Heydar Radjavi

Linear Maps Preserving Commutativity

Abstract Commutativity-preserving maps on the real space of all real symmetric or complex self-adjoint matrices are characterized. Related results are given for adjoint-preserving maps defined on all n × n matrices. These results are extended to infinite dimensions in the case of invertible maps.

Products of involutions

Abstract Every square matrix over a field, with determinant ±1, is the product of not more than four involutions.

On positive linear maps preserving invertibility

THEOREM: A positive linear map 4 between two C*-algebras is a Jordan homomorphism if d preserves invertibility and the range of ) is a C*-algebra. A counterexample is given for the case that the range of Q is not assumed to be a C*algebra; this answers a question raised by B. Russo (Proc. Amer. Math. Sot. 17

Commutativity-preserving operators on symmetric matrices

Abstract If φ is a nonsingular linear operator on n × n symmetric matrices over a formally real field, n ⩾ 3, and if φ preserves commutativity, then φ( A ) = cU -1 AU + f(A)I , where U t U and c are scalar and ƒ is a linear functional. This is an extension of a result of Chan and Lim.

Linear spaces of nilpotent matrices

Abstract We consider several questions on spaces of nilpotent matrices. We present sufficient conditions for triangularizability and give examples of irreducible spaces. We give a necessary and sufficient condition, in terms of the trace, for all linear combinations of a given set of operators to be nilpotent. We also consider the question of the dimension of a space L of nilpotents on F n . In particular, we give a simple new proof of a theorem due to M. Gerstenhaber concerning the maximal dimension of such spaces.

A nil algebra of bounded operators on Hilbert space with semisimple norm closure

An algebra of operators having the property of the title is constructed and it is used to give examples related to some recent invariant subspace results.

On matrix spaces with zero determinant

Let g be a linear space of n×n matrices of determinant zero over an infinite (or suitably large finite) field. It is proved that if the dimension of. L exceeds n 2−2n+2, then either L or its transpose has a common null vector. This extends a result due to Dieudonne and solves a recent research problem posed by S. Pierce in this journal. We also consider the problem of classifying all maximal matrix spaces with zero determinant, and offer some examples and observations.

On invariant operator ranges

A matricial representation is given for the algebra of operators leaving a given dense operator range invariant. It is shown that every operator on an infinite-dimensional Hilbert space has an uncountable family of invariant operator ranges, any two of which intersect only in (0).

Products of hermitian matrices and symmetries

It is the main purpose of this note to prove that every complex matrix with real determinant is the product of four hermitian matrices; Theorem 2 is an actually stronger result. Every real square matrix is the product of two real hermitian matrices [1]; this is a special case of our Theorem 1 which is of interest in itself, if it is indeed new. Theorem 3 was motivated by a theorem of Halmos and Kakutani [3 ] who proved that every unitary operator on an infinitedimensional Hilbert space is the product of four symmetries (i.e., operators that are hermitian and unitary). We also show that the number of factors in these results cannot be reduced in general.

An operator not satisfying Lomonosov's hypothesis☆

Abstract An example is presented of a Hilbert space operator such that no non-scalar operator that commutes with it commutes with a non-zero compact operator. This shows that Lomonosovs invariant subspace theorem does not apply to every operator.