# American Institute of Mathematical Sciences

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 & Continuous Dynamical Systems - A, 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. Google Scholar [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. doi: 10.1090/S0002-9904-1975-13761-X. Google Scholar [3] G. D. Allen and F. J. Narcowich, $R$-operators I. Representation theory and applications,, Indiana Univ. Math. J., 25 (1976), 945. doi: 10.1512/iumj.1976.25.25075. Google Scholar [4] Yu. Arlinskii, S. Belyi and E. Tsekanovskii, "Conservative Realizations of Herglotz-Nevanlinna Functions,", Operator Theory advances and Applications, 217 (2011). doi: 10.1007/978-3-7643-9996-2. Google Scholar [5] H. Baumgärtel and M. Wollenberg, "Mathematical Scattering Theory,", Operator Theory: Advances and Applications, 9 (1983). Google Scholar [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. Google Scholar [7] Ju. Berezanskiĭ, "Expansions in Eigenfunctions of Selfadjoint Operators,", Transl. Math. Mongraphs, 17 (1968). Google Scholar [8] Yu. Berezanskiĭ, "Selfadjoint Operators in Spaces of Functions of Infinitely Many Variables,", Transl. Math. Mongraphs, 63 (1986). Google Scholar [9] Y. M. Berezansky, Z. G. Sheftel and G. F. Us, "Functional Analysis Vol. II,", Operator Theory: Advances and Applications, 86 (1996). Google Scholar [10] M. Š. Birman and S. B. Èntina, The stationary method in the abstract theory of scattering theory,, Math. SSSR Izv., 1 (1967), 391. doi: 10.1070/IM1967v001n02ABEH000564. Google Scholar [11] M. S. Birman and M. Z. Solomjak, "Spectral Theory of Self-Adjoint Operators in Hilbert Space,", Reidel, (1987). doi: 10.1007/978-94-009-4586-9. Google Scholar [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. doi: 10.1007/BF01255563. Google Scholar [13] M. S. Brodskii, "Triangular and Jordan Representations of Linear Operators,", Transl. Math. Mongraphs, 32 (1971). Google Scholar [14] R. W. Carey, A unitary invariant for pairs of self-adjoint operators,, J. Reine Angew. Math., 283 (1976), 294. Google Scholar [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. doi: 10.1016/0022-1236(91)90024-Y. Google Scholar [16] V. A. Derkach and M. M. Malamud, The extension theory of Hermitian operators and the moment problem,, J. Math. Sci., 73 (1995), 141. doi: 10.1007/BF02367240. Google Scholar [17] V. A. Derkach and M. M. Malamud, On some classes of holomorphic operator functions with nonnegative imaginary part,, in, (1997), 113. Google Scholar [18] J. Diestel and J. J. Uhl, "Vector Measures,", Mathematical Surveys, 15 (1977). Google Scholar [19] J. Dixmier, "Les Algèbres d'Opérateurs dans l'Espace Hilbertien (Algebrès de von Neumann),", 2nd extended ed., (1969). Google Scholar [20] N. Dunford and J. T. Schwartz, "Linear Operators Part II: Spectral Theory,", Interscience, (1988). Google Scholar [21] P. A. Fuhrmann, "Linear Systems and Operators in Hilbert Space,", McGraw-Hill, (1981). Google Scholar [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. Google Scholar [23] I. M. Gel'fand and G. E. Shilov, "Generalized Functions. Volume 3: Theory of Differential Equations,", Academic Press, (1967). Google Scholar [24] F. Gesztesy, N. J. Kalton, K. A. Makarov and E. Tsekanovskii, Some applications of operator-valued Herglotz functions,, in, 123 (2001), 271. Google Scholar [25] F. Gesztesy, K. A. Makarov and E. Tsekanovskii, An Addendum to Krein's formula,, J. Math. Anal. Appl., 222 (1998), 594. doi: 10.1006/jmaa.1998.5948. Google Scholar [26] F. Gesztesy and M. Mitrea, Robin-to-Robin maps and Krein-type resolvent formulas for Schrödinger operators on bounded Lipschitz domains,, in, 191 (2009), 81. doi: 10.1007/978-3-7643-9921-4_6. Google Scholar [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, 79 (2008), 105. Google Scholar [28] F. Gesztesy, M. Mitrea and M. Zinchenko, Variations on a theme of Jost and Pais,, J. Funct. Anal., 253 (2007), 399. doi: 10.1016/j.jfa.2007.05.009. Google Scholar [29] F. Gesztesy and E. Tsekanovskii, On matrix-valued Herglotz functions,, Math. Nachr., 218 (2000), 61. doi: 10.1002/1522-2616(200010)218:1<61::AID-MANA61>3.0.CO;2-D. Google Scholar [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. Google Scholar [31] F. Gesztesy, R. Weikard and M. Zinchenko, On spectral theory for Schrödinger operators with operator-valued potentials,, , (). Google Scholar [32] M. L. Gorbačuk, On spectral functions of a second order differential operator with operator coefficients,, Ukrain. Math. J., 18 (1966), 3. Google Scholar [33] E. Heinz, Beiträge zur Störungstheorie der Spektralzerlegung,, Math. Ann., 123 (1951), 415. doi: 10.1007/BF02054965. Google Scholar [34] T. Kato, Notes on some operator inequalities for linear operators,, Math. Ann., 125 (1952), 208. doi: 10.1007/BF01343117. Google Scholar [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. Google Scholar [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. Google Scholar [37] M. G. Krein and I. E. Ovčarenko, $Q$-functions and sc-resolvents of nondensely defined Hermitian contractions,, Sib. Math. J., 18 (1977), 728. Google Scholar [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. Google Scholar [39] M. M. Malamud and S. M. Malamud, On the spectral theory of operator measures,, Funct. Anal. Appl., 36 (2002), 154. doi: 10.1023/A:1015655005114. Google Scholar [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. doi: 10.1090/S1061-0022-04-00812-X. Google Scholar [41] M. Malamud and H. Neidhardt, On the unitary equivalence of absolutely continuous parts of self-adjoint extensions,, J. Funct. Anal., 260 (2011), 613. doi: 10.1016/j.jfa.2010.10.021. Google Scholar [42] M. Malamud and H. Neidhardt, Sturm-Liouville boundary value problems with operator potentials and unitary equivalence,, J. Diff. Eq., 252 (2012), 5875. doi: 10.1016/j.jde.2012.02.018. Google Scholar [43] V. Mogilevskii, Boundary triplets and Titchmarsh-Weyl functions of differential operators with arbitrary deficiency indices,, Meth. Funct. Anal. Topology, 15 (2009), 280. Google Scholar [44] S. N. Naboko, Boundary values of analytic operator functions with a positive imaginary part,, J. Soviet Math., 44 (1989), 786. doi: 10.1007/BF01463185. Google Scholar [45] S. N. Naboko, Nontangential boundary values of operator-valued $R$-functions in a half-plane,, Leningrad Math. J., 1 (1990), 1255. Google Scholar [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, 30 (1994), 277. Google Scholar [47] F. J. Narcowich, Mathematical theory of the $R$ matrix. II. The $R$ matrix and its properties,, J. Math. Phys., 15 (1974), 1635. doi: 10.1063/1.1666518. Google Scholar [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. doi: 10.1512/iumj.1977.26.26038. Google Scholar [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. doi: 10.1007/BF01629245. Google Scholar [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. Google Scholar [51] M. Rosenberg, The square-integrability of matrix-valued functions with respect to a non-negative Hermitian measure,, Duke Math. J., 31 (1964), 291. doi: 10.1215/S0012-7094-64-03128-X. Google Scholar [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. Google Scholar [53] Yu. L. Shmul'yan, On operator $R$-functions,, Siberian Math. J., 12 (1971), 315. doi: 10.1007/BF00969054. Google Scholar [54] I. Trooshin, Asymptotics for the spectral and Weyl functions of the operator-valued Sturm-Liouville problem,, in, (2000), 189. Google Scholar [55] J. von Neumann, "Functional Operators. Vol. II: The Geometry of Orthogonal Spaces,", Ann. Math. Stud., 22 (1951). Google Scholar [56] J. Weidmann, "Linear Operators in Hilbert Spaces,", Graduate Texts in Mathematics, 68 (1980). Google Scholar [57] A. Wilansky, "Topology for Analysis,", Reprint of the 1970 edition with corrections, (1970). Google Scholar

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. Google Scholar [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. doi: 10.1090/S0002-9904-1975-13761-X. Google Scholar [3] G. D. Allen and F. J. Narcowich, $R$-operators I. Representation theory and applications,, Indiana Univ. Math. J., 25 (1976), 945. doi: 10.1512/iumj.1976.25.25075. Google Scholar [4] Yu. Arlinskii, S. Belyi and E. Tsekanovskii, "Conservative Realizations of Herglotz-Nevanlinna Functions,", Operator Theory advances and Applications, 217 (2011). doi: 10.1007/978-3-7643-9996-2. Google Scholar [5] H. Baumgärtel and M. Wollenberg, "Mathematical Scattering Theory,", Operator Theory: Advances and Applications, 9 (1983). Google Scholar [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. Google Scholar [7] Ju. Berezanskiĭ, "Expansions in Eigenfunctions of Selfadjoint Operators,", Transl. Math. Mongraphs, 17 (1968). Google Scholar [8] Yu. Berezanskiĭ, "Selfadjoint Operators in Spaces of Functions of Infinitely Many Variables,", Transl. Math. Mongraphs, 63 (1986). Google Scholar [9] Y. M. Berezansky, Z. G. Sheftel and G. F. Us, "Functional Analysis Vol. II,", Operator Theory: Advances and Applications, 86 (1996). Google Scholar [10] M. Š. Birman and S. B. Èntina, The stationary method in the abstract theory of scattering theory,, Math. SSSR Izv., 1 (1967), 391. doi: 10.1070/IM1967v001n02ABEH000564. Google Scholar [11] M. S. Birman and M. Z. Solomjak, "Spectral Theory of Self-Adjoint Operators in Hilbert Space,", Reidel, (1987). doi: 10.1007/978-94-009-4586-9. Google Scholar [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. doi: 10.1007/BF01255563. Google Scholar [13] M. S. Brodskii, "Triangular and Jordan Representations of Linear Operators,", Transl. Math. Mongraphs, 32 (1971). Google Scholar [14] R. W. Carey, A unitary invariant for pairs of self-adjoint operators,, J. Reine Angew. Math., 283 (1976), 294. Google Scholar [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. doi: 10.1016/0022-1236(91)90024-Y. Google Scholar [16] V. A. Derkach and M. M. Malamud, The extension theory of Hermitian operators and the moment problem,, J. Math. Sci., 73 (1995), 141. doi: 10.1007/BF02367240. Google Scholar [17] V. A. Derkach and M. M. Malamud, On some classes of holomorphic operator functions with nonnegative imaginary part,, in, (1997), 113. Google Scholar [18] J. Diestel and J. J. Uhl, "Vector Measures,", Mathematical Surveys, 15 (1977). Google Scholar [19] J. Dixmier, "Les Algèbres d'Opérateurs dans l'Espace Hilbertien (Algebrès de von Neumann),", 2nd extended ed., (1969). Google Scholar [20] N. Dunford and J. T. Schwartz, "Linear Operators Part II: Spectral Theory,", Interscience, (1988). Google Scholar [21] P. A. Fuhrmann, "Linear Systems and Operators in Hilbert Space,", McGraw-Hill, (1981). Google Scholar [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. Google Scholar [23] I. M. Gel'fand and G. E. Shilov, "Generalized Functions. Volume 3: Theory of Differential Equations,", Academic Press, (1967). Google Scholar [24] F. Gesztesy, N. J. Kalton, K. A. Makarov and E. Tsekanovskii, Some applications of operator-valued Herglotz functions,, in, 123 (2001), 271. Google Scholar [25] F. Gesztesy, K. A. Makarov and E. Tsekanovskii, An Addendum to Krein's formula,, J. Math. Anal. Appl., 222 (1998), 594. doi: 10.1006/jmaa.1998.5948. Google Scholar [26] F. Gesztesy and M. Mitrea, Robin-to-Robin maps and Krein-type resolvent formulas for Schrödinger operators on bounded Lipschitz domains,, in, 191 (2009), 81. doi: 10.1007/978-3-7643-9921-4_6. Google Scholar [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, 79 (2008), 105. Google Scholar [28] F. Gesztesy, M. Mitrea and M. Zinchenko, Variations on a theme of Jost and Pais,, J. Funct. Anal., 253 (2007), 399. doi: 10.1016/j.jfa.2007.05.009. Google Scholar [29] F. Gesztesy and E. Tsekanovskii, On matrix-valued Herglotz functions,, Math. Nachr., 218 (2000), 61. doi: 10.1002/1522-2616(200010)218:1<61::AID-MANA61>3.0.CO;2-D. Google Scholar [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. Google Scholar [31] F. Gesztesy, R. Weikard and M. Zinchenko, On spectral theory for Schrödinger operators with operator-valued potentials,, , (). Google Scholar [32] M. L. Gorbačuk, On spectral functions of a second order differential operator with operator coefficients,, Ukrain. Math. J., 18 (1966), 3. Google Scholar [33] E. Heinz, Beiträge zur Störungstheorie der Spektralzerlegung,, Math. Ann., 123 (1951), 415. doi: 10.1007/BF02054965. Google Scholar [34] T. Kato, Notes on some operator inequalities for linear operators,, Math. Ann., 125 (1952), 208. doi: 10.1007/BF01343117. Google Scholar [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. Google Scholar [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. Google Scholar [37] M. G. Krein and I. E. Ovčarenko, $Q$-functions and sc-resolvents of nondensely defined Hermitian contractions,, Sib. Math. J., 18 (1977), 728. Google Scholar [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. Google Scholar [39] M. M. Malamud and S. M. Malamud, On the spectral theory of operator measures,, Funct. Anal. Appl., 36 (2002), 154. doi: 10.1023/A:1015655005114. Google Scholar [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. doi: 10.1090/S1061-0022-04-00812-X. Google Scholar [41] M. Malamud and H. Neidhardt, On the unitary equivalence of absolutely continuous parts of self-adjoint extensions,, J. Funct. Anal., 260 (2011), 613. doi: 10.1016/j.jfa.2010.10.021. Google Scholar [42] M. Malamud and H. Neidhardt, Sturm-Liouville boundary value problems with operator potentials and unitary equivalence,, J. Diff. Eq., 252 (2012), 5875. doi: 10.1016/j.jde.2012.02.018. Google Scholar [43] V. Mogilevskii, Boundary triplets and Titchmarsh-Weyl functions of differential operators with arbitrary deficiency indices,, Meth. Funct. Anal. Topology, 15 (2009), 280. Google Scholar [44] S. N. Naboko, Boundary values of analytic operator functions with a positive imaginary part,, J. Soviet Math., 44 (1989), 786. doi: 10.1007/BF01463185. Google Scholar [45] S. N. Naboko, Nontangential boundary values of operator-valued $R$-functions in a half-plane,, Leningrad Math. J., 1 (1990), 1255. Google Scholar [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, 30 (1994), 277. Google Scholar [47] F. J. Narcowich, Mathematical theory of the $R$ matrix. II. The $R$ matrix and its properties,, J. Math. Phys., 15 (1974), 1635. doi: 10.1063/1.1666518. Google Scholar [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. doi: 10.1512/iumj.1977.26.26038. Google Scholar [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. doi: 10.1007/BF01629245. Google Scholar [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. Google Scholar [51] M. Rosenberg, The square-integrability of matrix-valued functions with respect to a non-negative Hermitian measure,, Duke Math. J., 31 (1964), 291. doi: 10.1215/S0012-7094-64-03128-X. Google Scholar [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. Google Scholar [53] Yu. L. Shmul'yan, On operator $R$-functions,, Siberian Math. J., 12 (1971), 315. doi: 10.1007/BF00969054. Google Scholar [54] I. Trooshin, Asymptotics for the spectral and Weyl functions of the operator-valued Sturm-Liouville problem,, in, (2000), 189. Google Scholar [55] J. von Neumann, "Functional Operators. Vol. II: The Geometry of Orthogonal Spaces,", Ann. Math. Stud., 22 (1951). Google Scholar [56] J. Weidmann, "Linear Operators in Hilbert Spaces,", Graduate Texts in Mathematics, 68 (1980). Google Scholar [57] A. Wilansky, "Topology for Analysis,", Reprint of the 1970 edition with corrections, (1970). Google Scholar
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