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

DOI:
https://doi.org/10.1090/S0002-9939-98-04921-1

MathSciNet review:
1610913

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Abstract | References | Similar Articles | Additional Information

Abstract: Let be an -dimensional complex linear space and the algebra of all linear transformations on . We prove that every linear map on , 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).

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Additional Information

**Ali A. Jafarian**

Affiliation:
Department of Mathematics, University of New Haven, West Haven, Connecticut 06516

Email:
jafarian@charger.newhaven.edu

**Leiba Rodman**

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

Email:
lxrodm@math.wm.edu

**Peter Semrl**

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

Email:
Peter.Semrl@uni-mb.si

DOI:
https://doi.org/10.1090/S0002-9939-98-04921-1

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