## Banach Journal of Mathematical Analysis

### On a J-polar decomposition of a bounded operator and matrices of J-symmetric and J-skew-symmetric operators

Sergey M. Zagorodnyuk

#### Abstract

In this paper we study a possibility of a decomposition of a bounded operator in a Hilbert space $H$ as a product of a J-unitary and a J-self-adjoint operators, where J is a conjugation (an antilinear involution). This decomposition shows an inner structure of a bounded operator in a Hilbert space. Some decompositions of J-unitary and unitary operators which generalize decompositions in the finite-dimensional case are also obtained. Matrix representations for J-symmetric and J-skew-symmetric operators are studied. Simple basic properties of J-symmetric, J-skew-symmetric and J-isometric operators are obtained.

#### Article information

Source
Banach J. Math. Anal. Volume 4, Number 2 (2010), 11-36.

Dates
First available in Project Euclid: 7 February 2011

https://projecteuclid.org/euclid.bjma/1297117238

Digital Object Identifier
doi:10.15352/bjma/1297117238

Mathematical Reviews number (MathSciNet)
MR2606479

Zentralblatt MATH identifier
1200.47050

Subjects
Primary: 47B99: None of the above, but in this section
Secondary: 33C05: Classical hypergeometric functions, $_2F_1$

#### Citation

Zagorodnyuk, Sergey M. On a J-polar decomposition of a bounded operator and matrices of J-symmetric and J-skew-symmetric operators. Banach J. Math. Anal. 4 (2010), no. 2, 11--36. doi:10.15352/bjma/1297117238. https://projecteuclid.org/euclid.bjma/1297117238

#### References

• N.I. Akhiezer, Classical moment problem and some questions of analysis related to it , Fizmatlit., Moscow, 1961 (Russian).
• N.I. Akhiezer and I.M. Glazman, Theory of linear operators in a Hilbert space. Vol.1, Izdat-vo pri KhGU izd. obyed. Vysha shkola, Kharkov, 1977 (Russian).
• Sh. Asadi and I.E. Lutsenko, Skew-unitary transformations of operator bundles, Vestnik Kharkovskogo universiteta, Mehanika i matematika 37 No.83 (1972), 21–27 (Russian).
• F.R. Gantmacher, Theory of matrices, Nauka, Moscow, 1967 (Russian).
• S.R. Garcia, The eigenstructure of complex symmetric operators, Operator Theory: Advances and Applications 179 (2008), 169–184.
• S.R. Garcia and M. Putinar, Complex symmetric operators and applications, Trans. Amer. Math. Soc. 358 (2006), 1285-1315.
• S.R. Garcia and M. Putinar, Complex symmetric operators and applications II, Trans. Amer. Math. Soc. 359 (2007), 3913–3931.
• S. Garcia and W.T. Ross, A non-linear extremal problem on the Hardy space, Computational Methods and Function Theory 9 (2009), No.2, 485–524.
• E. Prodan, S.R. Garcia and M. Putinar, Norm estimates of complex symmetric operators applied to quantum systems, J. Phys. A: Math. and Gen. 39 (2006), 389–400.
• I.M. Glazman, On an analog of the theory of extensions of Hermitian operators and a non-symmetric one-dimensional boundary problem on a semi-axis, DAN SSSR 115 No.2 (1957), 214–216 (Russian).
• I.M. Glazman, Direct methods of qualititative spectral analysis of singular differential operators, Gos. izdat. fiz.-mat. liter., Moscow, 1963 (Russian).
• I.T. Gohberg and M.G. Krein, Theory of Volterra operators in a Hilbert space and its applications, Nauka, Moscow, 1967 (Russian).
• T.B. Kalinina, On the theory of extensions of K-symmetric operators, Funkcionalniy analiz (Ulyanovsk) 7 (1976), 78–85 (Russian).
• T.B. Kalinina, On one perturbation of an operator in a Hilbert space with an antiunitary transformation, Funkcionalniy analiz (Ulyanovsk) 7 (1979), 87–90 (Russian).
• T.B. Kalinina, On extensions of an operator in a Hilbert space with a skew-unitary transformation, Funkcionalniy analiz (Ulyanovsk) 17 (1981), 68–75 (Russian).
• T.B. Kalinina, One extension of an operator in a Hilbert space with a skew-unitary transformation, Funkcionalniy analiz (Ulyanovsk) 18 (1982), 63–71 (Russian).
• T.B. Kalinina, Generalized resolvents of an operator which is skew-symmetric with respect to an antilinear transformation of a Hilbert space, Funkcionalniy analiz (Ulyanovsk) 20 (1983), 60–72 (Russian).
• L.A. Kamerina, Self-adjoint extensions of a symmetric operator in a space with an involution, Funkcionalniy analiz (Ulyanovsk) 2 (1974), 16–26 (Russian).
• L.A. Kamerina, Unitary equivalence of operators of the class $K_y$, Funkcionalniy analiz (Ulyanovsk) 5 (1975), 72–78 (Russian).
• L.A. Kamerina, A symmetric operator in the orthogonal sum of two Hilbert spaces with an involution, Funkcionalniy analiz (Ulyanovsk) 7 (1976), 86–94 (Russian).
• L.A. Kamerina, Quasi-unitary equivalence of operators in spaces with an involution, Funkcionalniy analiz (Ulyanovsk) 27 (1987), 72–78 (Russian).
• I. Knowles, On the boundary conditions characterizing $J$-selfadjoint extensions of $J$-symmetric operators, J. Diff. Equations 40 (1981), 193–216.
• A.N. Kochubey, On extensions of $J$-symmetric operators, Funkcionalniy analiz (Ulyanovsk) 10 (1978), 93–97 (Russian).
• M.G. Krein and Yu.L. Shmulyan, $J$-polar representation of plus-operators, Matem. issledovaniya (Kishinev) Vol.1 2 (1966), 172–210 (Russian).
• V.P. Li, On the theory of $J$-symmetric operators, Funkcionalniy analiz (Ulyanovsk) 3 (1974), 84–91 (Russian).
• V.P. Li, A $J$-adjoint operator in a Hilbert space with an involution $J$, Funkcionalniy analiz (Ulyanovsk) 3 (1974), 77–83 (Russian).
• V.P. Li, Quasi-dissipative and quasi-acummulative extensions of a $J$-symmetric operator, Funkcionalniy analiz (Ulyanovsk) 4 (1975), 62–69 (Russian).
• A.D. Makarova, Extensions of $J$-symmetric operators with a non-dense domain, Funkcionalniy analiz (Ulyanovsk) 8 (1977), 102–112 (Russian).
• A.D. Makarova, On the generalized resolvent of a $J$-symmetric operator, Funkcionalniy analiz (Ulyanovsk) 14 (1980), 99–104 (Russian).
• A.D. Makarova, Extensions and generalized resolvents of an operator which commutes with an involution, Funkcionalniy analiz (Ulyanovsk) 20 (1983), 83–92 (Russian).
• B.G. Mironov, On the theory of $J$-symmetric operators with a non-dense domain, Funkcionalniy analiz (Ulyanovsk) 27 (1987), 128–133 (Russian).
• B.G. Mironov, On extensions and regular points of one extension of a $J$-symmetric operator, Funkcionalniy analiz (Ulyanovsk) 29 (1989), 103–111 (Russian).
• U.V. Riss, Extension of the Hilbert space by $J$-unitary transformations, Helv. Phys. Acta 71 (1998), 288–313.
• D. Race, The theory of $J$-selfadjoint extensions of $J$-symmetric operators, J. Diff. Equations 57 (1985), 258–274.
• L.M. Rayh, On an extension of $J$-symmetricc operator with a non-dense domain, Matem. zametki 17 N.5 (1975), 737–743.
• L.M. Rayh and E.R. Tsekanovskii, Biinvolutive self-adjoint biextensions of $J$-symmetric operators, Teoriya funkciy, funcionalniy analiz i ih prilozheniya (Kharkov) 23 (1975), 79–93.
• M.H. Stone, Linear transformations in Hilbert space and their applications to analysis, AMS Colloquium Publications, Vol. 15, Providence, Rhode Island, 1932.
• S.M. Zagorodnyuk, On generalized Jacobi matrices and orthogonal polynomials, New York J. Math. 9 (2003), 117–136.
• S.M. Zagorodnyuk, Direct and inverse spectral problems for (2N+1)-diagonal, complex, symmetric non-Hermitian matrices, Serdica Math. J. 30 (2004), 471–482.
• S.M. Zagorodnyuk, The direct and inverse spectral problems for (2N+1)-diagonal complex transposition-antisymmetric matrices, Methods Funct. Anal. Topology 14 (2008), 124–131.
• S.M. Zagorodnyuk, Integral representations for spectral functions of some non-selfadjoint Jacobi matrices, Methods Funct. Anal. Topology 15 (2009), 91–100.