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1
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Maps on idempotents

100%
Šemrl P.
Studia Mathematica
|
2005
|
tom 169
|
nr 1
21-44
EN
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 bijective orthogonality preserving maps on the set of idempotents of a fixed finite rank. As an application we present a nonlinear extension of the structural result for bijective linear biseparating maps on B(X).
2
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Maps on idempotent operators

100%
Šemrl P.
Banach Center Publications
|
2007
|
tom 75
|
nr 1
289-301
EN
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 be explained. At the end we will prove a new result concerning bijective maps on idempotent operators preserving comparability.
3
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Two characterizations of automorphisms on B(X)

100%
Šemrl P.
Studia Mathematica
|
1993
|
tom 105
|
nr 2
143-149
EN
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.
4
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Spectral characterizations of central elements in Banach algebras

64%
Brešar M., Šemrl P.
Studia Mathematica
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1996
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tom 120
|
nr 1
47-52
EN
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).
5
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Derivations mapping into the socle, III

64%
Boudi N., Šemrl P.
Studia Mathematica
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2010
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tom 197
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nr 2
141-155
EN
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 of such maps in the semisimple case.
6
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From geometry to invertibility preservers

64%
Havlicek H., Šemrl P.
Studia Mathematica
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2006
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tom 174
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nr 1
99-109
EN
We characterize bijections on matrix spaces (operator algebras) preserving full rank (invertibility) of differences of matrix (operator) pairs in both directions.
7
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On local automorphisms and mappings that preserve idempotents

64%
Brešar M., Šemrl P.
Studia Mathematica
|
1995
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tom 113
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nr 2
101-108
EN
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.
8
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Linear preservers on ℬ(X)

64%
Brešar M., Šemrl P.
Banach Center Publications
|
1997
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tom 38
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nr 1
49-58
9
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Finite rank elements in semisimple Banach algebras

64%
Brešar M., Šemrl P.
Studia Mathematica
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1998
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tom 128
|
nr 3
287-298
EN
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.
10
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Linear maps preserving quasi-commutativity

64%
Radjavi H., Šemrl P.
Studia Mathematica
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2008
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tom 184
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nr 2
191-204
EN
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.
11
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Isometries of normed spaces

52%
Figiel T., Šemrl P., Väisälä J.
Colloquium Mathematicum
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2002
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tom 92
|
nr 1
153-154
EN
We improve the Mazur-Ulam theorem by relaxing the surjectivity condition.
12
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Maps on matrices that preserve the spectral radius distance

52%
Bhatia R., Šemrl P., Sourour A. R.
Studia Mathematica
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1999
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tom 134
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nr 2
99-110
EN
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.
13
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On algebraic generation of B(X) by two subalgebras with square zero

44%
Šemrl P.
Studia Mathematica
|
1990
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tom 97
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nr 2
139-142
14
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On operator bands

39%
Drnovšek R., Livshits L., MacDonald G. W., Mathes B., Radjavi H., Šemrl P.
Studia Mathematica
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2000
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tom 139
|
nr 1
91-100
EN
A multiplicative semigroup of idempotent operators is called an operator band. We prove that for each K>1 there exists an irreducible operator band on the Hilbert space $l^2$ which is norm-bounded by K. This implies that there exists an irreducible operator band on a Banach space such that each member has operator norm equal to 1. Given a positive integer r, we introduce a notion of weak r-transitivity of a set of bounded operators on a Banach space. We construct an operator band on $l^2$ that is weakly r-transitive and is not weakly (r+1)-transitive. We also study operator bands S satisfying a polynomial identity p(A, B) = 0 for all non-zero A,B ∈ S, where p is a given polynomial in two non-commuting variables. It turns out that the polynomial $p(A, B) = (A B - B A)^2$ has a special role in these considerations.
15
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Quadratic functionals and Jordan *-derivations

38%
Šemrl P.
Studia Mathematica
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1990
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tom 97
|
nr 3
157-165
16
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On Jordan *-derivations and an application

32%
Šemrl P.
Colloquium Mathematicum
|
1990
|
tom 59
|
nr 2
241-251
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