Finite operators and similarity orbits
HTML articles powered by AMS MathViewer
- by Lawrence A. Fialkow and Domingo A. Herrero PDF
- Proc. Amer. Math. Soc. 93 (1985), 601-609 Request permission
Abstract:
In this note we answer the question of J. P. Williams as to which Hilbert space operators $T$ have the property that every similarity transformation ${W^{ - 1}}TW$ is a finite operator: $T$ has this property if and only if its image in the Calkin algebra satisfies a quadratic equation.References
-
J. H. Anderson, Derivations, commutators, and the essential numerical range, Dissertation, Indiana University, 1971.
- J. H. Anderson, Derivation ranges and the identity, Bull. Amer. Math. Soc. 79 (1973), 705–708. MR 322518, DOI 10.1090/S0002-9904-1973-13271-9 C. Apóstol, L. A. Fialkow, D. A. Herrero and D. Voiculescu, Approximation of Hilbert space operators. II, Res. Notes in Math., Pitman, Boston, Mass., 1984.
- Constantin Apostol, Domingo A. Herrero, and Dan Voiculescu, The closure of the similarity orbit of a Hilbert space operator, Bull. Amer. Math. Soc. (N.S.) 6 (1982), no. 3, 421–426. MR 648526, DOI 10.1090/S0273-0979-1982-15007-8
- L. A. Fialkow, The similarity orbit of a normal operator, Trans. Amer. Math. Soc. 210 (1975), 129–137. MR 374956, DOI 10.1090/S0002-9947-1975-0374956-X
- P. R. Halmos, Quasitriangular operators, Acta Sci. Math. (Szeged) 29 (1968), 283–293. MR 234310
- P. R. Halmos, Irreducible operators, Michigan Math. J. 15 (1968), 215–223. MR 231233
- P. R. Halmos, Ten problems in Hilbert space, Bull. Amer. Math. Soc. 76 (1970), 887–933. MR 270173, DOI 10.1090/S0002-9904-1970-12502-2
- Domingo A. Herrero, Quasidiagonality, similarity and approximation by nilpotent operators, Indiana Univ. Math. J. 30 (1981), no. 2, 199–233. MR 604280, DOI 10.1512/iumj.1981.30.30017 —, Approximation of Hilbert space operators. I, Res. Notes in Math., Vol. 72, Pitman, Boston, Mass., 1982.
- Domingo A. Herrero, On derivation ranges and the identity operator, J. Operator Theory 7 (1982), no. 1, 139–148. MR 650198
- Domingo A. Herrero, On quasidiagonal weighted shifts and approximation of operators, Indiana Univ. Math. J. 33 (1984), no. 4, 549–571. MR 749314, DOI 10.1512/iumj.1984.33.33029
- V. P. Havin, S. V. Hruščëv, and N. K. Nikol′skiĭ (eds.), Linear and complex analysis problem book, Lecture Notes in Mathematics, vol. 1043, Springer-Verlag, Berlin, 1984. 199 research problems. MR 734178, DOI 10.1007/BFb0072183
- Stefan Hildebrandt, Über den numerischen Wertebereich eines Operators, Math. Ann. 163 (1966), 230–247 (German). MR 200725, DOI 10.1007/BF02052287
- G. Lumer, Semi-inner-product spaces, Trans. Amer. Math. Soc. 100 (1961), 29–43. MR 133024, DOI 10.1090/S0002-9947-1961-0133024-2
- Johann von Neumann, Eine Spektraltheorie für allgemeine Operatoren eines unitären Raumes, Math. Nachr. 4 (1951), 258–281 (German). MR 43386, DOI 10.1002/mana.3210040124
- Carl Pearcy and Norberto Salinas, Extensions of $C^*$-algebras and the reducing essential matricial spectra of an operator, $K$-theory and operator algebras (Proc. Conf., Univ. Georgia, Athens, Ga., 1975) Lecture Notes in Math., Vol. 575, Springer, Berlin, 1977, pp. 96–112. MR 0467334
- Norberto Salinas, Reducing essential eigenvalues, Duke Math. J. 40 (1973), 561–580. MR 390816
- Robert Schatten, Norm ideals of completely continuous operators, Ergebnisse der Mathematik und ihrer Grenzgebiete, (N.F.), Heft 27, Springer-Verlag, Berlin-Göttingen-Heidelberg, 1960. MR 0119112 R. A. Smucker, Quasidiagonal and quasitriangular operators, Dissertation, Indiana University, 1973.
- Joseph G. Stampfli, Derivations on ${\cal B}({\cal H})$: the range, Illinois J. Math. 17 (1973), 518–524. MR 318914
- J. G. Stampfli and J. P. Williams, Growth conditions and the numerical range in a Banach algebra, Tohoku Math. J. (2) 20 (1968), 417–424. MR 243352, DOI 10.2748/tmj/1178243070
- Helmut Wielandt, Über die Unbeschränktheit der Operatoren der Quantenmechanik, Math. Ann. 121 (1949), 21 (German). MR 30701, DOI 10.1007/BF01329611
- James P. Williams, Similarity and the numerical range, J. Math. Anal. Appl. 26 (1969), 307–314. MR 240664, DOI 10.1016/0022-247X(69)90154-1
- J. P. Williams, Finite operators, Proc. Amer. Math. Soc. 26 (1970), 129–136. MR 264445, DOI 10.1090/S0002-9939-1970-0264445-6
- Aurel Wintner, The unboundedness of quantum-mechanical matrices, Phys. Rev. (2) 71 (1947), 738–739. MR 20724
- Dan Voiculescu, A non-commutative Weyl-von Neumann theorem, Rev. Roumaine Math. Pures Appl. 21 (1976), no. 1, 97–113. MR 415338
Additional Information
- © Copyright 1985 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 93 (1985), 601-609
- MSC: Primary 47A66; Secondary 47A12, 47A65
- DOI: https://doi.org/10.1090/S0002-9939-1985-0776187-X
- MathSciNet review: 776187