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1997 | 66 | 1 | 223-238
Tytuł artykułu

$𝒞^∞$-vectors and boundedness

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EN
Abstrakty
EN
The following two questions as well as their relationship are studied: (i) Is a closed linear operator in a Banach space bounded if its $𝒞^∞$-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.
Twórcy
autor
  • Institute of Mathematics, Jagiellonian University, Reymonta 4, 30-059 Kraków, Poland
  • Institute of Mathematics, Jagiellonian University, Reymonta 4, 30-059 Kraków, Poland
Bibliografia
  • [1] P. R. Chernoff, Some remarks on quasi-analytic vectors, Trans. Amer. Math. Soc. 167 (1972), 105-113.
  • [2] P. R. Chernoff, A semibounded closed symmetric operator whose square has trivial domain, Proc. Amer. Math. Soc. 89 (1983), 289-290.
  • [3] J. Daneš, On local spectral radius, Časopis Pěst. Mat. 112 (1987), 177-187.
  • [4] W. G. Faris, Selfadjoint Operators, Lecture Notes in Math. 433, Springer, Berlin, 1975.
  • [5] T. Furuta, On the class of paranormal operators, Proc. Japan Acad. 43 (1967), 594-598.
  • [6] M. Hasegawa, On quasi-analytic vectors for dissipative operators, Proc. Amer. Math. Soc. 29 (1971), 81-84.
  • [7] W. Mlak, The Schrödinger type couples related to weighted shifts, Univ. Iagel. Acta Math. 27 (1988), 297-301.
  • [8] E. Nelson, Analytic vectors, Ann. of Math. 70 (1959), 572-615.
  • [9] A. E. Nussbaum, Quasi-analytic vectors, Ark. Mat. 6 (1967), 179-191.
  • [10] A. E. Nussbaum, A note on quasi-analytic vectors, Studia Math. 33 (1969), 305-309.
  • [11] Y. Okazaki, Boundedness of closed linear operator T satisfying R(T)⊂ D(T), Proc. Japan Acad. 62 (1986), 294-296.
  • [12] S. Ôta, Closed linear operators with domain containing their range, Proc. Edinburgh Math. Soc. 27 (1984), 229-233.
  • [13] S. Ôta, Unbounded nilpotents and idempotents, J. Math. Anal. Appl. 132 (1988), 300-308.
  • [14] K. Schmüdgen, On domains of powers of closed symmetric operators, J. Operator Theory 9 (1983), 53-75.
  • [15] K. Schmüdgen, Unbounded Operator Algebras and Representation Theory, Akademie-Verlag, Berlin, 1990.
  • [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.
  • [17] J. Stochel and F. H. Szafraniec, Boundedness of linear and related nonlinear maps, Part I, Exposition. Math. 1 (1983), 71-73.
  • [18] J. Stochel and F. H. Szafraniec, Boundedness of linear and related nonlinear maps, Part II, Exposition. Math. 2 (1984), 283-287.
  • [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.
  • [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.
  • [21] J. Stochel and F. H. Szafraniec, A few assorted questions about unbounded subnormal operators, Univ. Iagel. Acta Math. 28 (1991), 163-170.
  • [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.
  • [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.
  • [24] F. H. Szafraniec, Kato-Protter type inequalities, bounded vectors and the exponential function, Ann. Polon. Math. 51 (1990), 303-312.
Typ dokumentu
Bibliografia
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bwmeta1.element.bwnjournal-article-apmv66z1p223bwm
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