Peter Šemrl

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.

Some general techniques on linear preserver problems

Abstract Several general techniques on linear preserver problems are described. The first one is based on a transfer principle in Model Theoretic Algebra that allows one to extend linear preserver results on complex matrices to matrices over other algebraically closed fields of characteristic 0. The second one concerns the use of some simple geometric technique to reduce linear preserver problems to standard types so that known results can be applied. The third one is about solving linear preserver problems on more general (operator) algebras by reducing the problems to idempotent preservers. Numerous examples will be given to demonstrate the proposed techniques.

The functional equation of multiplicative derivation is superstable on standard operator algebras

LetX be a real or complex infinite dimensional Banach space andA a standard operator algebra onX. Denote byB(X) the algebra of all bounded linear operators onX. Let ϕ: ℝ+ → ℝ+ be a function with the property limt→∞ φ(t)t−1=0. Assume that a mappingD:A →B(X) satisfies ‖D(AB)−AD(B)−D(A)B‖<φ(‖A‖ ‖B‖) for all operatorsA, B ∈D (no linearity or continuity ofD is assumed). ThenD is of the formD(A)=AT−TA for someT∈B(X).

On locally linearly dependent operators and derivations

The first section of the paper deals with linear operators Ti : U −→ V , i = 1, . . . , n, where U and V are vector spaces over an infinite field, such that for every u ∈ U , the vectors T1u, . . . , Tnu are linearly dependent modulo a fixed finite dimensional subspace of V . In the second section, outer derivations of dense algebras of linear operators are discussed. The results of the first two sections of the paper are applied in the last section, where commuting pairs of continuous derivations d, g of a Banach algebra A such that (dg)(x) is quasi–nilpotent for every x ∈ A are characterized.

Orthogonality of matrices and some distance problems

Abstract If A and B are matrices such that ||A + zB|| ⩾ ||A|| for all complex numbers z, then A is said to be orthogonal to B. We find necessary and sufficient conditions for this to be the case. Some applications and generalisations are also discussed.

Some linear preserver problems on upper triangular matrices

We consider linear preserver problems on the algebra of upper triangular matrices. We obtain the general form of nonsingular linear maps preserving rank one idempotents or commutativity. We give examples to show that the nonsingularity assumption is essential in our results.

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 .

Hua’s fundamental theorems of the geometry of matrices and related results☆

Abstract We briefly survey some recent improvements of Hua’s fundamental theorem of the geometry of rectangular matrices. Then we discuss possible further generalizations as well as some related open problems in the theory of preservers. We solve one such open problem using Ovchinnikov’s characterization of automorphisms of the poset of idempotent matrices. Using Ovchinnikov’s result we obtain a short proof of the fundamental theorem of the geometry of square matrices.

Determinant preserving maps on matrix algebras

Abstract Let M n be the algebra of all n × n complex matrices. If φ : M n → M n is a surjective mapping satisfying det( A + λB )=det( φ ( A )+ λφ ( B )), A , B ∈ M n , λ∈ C , then either φ is of the form φ ( A )= MAN , A ∈ M n , or φ is of the form φ ( A )= MA t N , A ∈ M n , where M , N ∈ M n are nonsingular matrices with det( MN )=1.

Nonsurjective nearisometries of Banach spaces

Abstract We obtain sharp approximation results for into nearisometries between L p spaces and nearisometries into a Hilbert space. Our main theorem is the optimal approximation result for nearsurjective nearisometries between general Banach spaces.