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

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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

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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

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  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

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