A note on equivalence preserving maps Petek, Tatjana; Radić, Gordana
Linear & multilinear algebra,
11/1/2020, 2020-11-01, 20201101, Letnik:
68, Številka:
11
Journal Article
Recenzirano
Let
be the algebra of all bounded linear operators on complex Banach space
and let the relation
,
, denote B=TAS for some invertible
. We will give a complete description of surjective maps on
such ...that
if and only if
, for every
.
Glycolysis, one of the leading metabolic pathways, involves many different periodic oscillations emerging at positive steady states of the biochemical models describing this essential process. One of ...the models employing the molecular diffusion of intermediates is the Higgins biochemical model to explain sustained oscillations. In this paper, we investigate the center-focus problem for the minimal Higgins model for general values of the model parameters with the help of computational algebra. We demonstrate that the model always has a stable focus point by finding a general form of the first Lyapunov number. Then, varying two of the model parameters, we obtain the first three coefficients of the period function for the stable focus point of the model and prove that the singular point is actually a bi-weak monodromic equilibrium point of type 1, 2. Additionally, we prove that there are two (small) intervals for a chosen parameter a > 0 for which one critical period bifurcates from this singular point after small perturbations
We obtain a general form of a surjective (not assumed additive) mapping φ, preserving the nonzero idempotency of a certain product, being defined (a) on the algebra of all bounded linear operators ...B(X), where X is at least three-dimensional real or complex Banach space, (b) on the set of all rank-one idempotents in B(X) and (c) on the set of all idempotents in B(X). In any of the cases it turns out that φ is additive and either multiplicative or antimultiplicative.
Let \(\mathcal{X}\) be a real or complex Banach space with \( \dim \mathcal{X}\geq 3\). We give a complete description of surjective mappings on \(\mathcal{B(X)}\) that preserve the ascent of Jordan ...triple product of operators or, preserve the descent of Jordan triple product of operators.
In this paper, we give the complete description of maps on self-adjoint bounded operators on Hilbert space which preserve a triadic relation involving the difference of operators and either ...commutativity or quasi-commutativity in both directions. We show that those maps are implemented by unitary or antiunitary equivalence and possible additive perturbation by a scalar operator.
There are two ways to compute Poincaré-Dulac normal forms of systems of ODEs. Under the original approach used by Poincaré the normalizing transformation is explicitly computed. On each step, the ...normalizing procedure requires the substitution of a polynomial to a series. Under the other approach, a normal form is computed using Lie transformations. In this case, the changes of coordinates are performed as actions of certain infinitesimal generators. In both cases, on each step the homological equation is solved in the vector space of polynomial vector fields \(V^n_j\) where each component of the vector field is a homogeneous polynomial of degree \(j\). We present the third way of computing normal forms of polynomial systems of ODEs where the coefficients of all terms are parameters. Although we use Lie transforms the homological equation is solved not in \(V^n_j\) but in the vector space of polynomial vector fields where each component is a homogeneous polynomial in the parameters of the system. It is shown that the space of the parameters is a kind of dual space and, the computation of normal forms can be performed in the space of parameters treated as the space of generalized vector fields. The approach provides a simple way to parallelize the normal form computations opening the way to compute normal forms up to higher order than under previously known two approaches.
Let
H be an infinite-dimensional complex Hilbert space. We give the characterization of surjective mappings on
B(
H) that preserve unitary similarity in both directions.
We consider arbitrary block upper-triangular subalgebras \(\mathcal{A} \subseteq M_n\) (i.e. subalgebras of \(M_n\) which contain the algebra of upper-triangular matrices) and their Jordan ...embeddings. We first describe Jordan embeddings \(\phi : \mathcal{A} \to M_n\) as maps of the form $$ \phi(X)=TXT^{-1} \qquad \mbox{or} \qquad \phi(X)=TX^tT^{-1}, $$ where \(T\in M_n\) is an invertible matrix, and then we obtain a simple criteria of when one block upper-triangular subalgebra Jordan-embeds into another (and in that case we describe the form of such embeddings). As a main result, we characterize Jordan embeddings \(\phi : \mathcal{A} \to M_n\) (when \(n\geq 3\)) as continuous injective maps which preserve commutativity and spectrum. We show by counterexamples that all these assumptions are indispensable (unless \(\mathcal{A} = M_n\) when injectivity is superfluous).
PDF
