Matjaž Omladič

Additive mappings preserving operators of rank one

Abstract Let B ( X ) be the algebra of all bounded linear operators on a nontrivial real or complex Banach space, and let F ( X ) be the subalgebra of all finite-rank operators. A characterization of additive mappings on F ( X ) which preserve operators of rank one or projections of rank one is given. In the real case such mappings are automatically linear.

On operators preserving commutativity

Abstract Let L ( X ) be the algebra of all bounded operators on a non-trivial complex Banach space X and F : L ( X ) → L ( X ) a bijective linear operator such that F and F −1 both send commuting pairs of operators into commuting pairs. Then, either F ( A ) = σUAU −1 + p ( A ) I , or F ( A ) = σUA ′ U −1 + p ( A ) I , where p is a linear functional on L ( X ), U is a bounded linear bijective operator between the appropriate two spaces, σ is a complex constant, and A ′ is the adjoint of A . The form of an operator F for which F and F −1 both send projections of rank one into projections of rank one is also determined.

Spectrum-preserving additive maps☆

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

On operators preserving the numerical range

Abstract Let F be a surjective linear mapping between the algebras L(H) and L(K) of all bounded operators on nontrivial complex Hilbert spaces H and K respectively. For any positive integer k let Wk(A) denote the kth numerical range of an operator A on H. If k is strictly less than one-half the dimension of H and Wk(F(A>))=Wk(A) for all A from L(H), then there is a unitary mapping U:H→K such that either F(A)= UAU ∗ or F(A)=(UAU ∗ ) t for every A∈L(H), where the transposition is taken in any basis of K, fixed in advance. This generalizes the result of S. Pierce and W. Watkins on finite-dimensional spaces. The case of k greater than or equal to one-half of the dimension of H is also treated using our method. Our proofs depend on a characterization of those linear operators preserving projections of rank one, which is of independent interest.

Approximating commuting operators

The problem of approximating m-tuples of commuting n×n complex matrices by commuting m-tuples of generic matrices is studied. We narrow the gap for commuting triples by showing that they can be perturbed if n 29.

Irreducible semigroups with multiplicative spectral radius

Abstract Irreducible multiplicative semigroups of finite dimensional linear operators with some additional properties are studied. Each of the properties—submultiplicativity of the spectral radius, its multiplicativity, or is constancy, as well as a certain property called the Rota condition—implies that every member of such a semigroup is, except for a scalar coefficient, similar to the direct sum of an isometry and a nilpotent operator.

On semigroups of matrices with traces in a subfield

Abstract It is shown that an irreducible semigroup of n × n complex matrices with real spectra is simultaneously similar to a semigroup of real matrices. Weaker results are obtained for semigroups of matrices over a general field with traces in a subfield.

On the distance between normal matrices

The least upper bound for the norm distance between two normal matrices is given in terms of their eigenvalues exclusively, thus solving a problem which appears to be long open.

Linearly recursive sequences and operator polynomials

Abstract Operator polynomials (i.e., polynomials over endomorphism algebras of vector spaces) are studied in the framework of polynomial modules and their comodule duals, i.e., the comodules of linearly recursive sequences. The subcomodules, which replace the concept of Jordan matrix pairs, contain enough information to recover the polynomials up to unit factors.

ON SEMITRANSITIVE JORDAN ALGEBRAS OF MATRICES

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