Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



Linear maps preserving the isomorphism class
of lattices of invariant subspaces

Authors: Ali A. Jafarian, Leiba Rodman and Peter Semrl
Journal: Proc. Amer. Math. Soc. 126 (1998), 3607-3617
MSC (1991): Primary 47A15
MathSciNet review: 1610913
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Let ${\mathcal{V}}$ be an $n$-dimensional complex linear space and ${\mathcal{L}}({\mathcal{V}})$ the algebra of all linear transformations on ${\mathcal{V}}$. We prove that every linear map on ${\mathcal{L}}({\mathcal{V}})$, which maps every operator into an operator with isomorphic lattice of invariant subspaces, is an inner automorphism or an inner antiautomorphism multiplied by a nonzero constant and additively perturbed by a scalar type operator. The same result holds if we replace the lattice of invariant subspaces by the lattice of hyperinvariant subspaces or the set of reducing subspaces. Some of these results are extended to linear transformations of finite-dimensional linear spaces over fields other than the complex numbers. We also characterize linear bijective maps on the algebra of linear bounded operators on an infinite-dimensional complex Hilbert space which have similar properties with respect to the lattice of all invariant subpaces (not necessarily closed).

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

  • [BPW] P. Botta, S. Pierce, and W. Watkins, Linear transformations that preserve the nilpotent matrices, Pacific J. Math. 104 (1983), 39-46. MR 84c:15014
  • [BrS] M. Bre\v{s}ar and P. \v{S}emrl, On local automorphisms and mappings that preserve idempotents, Studia Math. 113 (1995), 101-108. MR 96i:47058
  • [Che] P.R. Chernoff, Representations, automorphisms and derivations of some operator algebras, J. Funct. Anal. 12 (1973), 275-289. MR 50:2934
  • [GLR] I. Gohberg, P. Lancaster, and L. Rodman, Invariant subspaces of matrices with applications, Wiley-Interscience, New York, 1986. MR 88a:15001
  • [GR] I. Gohberg and L. Rodman, On distance between lattices of invariant subspaces of matrices, Linear Algebra Appl. 76 (1986), 85-120. MR 87h:47014
  • [Her] I.N. Herstein, Topics in ring theory, University of Chicago Press, Chicago, 1969. MR 42:6018
  • [JaS] A.A. Jafarian and A.R. Sourour, Linear maps that preserve the commutant, double commutant or the lattice of invariant subspaces, Linear and Multilinear Algebra 38 (1994), 117-129. MR 95j:47043
  • [O] V. Olshevsky, A condition for the closeness of the sets of invariant subspaces of close matrices in terms of their Jordan structure, Siberian J. of Math. 30 (1989), 102-110. MR 91b:47010
  • [PeT] C. Pearcy and D. Topping, Sums of small number of idempotents, Michigan Math. J. 14 (1967), 453-465. MR 36:2006
  • [RaR] H. Radjavi and P. Rosenthal, Invariant subspaces, Springer-Verlag, Berlin, 1973. MR 51:3924
  • [Sem] P. \v{S}emrl, Characterization of matrices having rank $k$, Linear Algebra and Appl. 42 (1997), 233-238. CMP 98:06

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (1991): 47A15

Retrieve articles in all journals with MSC (1991): 47A15

Additional Information

Ali A. Jafarian
Affiliation: Department of Mathematics, University of New Haven, West Haven, Connecticut 06516

Leiba Rodman
Affiliation: Department of Mathematics, College of William and Mary, Williamsburg, Virginia 23187-8795

Peter Semrl
Affiliation: Faculty of Mechanical Engineering, University of Maribor, Smetanova 17, 2000 Maribor, Slovenia

Received by editor(s): April 21, 1997
Additional Notes: The first author was supported by a grant from the University of New Haven.
The second author was partially supported by NSF Grant DMS-9500924.
The third author was supported by a grant from the Ministry of Science of Slovenia.
Communicated by: David R. Larson
Article copyright: © Copyright 1998 American Mathematical Society

American Mathematical Society