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Examples Illustrating some Aspects of the Weak Deligne-Simpson Problem

Kostov, Vladimir (2001)

Serdica Mathematical Journal

Research partially supported by INTAS grant 97-1644We consider the variety of (p + 1)-tuples of matrices Aj (resp. Mj ) from given conjugacy classes cj ⊂ gl(n, C) (resp. Cj ⊂ GL(n, C)) such that A1 + . . . + A[p+1] = 0 (resp. M1 . . . M[p+1] = I). This variety is connected with the weak Deligne-Simpson problem: give necessary and sufficient conditions on the choice of the conjugacy classes cj ⊂ gl(n, C) (resp. Cj ⊂ GL(n, C)) so that there exist (p + 1)-tuples with trivial centralizers of matrices...

Facial structures of separable and PPT states

Seung-Hyeok Kye (2011)

Banach Center Publications

A positive semi-definite block matrix (a state if it is normalized) is said to be separable if it is the sum of simple tensors of positive semi-definite matrices. A state is said to be entangled if it is not separable. It is very difficult to detect the border between separable and entangled states. The PPT (positive partial transpose) criterion tells us that the partial transpose of a separable state is again positive semi-definite, as was observed by M. D. Choi in 1982 from...

Homomorphisms from the unitary group to the general linear group over complex number field and applications

Chong-Guang Cao, Xian Zhang (2002)

Archivum Mathematicum

Let M n be the multiplicative semigroup of all n × n complex matrices, and let U n and G L n be the n –degree unitary group and general linear group over complex number field, respectively. We characterize group homomorphisms from U n to G L m when n > m ≥ 1 or n = m ≥ 3 , and thereby determine multiplicative homomorphisms from U n to M m when n > m ≥ 1 or n = m ≥ 3 . This generalize Hochwald’s result in [Lin. Alg. Appl.  212/213:339-351(1994)]: if f : U n → M n is a spectrum–preserving multiplicative homomorphism, then there exists a matrix R in G L n such that f ( A ) = R A R for...

Identities and the group of isostrophisms

Aleš Drápal, Viktor Alekseevich Shcherbakov (2012)

Commentationes Mathematicae Universitatis Carolinae

In this paper we reexamine the concept of isostrophy. We connect it to the notion of term equivalence, and describe the action of dihedral groups that are associated with loops by means of isostrophy. We also use it to prove and present in a new way some well known facts on m -inverse loops and middle Bol loops.

Maps on matrices that preserve the spectral radius distance

Rajendra Bhatia, Peter Šemrl, A. Sourour (1999)

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.

Multiplicative maps that are close to an automorphism on algebras of linear transformations

L. W. Marcoux, H. Radjavi, A. R. Sourour (2013)

Studia Mathematica

Let be a complex, separable Hilbert space of finite or infinite dimension, and let ℬ() be the algebra of all bounded operators on . It is shown that if φ: ℬ() → ℬ() is a multiplicative map(not assumed linear) and if φ is sufficiently close to a linear automorphism of ℬ() in some uniform sense, then it is actually an automorphism; as such, there is an invertible operator S in ℬ() such that φ ( A ) = S - 1 A S for all A in ℬ(). When is finite-dimensional, similar results are obtained with the mere assumption that there...

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