On a property of weak resolvents and its application to a spectral problem

Yoichi Uetake

Annales Polonici Mathematici (1997)

  • Volume: 66, Issue: 1, page 263-268
  • ISSN: 0066-2216

Abstract

top
We show that the poles of a resolvent coincide with the poles of its weak resolvent up to their orders, for operators on Hilbert space which have some cyclic properties. Using this, we show that a theorem similar to the Mlak theorem holds under milder conditions, if a given operator and its adjoint have cyclic vectors.

How to cite

top

Yoichi Uetake. "On a property of weak resolvents and its application to a spectral problem." Annales Polonici Mathematici 66.1 (1997): 263-268. <http://eudml.org/doc/269991>.

@article{YoichiUetake1997,
abstract = {We show that the poles of a resolvent coincide with the poles of its weak resolvent up to their orders, for operators on Hilbert space which have some cyclic properties. Using this, we show that a theorem similar to the Mlak theorem holds under milder conditions, if a given operator and its adjoint have cyclic vectors.},
author = {Yoichi Uetake},
journal = {Annales Polonici Mathematici},
keywords = {weak resolvent; cyclic vector; spectral radius; Hardy class; operator model theory; scattering theory; control theory; poles of a resolvent; Mlak theorem; cyclic vectors},
language = {eng},
number = {1},
pages = {263-268},
title = {On a property of weak resolvents and its application to a spectral problem},
url = {http://eudml.org/doc/269991},
volume = {66},
year = {1997},
}

TY - JOUR
AU - Yoichi Uetake
TI - On a property of weak resolvents and its application to a spectral problem
JO - Annales Polonici Mathematici
PY - 1997
VL - 66
IS - 1
SP - 263
EP - 268
AB - We show that the poles of a resolvent coincide with the poles of its weak resolvent up to their orders, for operators on Hilbert space which have some cyclic properties. Using this, we show that a theorem similar to the Mlak theorem holds under milder conditions, if a given operator and its adjoint have cyclic vectors.
LA - eng
KW - weak resolvent; cyclic vector; spectral radius; Hardy class; operator model theory; scattering theory; control theory; poles of a resolvent; Mlak theorem; cyclic vectors
UR - http://eudml.org/doc/269991
ER -

References

top
  1. [1] H. R. Dowson, Spectral Theory of Linear Operators, Academic Press, London, 1978. Zbl0384.47001
  2. [2] N. Dunford and J. T. Schwartz, Linear Operators, Part I: General Theory, Intescience, New York, 1958. Zbl0084.10402
  3. [3] C. K. Fong, E. A. Nordgren, H. Radjavi and P. Rosenthal, Weak resolvents of linear operators, II, Indiana Univ. Math. J. 39 (1990), 67-83. Zbl0729.47002
  4. [4] J. W. Helton, Discrete time systems, operator models, and scattering theory, J. Funct. Anal. 16 (1974), 15-38. Zbl0282.93033
  5. [5] J. W. Helton, Systems with infinite-dimensional state space: the Hilbert space approach, Proc. IEEE 64 (1976), 145-160. 
  6. [6] P. Jakóbczak and J. Janas, On Nikolski theorem for several operators, Bull. Polish Acad. Sci. Math. 31 (1983), 369-374. Zbl0547.47001
  7. [7] J. Janas, On a theorem of Lebow and Mlak for several commuting operators, Studia Math. 76 (1983), 249-253. Zbl0535.47003
  8. [8] T. Kailath, Linear Systems, Prentice-Hall, Englewood Cliffs, N.J., 1980. 
  9. [9] R. E. Kalman, P. L. Falb and M. A. Arbib, Topics in Mathematical System Theory, McGraw-Hill, New York, 1969. Zbl0231.49001
  10. [10] T. Kato, Perturbation Theory for Linear Operators, 2nd ed., Springer, Berlin, 1976. Zbl0342.47009
  11. [11] P. D. Lax and R. S. Phillips, Scattering Theory, rev. ed., Academic Press, New York, 1989. 
  12. [12] A. Lebow, Spectral radius of an absolutely continuous operator, Proc. Amer. Math. Soc. 36 (1972), 511-514. Zbl0273.47001
  13. [13] W. Mlak, On a theorem of Lebow, Ann. Polon. Math. 35 (1977), 107-109. Zbl0371.47007
  14. [14] N. K. Nikol'skiĭ, A Tauberian theorem on the spectral radius, Sibirsk. Mat. Zh. 18 (1977), 1367-1372 (in Russian). 
  15. [15] E. Nordgren, H. Radjavi and P. Rosenthal, Weak resolvents of linear operators, Indiana Univ. Math. J. 36 (1987), 913-934. Zbl0644.47002
  16. [15] W. Rudin, Real and Complex Analysis, 2nd ed., McGraw-Hill, New York, 1974. Zbl0278.26001
  17. [16] B. Sz.-Nagy and C. Foiaş, Harmonic Analysis of Operators on Hilbert Space, North-Holland, Amsterdam, 1970. Zbl0201.45003

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.