential Equations," Academic Press, New York, 1967.

[24]

F. Gesztesy, N. J. Kalton, K. A. Makarov and E. Tsekanovskii, Some applications of operator-valued Herglotz functions, in "Operator Theory, System Theory and Related Topics," Oper. Theory Adv. Appl., 123, Birkhäuser, Basel, (2001), 271-321.

[25]

F. Gesztesy, K. A. Makarov and E. Tsekanovskii, An Addendum to Krein's formula, J. Math. Anal. Appl., 222 (1998), 594-606. doi: 10.1006/jmaa.1998.5948.

[26]

F. Gesztesy and M. Mitrea, Robin-to-Robin maps and Krein-type resolvent formulas for Schrödinger operators on bounded Lipschitz domains, in "Modern Analysis and Applications. The Mark Krein Centenary Conference" (eds. V. Adamyan, Yu. Berezansky, I. Gohberg, M. Gorbachuk, V. Gorbachuk, A. Kochubei, H. Langer and G. Popov), Operator Theory: Advances and Applications, 191, Birkhäuser, Basel, (2009), 81-113. doi: 10.1007/978-3-7643-9921-4_6.

[27]

F. Gesztesy and M. Mitrea, Generalized Robin boundary conditions, Robin-to-Dirichlet maps, and Krein-type resolvent formulas for Schrödinger operators on bounded Lipschitz domains, in "Perspectives in Partial Differential Equations, Harmonic Analysis and Applications: A Volume in Honor of Vladimir G. Maz'ya's 70th Birthday" (eds. D. Mitrea and M. Mitrea), Proceedings of Symposia in Pure Mathematics, 79, Amer. Math. Soc., Providence, RI, (2008), 105-173.

[28]

F. Gesztesy, M. Mitrea and M. Zinchenko, Variations on a theme of Jost and Pais, J. Funct. Anal., 253 (2007), 399-448. doi: 10.1016/j.jfa.2007.05.009.

[29]

F. Gesztesy and E. Tsekanovskii, On matrix-valued Herglotz functions, Math. Nachr., 218 (2000), 61-138. doi: 10.1002/1522-2616(200010)218:1<61::AID-MANA61>3.0.CO;2-D.

[30]

F. Gesztesy, R. Weikard and M. Zinchenko, Initial value problems and Weyl-Titchmarsh theory for Schrödinger operators with operator-valued potentials, Operators and Matrices, 7 (2013), 241-283.

[31]

F. Gesztesy, R. Weikard and M. Zinchenko, On spectral theory for Schrödinger operators with operator-valued potentials,, , (). 

[32]

M. L. Gorbačuk, On spectral functions of a second order differential operator with operator coefficients, Ukrain. Math. J., 18 (1966), 3-21. (Russian.) Engl. transl. in Amer. Math. Soc. Transl. (2), , 72 (1968), 177-202.

[33]

E. Heinz, Beiträge zur Störungstheorie der Spektralzerlegung, Math. Ann., 123 (1951), 415-438. doi: 10.1007/BF02054965.

[34]

T. Kato, Notes on some operator inequalities for linear operators, Math. Ann., 125 (1952), 208-212. doi: 10.1007/BF01343117.

[35]

I. S. Kats, On Hilbert spaces generated by monotone Hermitian matrix-functions, Zap. Mat. Otd. Fiz.-Mat. Fak. i Har'kov. Mat. Obšč. (4), 22 (1950), 95-113.

[36]

I. S. Kats, Linear relations generated by a canonical differential equation of dimension 2, and eigenfunction expansions, St. Petersburg Math. J., 14 (2003), 429-452.

[37]

M. G. Krein and I. E. Ovčarenko, $Q$-functions and sc-resolvents of nondensely defined Hermitian contractions, Sib. Math. J., 18 (1977), 728-746.

[38]

M. G. Krein and I. E. Ovčarenko, Inverse problems for $Q$-functions and resolvent matrices of positive Hermitian operators, Sov. Math. Dokl., 19 (1978), 1131-1134.

[39]

M. M. Malamud and S. M. Malamud, On the spectral theory of operator measures, Funct. Anal. Appl., 36 (2002), 154-158. doi: 10.1023/A:1015655005114.

[40]

M. M. Malamud and S. M. Malamud, On the spectral theory of operator measures in Hilbert space, St. Petersburg Math. J., 15 (2004), 323-373. doi: 10.1090/S1061-0022-04-00812-X.

[41]

M. Malamud and H. Neidhardt, On the unitary equivalence of absolutely continuous parts of self-adjoint extensions, J. Funct. Anal., 260 (2011), 613-638. doi: 10.1016/j.jfa.2010.10.021.

[42]

M. Malamud and H. Neidhardt, Sturm-Liouville boundary value problems with operator potentials and unitary equivalence, J. Diff. Eq., 252 (2012), 5875-5922. doi: 10.1016/j.jde.2012.02.018.

[43]

V. Mogilevskii, Boundary triplets and Titchmarsh-Weyl functions of differential operators with arbitrary deficiency indices, Meth. Funct. Anal. Topology, 15 (2009), 280-300.

[44]

S. N. Naboko, Boundary values of analytic operator functions with a positive imaginary part, J. Soviet Math., 44 (1989), 786-795. doi: 10.1007/BF01463185.

[45]

S. N. Naboko, Nontangential boundary values of operator-valued $R$-functions in a half-plane, Leningrad Math. J., 1 (1990), 1255-1278.

[46]

S. N. Naboko, The boundary behavior of ${\textfrakS}_p$-valued functions analytic in the half-plane with nonnegative imaginary part, Functional Analysis and Operator Theory, Banach Center Publications, 30, Institute of Mathematics, Polish Academy of Sciences, Warsaw, (1994), 277-285.

[47]

F. J. Narcowich, Mathematical theory of the $R$ matrix. II. The $R$ matrix and its properties, J. Math. Phys., 15 (1974), 1635-1642. doi: 10.1063/1.1666518.

[48]

F. J. Narcowich, $R$-operators II. On the approximation of certain operator-valued analytic functions and the Hermitian moment problem, Indiana Univ. Math. J., 26 (1977), 483-513. doi: 10.1512/iumj.1977.26.26038.

[49]

F. J. Narcowich and G. D. Allen, Convergence of the diagonal operator-valued Padé approximants to the Dyson expansion, Commun. Math. Phys., 45 (1975), 153-157. doi: 10.1007/BF01629245.

[50]

F. S. Rofe-Beketov, Expansions in eigenfunctions of infinite systems of differential equations in the non-self-adjoint and self-adjoint cases, Mat. Sb., 51 (1960), 293-342. (Russian.)

[51]

M. Rosenberg, The square-integrability of matrix-valued functions with respect to a non-negative Hermitian measure, Duke Math. J., 31 (1964), 291-298. doi: 10.1215/S0012-7094-64-03128-X.

[52]

Y. Saitō, Eigenfunction expansions associated with second-order differential equations for Hilbert space-valued functions,, Publ. RIMS, 7 (): 1.  doi: 10.2977/prims/1195193780.

[53]

Yu. L. Shmul'yan, On operator $R$-functions, Siberian Math. J., 12 (1971), 315-322. doi: 10.1007/BF00969054.

[54]

I. Trooshin, Asymptotics for the spectral and Weyl functions of the operator-valued Sturm-Liouville problem, in "Inverse Problems and Related Topics" (eds. G. Nakamura, S. Saitoh, J. K. Seo and M. Yamamoto), Chapman & Hall/CRC, Boca Raton, FL, (2000), 189-208.

[55]

J. von Neumann, "Functional Operators. Vol. II: The Geometry of Orthogonal Spaces," Ann. Math. Stud., 22, 2nd printing, Princeton University Press, Princeton, 1951.

[56]

J. Weidmann, "Linear Operators in Hilbert Spaces," Graduate Texts in Mathematics, 68, Springer, New York, 1980.

[57]

A. Wilansky, "Topology for Analysis," Reprint of the 1970 edition with corrections, Krieger, Melbourne, FL, 1983.

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