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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