A. R. Sourour

Spectrum-preserving linear maps☆

Abstract Let X and Y be Banach spaces. We show that a spectrum preserving surjective linear map φ from B ( X ) to B ( Y ) is either of the form φ ( T ) = ATA −1 for an isomorphism A of X onto Y or the form φ(T) = BT ∗ B −1 for an isomorphism B of X ∗ onto Y .

INVERTIBILITY PRESERVING LINEAR MAPS ON L(X)

For Banach spaces X and Y , we show that every unital bijective invertibility preserving linear map between L(X) and L(Y ) is a Jordan isomorphism. The same conclusion holds for maps between CI + K(X) and CI +K(Y ).

A factorization theorem for matrices

It is shown that a nonscalar invertible square matrix can be written as a product of two square matrices with prescribed eigenvalues subject only to the obvious determinant condition. As corollaries, we give short proofs of some known results such as Ballantines characterization of products of four or five positive definite matrices, the commutator theorem of Shoda-Thompson for fields with sufficiently many elements and other results.

Isometries of norm ideals of compact operators

Abstract Let J be a symmetric norm ideal of compact operators on Hilbert space H , and assume that the finite rank operators are dense in J and that J is not the ideal of Hilbert-Schmidt operators. A linear transformation τ on J is an isometry of J onto itself if and only if there are unitary operators U and V on H such that either τ(X) = UXV or τ(X) = UXtV, where Xt denotes the transpose of X with respect to a fixed orthonormal basis of H .

Commutativity preserving linear maps and Lie automorphisms of triangular matrix algebras

Abstract In this article we classify linear maps ϕ from the algebra T n of n × ? upper triangular matrices into itself satisfying ϕ(ab − ba) = 0 if and only if ab − ba = 0. In particular, we show that for n > 2, any such map is either the sum of an algebra automorphism and a map into the centre, or the sum of the negative of an algebra anti-automorphism and a map into the centre. As a consequence, Lie automorphisms of the algebra T n are also classified.

Lie and Jordan ideals of operators on Hilbert space

A linear manifold 2 in ©(§) is a Lie ideal in 33(§) if and only if there is an associative ideal 5 such that (S. ^(C)) C 2 c S + CI. Also S is a Lie ideal if and only if it contains the unitary orbit of every operator in it. On the other hand, a subset of 33(§) is a Jordan ideal if and only if it is an associative ideal.

Additive rank-one preserving mappings on triangular matrix algebras

Abstract We classify surjective additive maps on the space of block upper triangular matrices that preserve matrices of rank one as well as linear maps preserving matrices of rank one on fairly general subspaces of matrices.

LIE IDEALS IN TRIANGULAR OPERATOR ALGEBRAS

We study Lie ideals in two classes of triangular operator algebras: nest algebras and triangular UHF algebras. Our main results show that if 2 is a closed Lie ideal of the triangular operator algebra A, then there exist a closed associative ideal IC and a closed subalgebra Or of the diagonal A ni A* so that K C 2 C IC + Oc. INTRODUCTION Let A be an associative complex algebra. Under the Lie multiplication [x, y] xy yx, A becomes a Lie algebra. A Lie ideal in A is a linear manifold Z in A for which [a, k] c ? for every a E A and k c 2. In many instances, there is a close connection between the Lie ideal structure and the associative ideal structure of A. This connection has been investigated for prime rings in [6], in [3] for B(

Nilpotent factorization of matrices

5) the set of bounded operators on a Hilbert space Si, and in [10] for certain von Neumann algebras. (See also [9, 4, 11].) In this paper we pursue this line of investigation for two classes of triangular operator algebras, namely nest algebras anld triangular UHF algebras. The authors would like to thank Frank A. Zorzitto for many helpful conversations. 1. WEAKLY CLOSED LIE IDEALS IN NEST ALGEBRAS Recall that a nest JK on a Hilbert space Si is a chain of closed subspaces of Si which is closed under the operations of arbitrary intersections and closed linear spans, and which includes {O} and S5. The nest algebra T(KJ) is the algebra of all operators on Se leaving every member of JK invariant. This is always closed in the weak operator topology. The diagonal D(KJ) of a nest algebra T(JK) is the von Neumann subalgebra 7(K) n T(K)*. If E, F E KV with E < F, then F E is called an interval of the nest. The nonzero minimal intervals are called atoms. A nest is atomic if the atoms of the nest span 55. We refer the reader to [1] for more information on nest algebras. Our main result, Theorem 12, shows that for every weakly closed Lie ideal Z in 7(JK), there exist a corresponding weakly closed associative ideal IC and a von Neumann subalgebra SK of O(A/) such that

Linear maps that preserve the commutant, double commutant or the lattice of invariant subspaces

We show that, with the exception of 2 × 2 nonzero nilpotent matrices, every singular square matrix over an arbitrary field is a product of two nilpotent matrices.