Abstract
A local lifting theorem for bounded operators that intertwine a pair of jointly subnormal families of unbounded operators is proved. Each family in question is assumed to be composed of operators defined on a common invariant domain consisting of “joint” analytic vectors. This result can be viewed as a generalization of the local lifting theorem proved by Sebestyén, Thomson and the present authors for pairs of bounded subnormal operators.
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References
Birman M.Sh., Solomjak M.Z.: Spectral Theory of Selfadjoint Operators in Hilbert Space. D. Reidel Publishing Co., Dordrecht (1987)
Bram J.: Subnormal operators. Duke Math. J. 22, 75–94 (1955)
Conway, J.B.: The theory of subnormal operators, Mathematical Surveys and Monographs, Providence, Rhode Island (1991)
Conway, J.B.: A course in operator theory, Graduate Studies in Mathematics 21, American Mathematical Society, Providence, Rhode Island (2000)
Deddens J.A., Fillmore P.A.: Reflexive linear transformations. Linear Algebra Appl. 10, 89–93 (1975)
Majdak W., Sebestyén Z., Stochel J., Thomson J.E.: A local lifting theorem for subnormal operators. Proc. Amer. Math. Soc. 134, 1687–1699 (2006)
Maserick P.H.: Spectral theory of operator-valued transformations. J. Math. Anal. Appl. 41, 497–507 (1973)
Nelson E.: Analytic vectors. Ann. Math. 70, 572–615 (1959)
Olin R.F., Thomson J.E.: Algebras of subnormal operators. J. Funct. Anal. 37, 271–301 (1980)
Putnam C.R.: On normal operators in Hilbert space. Amer. J. Math. 73, 357–362 (1951)
Stochel J.: Lifting strong commutants of unbounded subnormal operators. Integr. Equ. Oper. Theory 43, 189–214 (2002)
Stochel J., Szafraniec F.H.: On normal extensions of unbounded operators. I. J. Oper. Theory 14, 31–55 (1985)
Stochel J., Szafraniec F.H.: On normal extensions of unbounded operators. II. Acta Sci. Math. (Szeged) 53, 153–177 (1989)
Stochel J., Szafraniec F.H.: On normal extensions of unbounded operators. III. Spectral properties. Publ. Res. Inst. Math. Sci. 25, 105–139 (1989)
Stochel J., Szafraniec F.H.: The normal part of an unbounded operator. Nederl. Akad. Wetensch. Indag. Math. 51, 495–503 (1989)
Stochel J., Szafraniec F.H.: \({\mathcal{C}^\infty}\) -vectors and boundedness. Ann. Polon. Math. 66, 223–238 (1997)
Stochel J., Szafraniec F.H.: The complex moment problem and subnormality: a polar decomposition approach. J. Funct. Anal. 159, 432–491 (1998)
Stochel J., Szafraniec F.H.: Domination of unbounded operators and commutativity. J. Math. Soc. Jpn. 55(2), 405–437 (2003)
Wogen W.R.: Some counterexamples in nonselfadjoint algebras. Ann. Math. 126, 415–427 (1987)
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The authors would like to thank the referee for a careful reading of the manuscript and for many suggestions that helped to improve the final version of the paper.
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W. Majdak was supported by the AGH local grant 10.420.03. J. Stochel was supported by the MNiSzW grant N201 026 32/1350.
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Open Access This is an open access article distributed under the terms of the Creative Commons Attribution Noncommercial License (https://creativecommons.org/licenses/by-nc/2.0), which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.
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Majdak, W., Stochel, J. A Local Lifting Theorem for Jointly Subnormal Families of Unbounded Operators. Integr. Equ. Oper. Theory 69, 233–246 (2011). https://doi.org/10.1007/s00020-010-1836-1
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DOI: https://doi.org/10.1007/s00020-010-1836-1