arXiv subjects
Publications and source records attributed to Tatjana Petek.
Let B(X) be the algebra of all bounded linear operators on a complex Banach space X of dimension at least three. For an arbitrary nonzero complex number t we determine the form of mappings f: B(X)-->B(X) with sufficiently large range such that t(AB+BA) is idempotent if and only if t(f(A)f(B)+f(B)f(A)) is idempotent, for all A, B in B(X). Note that f is not assumed to be linear or additive.
We investigate the integrability of polynomial vector fields through the lens of duality in parameter spaces. We examine formal power series solutions annihilated by differential operators and explore the properties of the integrability variety in relation to the invariants of the associated Lie group. Our study extends to differential operators on affine algebraic varieties, highlighting the intrinsic connection between these operators and local analytic first integrals. To illustrate the duality the case of quadratic vector fields is considered in detail.
Let $M_n$ be the algebra of $n \times n$ complex matrices. We consider arbitrary subalgebras $\mathcal{A}$ of $M_n$ which contain the algebra of all upper-triangular matrices (i.e.\ block upper-triangular subalgebras), and their Jordan embeddings. We first describe Jordan embeddings $ϕ: \mathcal{A} \to M_n$ as maps of the form $ϕ(X)=TXT^{-1}$ or $ϕ(X)=TX^tT^{-1}$, 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 $ϕ: \mathcal{A} \to M_n$ (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).
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.