Book Review
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MathSciNet review:
1567933
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Book Information:
Authors:
T. Ya. Azizov and
I. S. Iokhvidov
Title:
Linear operators in spaces with an indefinite metric
Additional book information:
John Wiley & Sons, 1989, 300 pp., $82.95. ISBN 0-471-92129-7.
T. Ando, Linear operators on Kreĭn spaces, Hokkaido University, Research Institute of Applied Electricity, Division of Applied Mathematics, Sapporo, 1979. MR 560903
[AI] T. Ya Azzizov and I. Iokhvidov, Linear operators in spaces with an indefinite metric, Mathematical Analysis 17 (Itogi nauki i tekhniki), 113-205, Moscow, 1979; (English Transl. J. Soviet Math. 15 (1981), 438-490. (Russian)
[B] H. Bercovici, Review on Introduction to operator theory and invariant subspaces by Bernard Beauzamy, Bull, of Amer. Math. Soc. (N. S. ) 22 (1990), 148-152.
Joseph A. Ball and William Greenberg, A Pontrjagin space analysis of the supercritical transport equation, Transport Theory Statist. Phys. 4 (1975), no. 4, 143–154. MR 456206, DOI 10.1080/00411457508204840
Joseph A. Ball, Israel Gohberg, and Leiba Rodman, Interpolation of rational matrix functions, Operator Theory: Advances and Applications, vol. 45, Birkhäuser Verlag, Basel, 1990. MR 1083145, DOI 10.1007/978-3-0348-7709-1
Joseph A. Ball and J. William Helton, A Beurling-Lax theorem for the Lie group $\textrm {U}(m,\,n)$ which contains most classical interpolation theory, J. Operator Theory 9 (1983), no. 1, 107–142. MR 695942
János Bognár, Indefinite inner product spaces, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 78, Springer-Verlag, New York-Heidelberg, 1974. MR 0467261
Harry Dym, $J$ contractive matrix functions, reproducing kernel Hilbert spaces and interpolation, CBMS Regional Conference Series in Mathematics, vol. 71, Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI, 1989. MR 1004239, DOI 10.1090/cbms/071
[DK] Yu. L. Daletskiy and M. G. Krein, Stable solutions of differential equations in a Banach space (Russian), Nauka, Moscow, 1970; (English Transl.: Amer. Math. Soc. Transl. Math. Monographs, vol. 43, Providence, RI, 1974.
K. Daho and H. Langer, Sturm-Liouville operators with an indefinite weight function, Proc. Roy. Soc. Edinburgh Sect. A 78 (1977/78), no. 1-2, 161–191. MR 492490, DOI 10.1017/S0308210500009914
Bruce A. Francis, A course in $H_\infty$ control theory, Lecture Notes in Control and Information Sciences, vol. 88, Springer-Verlag, Berlin, 1987. MR 932459, DOI 10.1007/BFb0007371
I. Gohberg, P. Lancaster, and L. Rodman, Matrices and indefinite scalar products, Operator Theory: Advances and Applications, vol. 8, Birkhäuser Verlag, Basel, 1983. MR 859708
W. Greenberg, C. van der Mee, and V. Protopopescu, Boundary value problems in abstract kinetic theory, Operator Theory: Advances and Applications, vol. 23, Birkhäuser Verlag, Basel, 1987. MR 896904, DOI 10.1007/978-3-0348-5478-8
J. William Helton, Joseph A. Ball, Charles R. Johnson, and John N. Palmer, Operator theory, analytic functions, matrices, and electrical engineering, CBMS Regional Conference Series in Mathematics, vol. 68, Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI, 1987. MR 896034, DOI 10.1090/cbms/068
I. S. Iohvidov, On a lemma of Ky Fan generalizing the fixed-point principle of A. N. Tihonov, Dokl. Akad. Nauk SSSR 159 (1964), 501–504 (Russian). MR 0173158
I. S. Iohvidov and M. G. Kreĭn, Spectral theory of operators in space with indefinite metric. I, Trudy Moskov. Mat. Obšč. 5 (1956), 367–432 (Russian). MR 0080879
I. S. Iohvidov, M. G. Kreĭn, and H. Langer, Introduction to the spectral theory of operators in spaces with an indefinite metric, Mathematical Research, vol. 9, Akademie-Verlag, Berlin, 1982. MR 691137
Peter Jonas, On the functional calculus and the spectral function for definitizable operators in Kreĭn space, Beiträge Anal. 16 (1981), 121–135. MR 663475
M. G. Kreĭn, A new application of the fixed-point principle in the theory of operators in a space with indefinite metric, Dokl. Akad. Nauk SSSR 154 (1964), 1023–1026 (Russian). MR 0169059
M. G. Kreĭn, Introduction to the geometry of indefinite $J$-spaces and to the theory of operators in those spaces, Second Math. Summer School, Part I (Russian), “Naukova Dumka”, Kiev, 1965, pp. 15–92 (Russian). MR 0192348
[K] L. Kronecker, Ueber die congruenten Transformationen der bilinearen Formen, Monatsber. Akad. Wiss., Berlin, 1874, 397-447.
M. G. Kreĭn and H. Langer, On some mathematical principles in the linear theory of damped oscillations of continua. I, Integral Equations Operator Theory 1 (1978), no. 3, 364–399. Translated from the Russian by R. Troelstra. MR 511976, DOI 10.1007/BF01682844
[L1] H. Langer, Spectral theory of linear operators in J-spaces and some applications to the bundle L(λ) = λ2I + λB + C, Habilitationschrift, Dresden, 1965. (German)
Heinz Langer, Spectral functions of definitizable operators in Kreĭn spaces, Functional analysis (Dubrovnik, 1981) Lecture Notes in Math., vol. 948, Springer, Berlin-New York, 1982, pp. 1–46. MR 672791
A. S. Markus, Introduction to the spectral theory of polynomial operator pencils, Translations of Mathematical Monographs, vol. 71, American Mathematical Society, Providence, RI, 1988. Translated from the Russian by H. H. McFaden; Translation edited by Ben Silver; With an appendix by M. V. Keldysh. MR 971506, DOI 10.1090/mmono/071
L. Pontrjagin, Hermitian operators in spaces with indefinite metric, Bull. Acad. Sci. URSS. Sér. Math. [Izvestia Akad. Nauk SSSR] 8 (1944), 243–280 (Russian, with English summary). MR 0012200
R. S. Phillips, Dissipative operators and hyperbolic systems of partial differential equations, Trans. Amer. Math. Soc. 90 (1959), 193–254. MR 104919, DOI 10.1090/S0002-9947-1959-0104919-1
Leiba Rodman, An introduction to operator polynomials, Operator Theory: Advances and Applications, vol. 38, Birkhäuser Verlag, Basel, 1989. MR 997092, DOI 10.1007/978-3-0348-9152-3
R. M. Redheffer, On a certain linear fractional transformation, J. Math. and Phys. 39 (1960), 269–286. MR 141990
Marshall Harvey Stone, Linear transformations in Hilbert space, American Mathematical Society Colloquium Publications, vol. 15, American Mathematical Society, Providence, RI, 1990. Reprint of the 1932 original. MR 1451877, DOI 10.1090/coll/015
[vN] J. von Neumann, Allgemeine Eigenwettheorie Hermitscher Funktionaloperatoren, Math. Ann. 102 (1929-1930), 49-131.
[W] G. Wittstock, On invariant subspaces of positive transformations in spaces with indefinite metric, Math. Ann. 172 (1967), 167-175. (German)
[We] K. Weierstrass, Zür Theorie der bilinearen und quadratischen Formen, Monatsber. Akad. Wiss., Berlin, 1868, pp. 310-338.
- [A] T. Ando, Linear operators on Krein spaces, Sapporo, Japan, 1979. MR 0560903
- [AI] T. Ya Azzizov and I. Iokhvidov, Linear operators in spaces with an indefinite metric, Mathematical Analysis 17 (Itogi nauki i tekhniki), 113-205, Moscow, 1979; (English Transl. J. Soviet Math. 15 (1981), 438-490. (Russian)
- [B] H. Bercovici, Review on Introduction to operator theory and invariant subspaces by Bernard Beauzamy, Bull, of Amer. Math. Soc. (N. S. ) 22 (1990), 148-152.
- [BG] J. A. Ball and W. Greenberg, A Pontryagin space analysis of the supercritical transport equation, Transport Theory and Stat. Physics 4 (1975), 143-154. MR 456206
- [BGR] J. A. Ball, I. Gohberg, and L. Rodman, Interpolation of rational matrix functions, OT45, Birkhäuser, 1990. MR 1083145
- [BH] J. A. Ball and J. W. Helton, A Beurling-Lax theorem for the Lie group U(m, n) which contains most classical interpolation theory, J. Operator Theory 9 (1983), 107-142. MR 695942
- [Bo] J. Bognar, Indefinite inner product spaces, Springer-Verlag, Berlin, 1974. MR 467261
- [D] H. Dym, J-contractive matrix functions, reproducing kernel Hilbert spaces and interpolation, CBMS Regional Conf. Ser. in Math., vol. 71, Amer. Math. Soc. Providence, RI, 1989. MR 1004239
- [DK] Yu. L. Daletskiy and M. G. Krein, Stable solutions of differential equations in a Banach space (Russian), Nauka, Moscow, 1970; (English Transl.: Amer. Math. Soc. Transl. Math. Monographs, vol. 43, Providence, RI, 1974.
- [DL] K. Daho and H. Langer, Sturm-Liouville operators with an indefinite weight function, Proc. Royal Soc. Edinburgh 78A (1977), 161-191. MR 492490
- [Fr] B. Francis, A course in H∞ control theory, Lecture Notes in Control and Information Sciences, vol. 88, Springer-Verlag, Berlin and New York, 1987. MR 932459
- [GLR] I. Gohberg, P. Lancaster, and L. Rodman, Matrices and indefinite scalar products, OT8, Birkhäuser, Verlag, Basel, 1983. MR 859708
- [GMP] W. Greenberg, C. V. M. van der Mee, and V. Protopopescu, Boundary value problems in abstract kinetic theory, OT23, Birkhäuser, Basel, 1987. MR 896904
- [H] J. W. Helton, Operator theory, analytic functions, matrices and electrical engineering, CBMS Regional Conf. Ser. in Math., vol. 68, Amer. Math. Soc. Providence, RI, 1987. MR 896034
- [I] I. S. Iokhvidov, On a lemma of K. Fan generalizing A. N. Tikhonov's principle of fixed points, Dokl. Akad. Nauk USSR 159 (1964), 501-504. (Russian) MR 173158
- [IK] I. S. Iokhvidov and M. G. Krein, Spectral theory of operators in spaces with an indefinite metric. I (Russian), Trudy Moskov., Mat. Obsch. 5 (1965), 367-432; Amer. Math. Soc. Translations (2), 1960, 105-75; II (Russian), Trudy Moskov, Mat. Obshch, 8 (1969), 413-96; Amer. Math. Soc. Translations (2), 34, 283-373. MR 80879
- [IKL] I. S. Iokhvidov, M. G. Krein, and H. Langer, Introduction to the spectral theory of operators in spaces with an indefinite metric, Akademie-Verlag, Berlin, 1982. MR 691137
- [J] P. Jonas, On the functional calculus and the spectral function for definitizable operators in Krein space, Belträge Anal. 16 (1981), 121-135. MR 663475
- [K1] M. G. Krein, A new application of the fixed-point principle in the theory of linear operators in a space with an indefinite metric, Dokl. Akad. Nauk USSR 1964 154 (5) 1023-1026; (English Transl.: Soviet Math. Dokl. 5 (1964), 224-227. (Russian) MR 169059
- [K2] M. G. Krein, Introduction to the geometry of indefinite J-spaces and the theory of operators in those spaces, Second Summer Mathematical School, I, Kiev, 1965, 15-92; English Transl: AMS Transl. (2) 93 (1970), 103-176. (Russian) MR 192348
- [K] L. Kronecker, Ueber die congruenten Transformationen der bilinearen Formen, Monatsber. Akad. Wiss., Berlin, 1874, 397-447.
- [KL] M. G. Krein, and H. Langer, On some mathematical principles in the linear theory of damped oscillations of continua, Trudy mezhdunarod. simposiuma po prilozheniyam teorii funktsiy v mechanike sploshnoy sredy, vol. 2, Moscow, 1966, 283-322; (English transl.: Integral Equations and Operator Theory 1 (1978), 364-399; 539-566. MR 511976
- [L1] H. Langer, Spectral theory of linear operators in J-spaces and some applications to the bundle L(λ) = λ2I + λB + C, Habilitationschrift, Dresden, 1965. (German)
- [L2] H. Langer, Spectral functions of definitizable operators in Krein spaces, Lecture Notes in Math., vol. 948, 1982, pp. 1-46. MR 672791
- [M] A. S. Markus, Introduction to spectral theory of polynomial operator pencils, Stiinca, Kishinev, 1986; (English Transl.: Amer. Math. Soc. Transl. of Math. Monographs, vol. 71, 1988. (Russian) MR 971506
- [P] L. S. Pontryagin, Hermitian operators in spaces with indefinite metric, Izves tiya Akad. Nauk USSR, Ser. Matem. 8 (1944), 243-280. (Russian) MR 12200
- [Ph] R. Phillips, Dissipative operators and hyperbolic systems of partial differential equations, Trans. Amer. Math. Soc. 90 (1959), 193-254. MR 104919
- [R] L. Rodman, An introduction to operator polynomials, OT38, Birkhäuser, Basel, 1989. MR 997092
- [Re] R. M. Redheffer, On a certain linear fractional transformation, J. Math. Phys. 39 (1960), 260-286. MR 141990
- [S] M. H. Stone, Linear transformations in Hilbert space and their applications to analysis, Amer. Math. Soc. Colloq. Publ., vol. 15, New York, 1932. MR 1451877
- [vN] J. von Neumann, Allgemeine Eigenwettheorie Hermitscher Funktionaloperatoren, Math. Ann. 102 (1929-1930), 49-131.
- [W] G. Wittstock, On invariant subspaces of positive transformations in spaces with indefinite metric, Math. Ann. 172 (1967), 167-175. (German)
- [We] K. Weierstrass, Zür Theorie der bilinearen und quadratischen Formen, Monatsber. Akad. Wiss., Berlin, 1868, pp. 310-338.
Review Information:
Reviewer:
Leiba Rodman
Journal:
Bull. Amer. Math. Soc.
25 (1991), 111-123
DOI:
https://doi.org/10.1090/S0273-0979-1991-16039-8