Factorization of Operators in Krein Spaces and Linear-Fractional Relations of Operator Balls

Affiliation(s)

Department of Mathematics, ORT Braude Academic College, Karmiel, Israel.

Pyatnitskoe Highway, 23-2-156, Moscow, Russia.

Department of Mathematics, ORT Braude Academic College, Karmiel, Israel.

Pyatnitskoe Highway, 23-2-156, Moscow, Russia.

ABSTRACT

We consider plus-operators in Krein spaces and generated operator linear fractional relations of the following form:

.

We study some special type of factorization for plus-operators *T*, among them the following one: *T* = *BU*, where *B* is a lower triangular plus-operator, *U* is a *J*-unitary operator. We apply the above factorization to the study of basical properties of relations (1), in particular, convexity and compactness of their images with respect to the weak operator topology. Obtained results we apply to the known Koenigs embedding problem, the Krein-Phillips problem of existing of invariant semidefinite subspaces for some families of plus-operators and to some other fields.

Cite this paper

V. Khatskevich and V. Senderov, "Factorization of Operators in Krein Spaces and Linear-Fractional Relations of Operator Balls,"*Advances in Pure Mathematics*, Vol. 3 No. 1, 2013, pp. 29-33. doi: 10.4236/apm.2013.31006.

V. Khatskevich and V. Senderov, "Factorization of Operators in Krein Spaces and Linear-Fractional Relations of Operator Balls,"

References

[1] S. L. Sobolev, “On the Motion of a Symmetric Top with a Cavity Filled with a Liquid,” Zh. Prikl. Mekhan. i Tekhn. Fiz., No. 3, 1960, pp. 20-55.

[2] L. S. Pontryagin, “Hermitian Operators in Spaces with Indefinite Metric,” Izvestiya Akademii Nauk SSSR, Seriya Matematicheskaya, Vol. 8, 1944, pp. 243-280.

[3] R. S. Phillips, “Dissipative Operators and Hyperbolic Systems of Partial Differential Equations,” Transactions of the American Mathematical Society, Vol. 90, No. 2, 1959, pp. 193-254. doi:10.1090/S0002-9947-1959-0104919-1

[4] R. S. Phillips, “Dissipative Operators and Parabolic Partial Differential Equations,” Communications on Pure and Applied Mathematics, Vol. 12, No. 2, 1959, pp. 249-276. doi:10.1002/cpa.3160120206

[5] R. S. Phillips, “The Extensions of Dual Subspaces Invariant under an Algebra,” Proceedings of the International Symposium on Linear Spaces, Jerusalem, 5-12 July 1960, pp. 366-398.

[6] T. Ya. Azizov and I. S. Iokhvidov, “Foundations of Theory of Linear Operators in Spaces with Indefinite Metric,” Nauka, Moscow, 1986.

[7] M. G. Krein and Yu. L. Shmul’yan, “J-Polar Representation of Plus Operators,” Materialy Issledovaniya, Vol. 1, No. 2, 1966, pp. 172-210.

[8] V. A. Khatskevich, “Some Global Properties of Fractional-Linear Transformations,” Operator Theory, Vol. 73, 1994, pp. 355-361.

[9] V. A. Khatskevich and V. S. Shulman, “Operator Fractional-Linear Transformations: Convexity and Compactness of Image, Application,” Studia Mathematica, Vol. 116, No. 2, 1995, pp. 189-195.

[10] V. A. Khatskevich and L. Zelenko, “Indefinite Metrics and Dichotomy of Solutions for Linear Differential Equations in Hilbert Spaces,” Chinese Journal of Mathematics, Vol. 24, No. 2, 1996, pp. 99-112.

[11] V. A. Khatskevich and L. Zelenko, “The Fractional-Linear Transformations of the Operator Ball and Dichotomy of Solutions to Evaluation Equations,” Contemporary Mathematics, Vol. 204, 1997, pp. 149-154. doi:10.1090/conm/204/02628

[12] V. A. Khatskevich, “Generalized Fractional Linear Transformations: Convexity and Compactness of the Image and the Pre-Image; Applications,” Studia Mathematica, Vol. 137, No. 2, 1999, pp. 169-175.

[13] V. A. Khatskevich and L. Zelenko, “Bistrict Plus-Operators in Krein Spaces and Dichotomous Behavior of Irreversible Dynamical Systems,” Operator Theory: Advances and Applications, Vol. 118, 2000, pp. 191-203.

[14] V. Khatskevich and V. A. Senderov, “On Convexity, Compactness, and Non-Emptiness of Images and Preimages of Operator Linear-Fractional Relations,” Doklady Akademii Nauk, Vol. 69, No. 3, 2004, pp. 409-411.

[15] T. Ya. Azizov, “On Extension of Invariant Dual Pairs,” Ukrainian Mathematical Journal, Vol. 41, No. 7, 1989, pp. 958-961. doi:10.1007/BF01060700

[16] V. Khatskevich, V. Senderov and V. Shulman, “On Operator Matrices Generating Linear Fractional Maps of Operator Balls,” Contemporary Mathematics, Vol. 364, 2004, pp. 93-102. doi:10.1090/conm/364/06679

[17] N. Dunford and J. Schwartz, “Linear Operators,” Wiley, New York, 1958.

[18] V. Khatskevich and V. A. Senderov, “On Operator Sets Generated by Plus-Operators,” Vestnik Voronezhskogo Gosudarstvennogo Universiteta, Seriya Fizika, Matematika, No. 2, 2010, pp. 170-174.

[19] V. Khatskevich, M. Ostrovskii and V. Shulman, “Linear Fractional Relations for Hilbert Space Operators,” Mathematische Nachrichten, Vol. 279, No. 8, 2006, pp. 875- 890. doi:10.1002/mana.200310400

[20] T. Azizov and V. Khatskevich, “A Theorem on Existence of Invariant Subspaces for J-Bi-Expansive Operators,” Operator Theory: Advances and Applications, Vol. 198, 2009, pp. 41-48.

[1] S. L. Sobolev, “On the Motion of a Symmetric Top with a Cavity Filled with a Liquid,” Zh. Prikl. Mekhan. i Tekhn. Fiz., No. 3, 1960, pp. 20-55.

[2] L. S. Pontryagin, “Hermitian Operators in Spaces with Indefinite Metric,” Izvestiya Akademii Nauk SSSR, Seriya Matematicheskaya, Vol. 8, 1944, pp. 243-280.

[3] R. S. Phillips, “Dissipative Operators and Hyperbolic Systems of Partial Differential Equations,” Transactions of the American Mathematical Society, Vol. 90, No. 2, 1959, pp. 193-254. doi:10.1090/S0002-9947-1959-0104919-1

[4] R. S. Phillips, “Dissipative Operators and Parabolic Partial Differential Equations,” Communications on Pure and Applied Mathematics, Vol. 12, No. 2, 1959, pp. 249-276. doi:10.1002/cpa.3160120206

[5] R. S. Phillips, “The Extensions of Dual Subspaces Invariant under an Algebra,” Proceedings of the International Symposium on Linear Spaces, Jerusalem, 5-12 July 1960, pp. 366-398.

[6] T. Ya. Azizov and I. S. Iokhvidov, “Foundations of Theory of Linear Operators in Spaces with Indefinite Metric,” Nauka, Moscow, 1986.

[7] M. G. Krein and Yu. L. Shmul’yan, “J-Polar Representation of Plus Operators,” Materialy Issledovaniya, Vol. 1, No. 2, 1966, pp. 172-210.

[8] V. A. Khatskevich, “Some Global Properties of Fractional-Linear Transformations,” Operator Theory, Vol. 73, 1994, pp. 355-361.

[9] V. A. Khatskevich and V. S. Shulman, “Operator Fractional-Linear Transformations: Convexity and Compactness of Image, Application,” Studia Mathematica, Vol. 116, No. 2, 1995, pp. 189-195.

[10] V. A. Khatskevich and L. Zelenko, “Indefinite Metrics and Dichotomy of Solutions for Linear Differential Equations in Hilbert Spaces,” Chinese Journal of Mathematics, Vol. 24, No. 2, 1996, pp. 99-112.

[11] V. A. Khatskevich and L. Zelenko, “The Fractional-Linear Transformations of the Operator Ball and Dichotomy of Solutions to Evaluation Equations,” Contemporary Mathematics, Vol. 204, 1997, pp. 149-154. doi:10.1090/conm/204/02628

[12] V. A. Khatskevich, “Generalized Fractional Linear Transformations: Convexity and Compactness of the Image and the Pre-Image; Applications,” Studia Mathematica, Vol. 137, No. 2, 1999, pp. 169-175.

[13] V. A. Khatskevich and L. Zelenko, “Bistrict Plus-Operators in Krein Spaces and Dichotomous Behavior of Irreversible Dynamical Systems,” Operator Theory: Advances and Applications, Vol. 118, 2000, pp. 191-203.

[14] V. Khatskevich and V. A. Senderov, “On Convexity, Compactness, and Non-Emptiness of Images and Preimages of Operator Linear-Fractional Relations,” Doklady Akademii Nauk, Vol. 69, No. 3, 2004, pp. 409-411.

[15] T. Ya. Azizov, “On Extension of Invariant Dual Pairs,” Ukrainian Mathematical Journal, Vol. 41, No. 7, 1989, pp. 958-961. doi:10.1007/BF01060700

[16] V. Khatskevich, V. Senderov and V. Shulman, “On Operator Matrices Generating Linear Fractional Maps of Operator Balls,” Contemporary Mathematics, Vol. 364, 2004, pp. 93-102. doi:10.1090/conm/364/06679

[17] N. Dunford and J. Schwartz, “Linear Operators,” Wiley, New York, 1958.

[18] V. Khatskevich and V. A. Senderov, “On Operator Sets Generated by Plus-Operators,” Vestnik Voronezhskogo Gosudarstvennogo Universiteta, Seriya Fizika, Matematika, No. 2, 2010, pp. 170-174.

[19] V. Khatskevich, M. Ostrovskii and V. Shulman, “Linear Fractional Relations for Hilbert Space Operators,” Mathematische Nachrichten, Vol. 279, No. 8, 2006, pp. 875- 890. doi:10.1002/mana.200310400

[20] T. Azizov and V. Khatskevich, “A Theorem on Existence of Invariant Subspaces for J-Bi-Expansive Operators,” Operator Theory: Advances and Applications, Vol. 198, 2009, pp. 41-48.