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Linear maps preserving A -unitary operators

Abdellatif Chahbi, Samir Kabbaj, Ahmed Charifi (2016)

Mathematica Bohemica

Let be a complex Hilbert space, A a positive operator with closed range in ( ) and A ( ) the sub-algebra of ( ) of all A -self-adjoint operators. Assume φ : A ( ) onto itself is a linear continuous map. This paper shows that if φ preserves A -unitary operators such that φ ( I ) = P then ψ defined by ψ ( T ) = P φ ( P T ) is a homomorphism or an anti-homomorphism and ψ ( T ) = ψ ( T ) for all T A ( ) , where P = A + A and A + is the Moore-Penrose inverse of A . A similar result is also true if φ preserves A -quasi-unitary operators in both directions such that there exists an...

Linear maps preserving quasi-commutativity

Heydar Radjavi, Peter Šemrl (2008)

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.

Linear maps that strongly preserve regular matrices over the Boolean algebra

Kyung-Tae Kang, Seok-Zun Song (2011)

Czechoslovak Mathematical Journal

The set of all m × n Boolean matrices is denoted by 𝕄 m , n . We call a matrix A 𝕄 m , n regular if there is a matrix G 𝕄 n , m such that A G A = A . In this paper, we study the problem of characterizing linear operators on 𝕄 m , n that strongly preserve regular matrices. Consequently, we obtain that if min { m , n } 2 , then all operators on 𝕄 m , n strongly preserve regular matrices, and if min { m , n } 3 , then an operator T on 𝕄 m , n strongly preserves regular matrices if and only if there are invertible matrices U and V such that T ( X ) = U X V for all X 𝕄 m , n , or m = n and T ( X ) = U X T V for all X 𝕄 n .

Linear operators that preserve Boolean rank of Boolean matrices

LeRoy B. Beasley, Seok-Zun Song (2013)

Czechoslovak Mathematical Journal

The Boolean rank of a nonzero m × n Boolean matrix A is the minimum number k such that there exist an m × k Boolean matrix B and a k × n Boolean matrix C such that A = B C . In the previous research L. B. Beasley and N. J. Pullman obtained that a linear operator preserves Boolean rank if and only if it preserves Boolean ranks 1 and 2 . In this paper we extend this characterizations of linear operators that preserve the Boolean ranks of Boolean matrices. That is, we obtain that a linear operator preserves Boolean rank...

Linear operators that preserve graphical properties of matrices: isolation numbers

LeRoy B. Beasley, Seok-Zun Song, Young Bae Jun (2014)

Czechoslovak Mathematical Journal

Let A be a Boolean { 0 , 1 } matrix. The isolation number of A is the maximum number of ones in A such that no two are in any row or any column (that is they are independent), and no two are in a 2 × 2 submatrix of all ones. The isolation number of A is a lower bound on the Boolean rank of A . A linear operator on the set of m × n Boolean matrices is a mapping which is additive and maps the zero matrix, O , to itself. A mapping strongly preserves a set, S , if it maps the set S into the set S and the complement of...

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