∞-vectors and boundedness
Jan Stochel ; F. H. Szafraniec
Annales Polonici Mathematici, Tome 66 (1997), p. 223-238 / Harvested from The Polish Digital Mathematics Library
Access to full text
Full (PDF)

The following two questions as well as their relationship are studied: (i) Is a closed linear operator in a Banach space bounded if its ∞-vectors coincide with analytic (or semianalytic) ones? (ii) When are the domains of two successive powers of the operator in question equal? The affirmative answer to the first question is established in case of paranormal operators. All these investigations are illustrated in the context of weighted shifts.

Publié le : 1997-01-01
EUDML-ID : urn:eudml:doc:270026 
@article{bwmeta1.element.bwnjournal-article-apmv66z1p223bwm,
     author = {Jan Stochel and F. H. Szafraniec},
     title = {$^$\infty$$-vectors and boundedness},
     journal = {Annales Polonici Mathematici},
     volume = {66},
     year = {1997},
     pages = {223-238},
     zbl = {0874.47002},
     language = {en},
     url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-apmv66z1p223bwm}
}

Jan Stochel; F. H. Szafraniec. $^∞$-vectors and boundedness. Annales Polonici Mathematici, Tome 66 (1997) pp. 223-238. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-apmv66z1p223bwm/

[000] [1] P. R. Chernoff, Some remarks on quasi-analytic vectors, Trans. Amer. Math. Soc. 167 (1972), 105-113. | Zbl 0259.47025

[001] [2] P. R. Chernoff, A semibounded closed symmetric operator whose square has trivial domain, Proc. Amer. Math. Soc. 89 (1983), 289-290. | Zbl 0526.47011

[002] [3] J. Daneš, On local spectral radius, Časopis Pěst. Mat. 112 (1987), 177-187. | Zbl 0645.47002

[003] [4] W. G. Faris, Selfadjoint Operators, Lecture Notes in Math. 433, Springer, Berlin, 1975.

[004] [5] T. Furuta, On the class of paranormal operators, Proc. Japan Acad. 43 (1967), 594-598. | Zbl 0163.37706

[005] [6] M. Hasegawa, On quasi-analytic vectors for dissipative operators, Proc. Amer. Math. Soc. 29 (1971), 81-84. | Zbl 0219.47025

[006] [7] W. Mlak, The Schrödinger type couples related to weighted shifts, Univ. Iagel. Acta Math. 27 (1988), 297-301. | Zbl 0695.47022

[007] [8] E. Nelson, Analytic vectors, Ann. of Math. 70 (1959), 572-615. | Zbl 0091.10704

[008] [9] A. E. Nussbaum, Quasi-analytic vectors, Ark. Mat. 6 (1967), 179-191. | Zbl 0182.46102

[009] [10] A. E. Nussbaum, A note on quasi-analytic vectors, Studia Math. 33 (1969), 305-309. | Zbl 0189.43903

[010] [11] Y. Okazaki, Boundedness of closed linear operator T satisfying R(T)⊂ D(T), Proc. Japan Acad. 62 (1986), 294-296. | Zbl 0618.47002

[011] [12] S. Ôta, Closed linear operators with domain containing their range, Proc. Edinburgh Math. Soc. 27 (1984), 229-233. | Zbl 0537.47018

[012] [13] S. Ôta, Unbounded nilpotents and idempotents, J. Math. Anal. Appl. 132 (1988), 300-308. | Zbl 0656.47039

[013] [14] K. Schmüdgen, On domains of powers of closed symmetric operators, J. Operator Theory 9 (1983), 53-75. | Zbl 0507.47009

[014] [15] K. Schmüdgen, Unbounded Operator Algebras and Representation Theory, Akademie-Verlag, Berlin, 1990.

[015] [16] B. Simon, The theory of semi-analytic vectors: a new proof of a theorem of Masson and McClary, Indiana Univ. Math. J. 20 (1971), 1145-1151. | Zbl 0244.47022

[016] [17] J. Stochel and F. H. Szafraniec, Boundedness of linear and related nonlinear maps, Part I, Exposition. Math. 1 (1983), 71-73. | Zbl 0533.47037

[017] [18] J. Stochel and F. H. Szafraniec, Boundedness of linear and related nonlinear maps, Part II, Exposition. Math. 2 (1984), 283-287. | Zbl 0598.47071

[018] [19] J. Stochel and F. H. Szafraniec, Bounded vectors and formally normal operators, in: Dilation Theory, Toeplitz Operators, and Other Topics, Proc. 7th Internat. Conf. on Oper. Theory, Timişoara and Herculane (Romania), June 7-17, 1982, C. Apostol, C. M. Pearcy, B. Sz.-Nagy and D. Voiculescu (eds.), Oper. Theory Adv. Appl. 11, Birkhäuser, Basel, 1983, 363-370. | Zbl 0544.47020

[019] [20] J. Stochel and F. H. Szafraniec, The normal part of an unbounded operator, Nederl. Akad. Wetensch. Proc. Ser. A 92 (1989), 495-503 = Indag. Math. 51 (1989), 495-503. | Zbl 0699.47020

[020] [21] J. Stochel and F. H. Szafraniec, A few assorted questions about unbounded subnormal operators, Univ. Iagel. Acta Math. 28 (1991), 163-170. | Zbl 0748.47015

[021] [22] F. H. Szafraniec, On the boundedness condition involved in dilation theory, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 24 (1976), 877-881. | Zbl 0343.47003

[022] [23] F. H. Szafraniec, Boundedness of the shift operator related to positive definite forms: an application to moment problems, Ark. Mat. 19 (1981), 251-259. | Zbl 0504.47030

[023] [24] F. H. Szafraniec, Kato-Protter type inequalities, bounded vectors and the exponential function, Ann. Polon. Math. 51 (1990), 303-312. | Zbl 0733.47038