Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

On the distance between unitary orbits of weighted shifts


Author: Laurent Marcoux
Journal: Trans. Amer. Math. Soc. 326 (1991), 585-612
MSC: Primary 47B37; Secondary 47A30, 47C99
DOI: https://doi.org/10.1090/S0002-9947-1991-1010887-9
MathSciNet review: 1010887
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: In this paper, we consider invertible bilateral weighted shift operators acting on a complex separable Hilbert space $ \mathcal{H}$. They have the property that there exist a constant $ \tau > 0$ and an orthonormal basis $ {\{ {{e_i}} \}_{i \in \mathbb{Z}}}$ for $ \mathcal{H}$ with respect to which a shift $ V$ acts by $ W{e_i}= {w_i}{e_{i + 1}},i \in \mathbb{Z}$ and $ {\mathbf{\vert}}{w_i}{\mathbf{\vert}} \geq \tau $. The equivalence class $ \mathcal{U}(W)= \{ {U^{\ast}}\;WU:U \in \mathcal{B}(\mathcal{H}),U\;{\text{unitary}}\} $ of weighted shifts with weight sequence (with respect to the basis $ {\{ {U^{\ast}}{e_i}\} _{i \in \mathbb{Z}}}$ for $ \mathcal{H})$ identical to that of $ W$ forms the unitary orbit of $ W$.

Given two shifts $ W$ and $ V$, one can define a distance $ d(\mathcal{U}(V),\mathcal{U}(W))= \inf \{\parallel \,X - Y\parallel :X \in \mathcal{U}(V),Y \in \mathcal{U}(W)\} $ between the unitary orbits of $ W$ and $ V$. We establish numerical estimates for upper and lower bounds on this distance which depend upon information drawn from finite dimensional restrictions of these operators.


References [Enhancements On Off] (What's this?)

  • [AHHK] W. B. Arveson, D. W. Hadwin, T. B. Hoover, and E. E. Kymala, Circular operators, Indiana Univ. Math. J. 33 (1984), 583-595. MR 749316 (86b:47050)
  • [AD] E. Azoff and C. Davis, On the distance between unitary orbits of self-adjoint operators, Acta Sci. Math. (Szeged) 47 (1984), 419-439. MR 783316 (86g:47020)
  • [Arv] W. B. Arveson, Notes on extensions of $ {C^{\ast}}$-algebras, Duke Math. J. 44 (1977), 329-355. MR 0438137 (55:11056)
  • [Brgl] I. D. Berg, On approximation of normal operators by weighted shifts, Michigan Math. J. 21 (1974), 377-383. MR 0370235 (51:6462)
  • [Brg2] -, Index theory for perturbations of direct sums of normal operators and weighted shifts, Canad. J. Math. 30 (1978), 1152-1165. MR 511553 (81a:47029)
  • [BDM] R. Bhatia, C. Davis, and A McIntosh, Perturbation of spectral subspaces and solution of linear operator equations, Linear Algebra Appl. 52:53 (1983), 45-67. MR 709344 (85a:47020)
  • [BDF] L. G. Brown, R. G. Douglas, and P. A. Fillmore, Unitary equivalence modulo the compact operators and extensions of $ {C^{\ast}}$-algebras, (Proc. Conf. Operator Theory, Halifax, Nova Scotia, 1973), Lecture Notes in Math., vol. 345, Springer-Verlag, Berlin, Heidelberg and New York, 1973, pp. 58-128. MR 0380478 (52:1378)
  • [Davl] K. R. Davidson, Estimating the distance between unitary orbits, J. Operator Theory (to appear). MR 972178 (90c:47025)
  • [Dav2] -, The distance between unitary orbits of normal operators, Acta Sci. Math. (Szeged) 50 (1986), 213-223. MR 862194 (88c:47040)
  • [Dav3] -, The distance between unitary orbits of normal elements in the Calkin algebra, Proc. Roy. Soc. Edinburgh 99A (1984), 35-43. MR 781083 (86d:47017)
  • [Dav4] -, Berg's technique and irrational rotation algebras, Proc. Roy. Irish Acad. 84A (1984), 117-123.
  • [Dve] A. M. Davie, Classification of essentially normal operators, Lecture Notes in Math., vol. 512, Springer-Verlag, 1975, pp. 29-55. MR 0493473 (58:12478)
  • [Gel] R. Gellar, Circular symmetric operators and subnormal operators, J. Analyse Math. 32 (1977), 93-117. MR 0493474 (58:12479)
  • [GP] R. Gellar and L. Page, Limits of unitarily equivalent normal operators, Duke Math. J. 41 (1974), 319-322. MR 0338817 (49:3581)
  • [Had] D. W. Hadwin, An operator-valued spectrum, Indiana Univ. Math. J. (2) 26 (1977), 329-340. MR 0428089 (55:1118)
  • [HH] D. W. Hadwin and T. B. Hoover, Weighted translation and weighted shift operators, Lecture Notes in Math., vol. 693, Springer-Verlag, 1978, pp. 93-99. MR 526536 (80g:47034)
  • [Hal1] P. R. Halmos, Limits of shifts, Acta Sci. Math. (Szeged) 34 (1973), 131-139. MR 0338812 (49:3576)
  • [Hal2] -, A Hilbert space problem book, Graduate Texts in Math., vol. 19, Springer-Verlag, 1974.
  • [Her1] D. A. Herrero, On quasidiagonal weighted shifts and approximation of operators, Indiana Univ. Math. J. (4) 33 (1984), 549-571. MR 749314 (86m:47036)
  • [Her2] -, Approximation of Hilbert space operators, Research Notes in Math. 72, Pitman, 1982. [Lam] A. Lambert, Unitary equivalence and reducibility of invertibly weighted shifts, Bull. Austral. Math. Soc. 5 (1971).
  • [Mar] L. Marcoux, Ph. D. Thesis, Univ. of Waterloo, 1988.
  • [McP] A. McIntosh and A. Pryde, The solution of systems of operator equations using Clifford algebras, Proc. Centre Math. Anal. Austral. Nat. Univ., Canberra, vol. 9, 1985, pp. 212-222. MR 825528 (88a:47018)
  • [MM] B. B. Morrell and P. S. Muhly, Centered operators, Studia Math. 51 (1974), 251-263. MR 0355658 (50:8132)
  • [O'Dn] D. O'Donovan, Weighted shifts and covariance algebras, Trans. Amer. Math. Soc. 208 (1975), 1-25. MR 0385632 (52:6492)
  • [Shd] A. L. Shields, Weighted shift operators and analytic function theory, Math. Surveys, no. 13, Amer. Math. Soc., Providence, R. I., 1974, pp. 49-128. MR 0361899 (50:14341)
  • [Vcu] D. Voiculescu's, A non-commutative Weyl-von Neumann theorem, Rev. Roumaine Math. Pures Appl. 21 (1976), 97-113. MR 0415338 (54:3427)
  • [Vgt] J. Voigt, Perturbation theory for commutative $ m$-tuples of self-adjoint operators, J. Funct. Anal. 25 (1977), 317-334. MR 0451011 (56:9301)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 47B37, 47A30, 47C99

Retrieve articles in all journals with MSC: 47B37, 47A30, 47C99


Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1991-1010887-9
Article copyright: © Copyright 1991 American Mathematical Society

American Mathematical Society