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### On quadratic and sesquilinear functionals.

Aequationes mathematicae

### On ring derivations and quadratic functionals.

Aequationes mathematicae

### On Jordan *-derivations and an application

Colloquium Mathematicae

### Two characterizations of automorphisms on B(X)

Studia Mathematica

Let X be an infinite-dimensional Banach space, and let ϕ be a surjective linear map on B(X) with ϕ(I) = I. If ϕ preserves injective operators in both directions then ϕ is an automorphism of the algebra B(X). If X is a Hilbert space, then ϕ is an automorphism of B(X) if and only if it preserves surjective operators in both directions.

### On algebraic generation of B(X) by two subalgebras with square zero

Studia Mathematica

### Quadratic functionals and Jordan *-derivations

Studia Mathematica

### Maps on idempotents

Studia Mathematica

Let X be an infinite-dimensional real or complex Banach space, B(X) the algebra of all bounded linear operators on X, and P(X) ⊂ B(X) the subset of all idempotents. We characterize bijective maps on P(X) preserving commutativity in both directions. This unifies and extends the characterizations of two types of automorphisms of P(X), with respect to the orthogonality relation and with respect to the usual partial order; the latter have been previously characterized by Ovchinnikov. We also describe...

### Maps on idempotent operators

Banach Center Publications

The set of all bounded linear idempotent operators on a Banach space X is a poset with the partial order defined by P ≤ Q if PQ = QP = P. Another natural relation on the set of idempotent operators is the orthogonality relation defined by P ⊥ Q ⇔ PQ = QP = 0. We briefly survey known theorems on maps on idempotents preserving order or orthogonality. We discuss some related results and open problems. The connections with physics, geometry, theory of automorphisms, and linear preserver problems will...

### On non linear perturbations of isometries.

Mathematische Annalen

### Multiplicative Derivations on C(X).

Monatshefte für Mathematik

### Generalized Cauchy functional equation and characterizations of inner product spaces.

Aequationes mathematicae

### Generalized Cauchy functional equation and characterizations of inner product spaces. (Summary).

Aequationes mathematicae

### Spectral characterizations of central elements in Banach algebras

Studia Mathematica

Let A be a complex unital Banach algebra. We characterize elements belonging to Γ(A), the set of elements central modulo the radical. Our result extends and unifies several known characterizations of elements in Γ(A).

### On local automorphisms and mappings that preserve idempotents

Studia Mathematica

Let B(H) be the algebra of all bounded linear operators on a Hilbert space H. Automorphisms and antiautomorphisms are the only bijective linear mappings θ of B(H) with the property that θ(P) is an idempotent whenever P ∈ B(H) is. In case H is separable and infinite-dimensional, every local automorphism of B(H) is an automorphism.

### Finite rank elements in semisimple Banach algebras

Studia Mathematica

Let A be a semisimple Banach algebra. We define the rank of a nonzero element a in the socle of A to be the minimum of the number of minimal left ideals whose sum contains a. Several characterizations of rank are proved.

### Linear maps preserving quasi-commutativity

Studia Mathematica

Let X and Y be Banach spaces and ℬ(X) and ℬ(Y) the algebras of all bounded linear operators on X and Y, respectively. We say that A,B ∈ ℬ(X) quasi-commute if there exists a nonzero scalar ω such that AB = ωBA. We characterize bijective linear maps ϕ : ℬ(X) → ℬ(Y) preserving quasi-commutativity. In fact, such a characterization can be proved for much more general algebras. In the finite-dimensional case the same result can be obtained without the bijectivity assumption.

### From geometry to invertibility preservers

Studia Mathematica

We characterize bijections on matrix spaces (operator algebras) preserving full rank (invertibility) of differences of matrix (operator) pairs in both directions.

### Derivations mapping into the socle, III

Studia Mathematica

Let A be a Banach algebra, and let d: A → A be a continuous derivation such that each element in the range of d has a finite spectrum. In a series of papers it has been proved that such a derivation is an inner derivation implemented by an element from the socle modulo the radical of A (a precise formulation of this statement can be found in the Introduction). The aim of this paper is twofold: we extend this result to the case where d is not necessarily continuous, and we give a complete description...

### Maps on matrices that preserve the spectral radius distance

Studia Mathematica

Let ϕ be a surjective map on the space of n×n complex matrices such that r(ϕ(A)-ϕ(B))=r(A-B) for all A,B, where r(X) is the spectral radius of X. We show that ϕ must be a composition of five types of maps: translation, multiplication by a scalar of modulus one, complex conjugation, taking transpose and (simultaneous) similarity. In particular, ϕ is real linear up to a translation.

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