Selfadjoint operator matrices with finite rows

Jan Janas; Jan Stochel

Annales Polonici Mathematici (1997)

  • Volume: 66, Issue: 1, page 155-172
  • ISSN: 0066-2216

Abstract

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A generalization of the Carleman criterion for selfadjointness of Jacobi matrices to the case of symmetric matrices with finite rows is established. In particular, a new proof of the Carleman criterion is found. An extension of Jørgensen's criterion for selfadjointness of symmetric operators with "almost invariant" subspaces is obtained. Some applications to hyponormal weighted shifts are given.

How to cite

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Jan Janas, and Jan Stochel. "Selfadjoint operator matrices with finite rows." Annales Polonici Mathematici 66.1 (1997): 155-172. <http://eudml.org/doc/269958>.

@article{JanJanas1997,
abstract = {A generalization of the Carleman criterion for selfadjointness of Jacobi matrices to the case of symmetric matrices with finite rows is established. In particular, a new proof of the Carleman criterion is found. An extension of Jørgensen's criterion for selfadjointness of symmetric operators with "almost invariant" subspaces is obtained. Some applications to hyponormal weighted shifts are given.},
author = {Jan Janas, Jan Stochel},
journal = {Annales Polonici Mathematici},
keywords = {selfadjoint operator; band matrix; weighted shift; selfadjointness of symmetric operators with almost invariant subspaces; Carleman criterion; selfadjointness of Jacobi matrices; symmetric matrices with finite rows; Jørgensen’s criterion; hyponormal weighted shifts},
language = {eng},
number = {1},
pages = {155-172},
title = {Selfadjoint operator matrices with finite rows},
url = {http://eudml.org/doc/269958},
volume = {66},
year = {1997},
}

TY - JOUR
AU - Jan Janas
AU - Jan Stochel
TI - Selfadjoint operator matrices with finite rows
JO - Annales Polonici Mathematici
PY - 1997
VL - 66
IS - 1
SP - 155
EP - 172
AB - A generalization of the Carleman criterion for selfadjointness of Jacobi matrices to the case of symmetric matrices with finite rows is established. In particular, a new proof of the Carleman criterion is found. An extension of Jørgensen's criterion for selfadjointness of symmetric operators with "almost invariant" subspaces is obtained. Some applications to hyponormal weighted shifts are given.
LA - eng
KW - selfadjoint operator; band matrix; weighted shift; selfadjointness of symmetric operators with almost invariant subspaces; Carleman criterion; selfadjointness of Jacobi matrices; symmetric matrices with finite rows; Jørgensen’s criterion; hyponormal weighted shifts
UR - http://eudml.org/doc/269958
ER -

References

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  1. [1] V. Bargmann, On a Hilbert space of analytic functions and an associated integral transform, Part I, Comm. Pure Appl. Math. 14 (1961), 187-214. Zbl0107.09102
  2. [2] Yu. M. Berezanskiĭ, Eigenfunction Expansions for Selfadjoint Operators, Naukova Dumka, Kiev, 1965 (in Russian). 
  3. [3] T. Carleman, Sur la théorie mathématique de l'équation de Schrödinger, Ark. Mat. Astr. Fys. 24 (1934), 1-7. Zbl0009.35702
  4. [4] W. Dahmen and C. A. Micchelli, Banded matrices with banded inverses, II: locally finite decomposition of spline spaces, Constr. Approx 9 (1993), 263-281. Zbl0784.15005
  5. [5] S. Demko, Inverses of band matrices and local convergence of spline projectors, SIAM J. Numer. Anal. 14 (1977), 616-619. Zbl0367.65024
  6. [6] P. R. Halmos, A Hilbert Space Problem Book, Van Nostrand, Princeton, N.J., 1967. 
  7. [7] J. Janas, Unbounded Toeplitz operators in the Bargmann-Segal space, Studia Math. 99 (1991), 87-99. Zbl0766.47004
  8. [8] J. Janas and J. Stochel, Unbounded Toeplitz operators in the Segal-Bargmann space. II, J. Funct. Anal. 126 (1994), 418-447. Zbl0824.47021
  9. [9] P. E. T. Jørgensen, Approximately reducing subspaces for unbounded linear operators, J. Funct. Anal. 23 (1976), 392-414. Zbl0341.47024
  10. [10] P. E. T. Jørgensen, Essential self-adjointness of semibounded operators, Math. Ann. 237 (1978), 187-192. Zbl0386.47010
  11. [11] W. Mlak, The Schrödinger type couples related to weighted shifts, Univ. Iagel. Acta Math. 27 (1988), 297-301. Zbl0695.47022
  12. [12] M. H. Stone, Linear Transformations in Hilbert Space and Their Applications to Analysis, Amer. Math. Soc. Colloq. Publ. 15, Amer. Math. Soc., Providence, R.I., 1932. 
  13. [13] O. Toeplitz, Zur Theorie der quadratischen Formen von unendlichvielen Veränderlichen, Göttingen Nachr. 1910, 489-506. Zbl41.0381.02
  14. [14] J. Weidmann, Linear Operators in Hilbert Spaces, Springer, New York, 1980. Zbl0434.47001

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