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.
@article{bwmeta1.element.bwnjournal-article-apmv66z1p263bwm,
author = {Yoichi Uetake},
title = {On a property of weak resolvents and its application to a spectral problem},
journal = {Annales Polonici Mathematici},
volume = {66},
year = {1997},
pages = {263-268},
zbl = {0872.47001},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-apmv66z1p263bwm}
}
Yoichi Uetake. On a property of weak resolvents and its application to a spectral problem. Annales Polonici Mathematici, Tome 66 (1997) pp. 263-268. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-apmv66z1p263bwm/
[000] [1] H. R. Dowson, Spectral Theory of Linear Operators, Academic Press, London, 1978. | Zbl 0384.47001
[001] [2] N. Dunford and J. T. Schwartz, Linear Operators, Part I: General Theory, Intescience, New York, 1958. | Zbl 0084.10402
[002] [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. | Zbl 0729.47002
[003] [4] J. W. Helton, Discrete time systems, operator models, and scattering theory, J. Funct. Anal. 16 (1974), 15-38. | Zbl 0282.93033
[004] [5] J. W. Helton, Systems with infinite-dimensional state space: the Hilbert space approach, Proc. IEEE 64 (1976), 145-160.
[005] [6] P. Jakóbczak and J. Janas, On Nikolski theorem for several operators, Bull. Polish Acad. Sci. Math. 31 (1983), 369-374. | Zbl 0547.47001
[006] [7] J. Janas, On a theorem of Lebow and Mlak for several commuting operators, Studia Math. 76 (1983), 249-253. | Zbl 0535.47003
[007] [8] T. Kailath, Linear Systems, Prentice-Hall, Englewood Cliffs, N.J., 1980.
[008] [9] R. E. Kalman, P. L. Falb and M. A. Arbib, Topics in Mathematical System Theory, McGraw-Hill, New York, 1969. | Zbl 0231.49001
[009] [10] T. Kato, Perturbation Theory for Linear Operators, 2nd ed., Springer, Berlin, 1976. | Zbl 0342.47009
[010] [11] P. D. Lax and R. S. Phillips, Scattering Theory, rev. ed., Academic Press, New York, 1989.
[011] [12] A. Lebow, Spectral radius of an absolutely continuous operator, Proc. Amer. Math. Soc. 36 (1972), 511-514. | Zbl 0273.47001
[012] [13] W. Mlak, On a theorem of Lebow, Ann. Polon. Math. 35 (1977), 107-109. | Zbl 0371.47007
[013] [14] N. K. Nikol'skiĭ, A Tauberian theorem on the spectral radius, Sibirsk. Mat. Zh. 18 (1977), 1367-1372 (in Russian).
[014] [15] E. Nordgren, H. Radjavi and P. Rosenthal, Weak resolvents of linear operators, Indiana Univ. Math. J. 36 (1987), 913-934. | Zbl 0644.47002
[015] [15] W. Rudin, Real and Complex Analysis, 2nd ed., McGraw-Hill, New York, 1974. | Zbl 0278.26001
[016] [16] B. Sz.-Nagy and C. Foiaş, Harmonic Analysis of Operators on Hilbert Space, North-Holland, Amsterdam, 1970. | Zbl 0201.45003