arXiv subjects
Publications and source records attributed to Gyula Lakos.
The conformal range (or the real Davis--Wielandt shell), which is a particular planar projection of the Davis--Wielandt shell, can be considered as the hyperbolic version of the numerical range; i. e. it is a ``field of values'' which can be interpreted as a subset of the asymptotically closed hyperbolic plane. Here we explain the analogue of the elliptical range theorem of $2\times2$ complex matrices for the conformal range.
Hyperbolic elliptic parabolic disks can be described by the inequality $\frac{x^2}{C^2}+2y^2-2y\leq0$ ($0<C<1$) in the unit disk based Beltrami--Cayley--Klein model of the hyperbolic geometry, up to hyperbolic congruences. The hyperbolic elliptic parabolic disks considered above are sort of close to their supporting half distance bands given by the inequalities $\frac{x^2}{C^2}+ y^2-1\leq0$ and $y\geq0$. Here we consider what `close' might mean, and we look for even more precise approximations, in terms of area and circumference.
We review and provide simplified proofs related to the Magnus expansion, and improve convergence estimates. Observations and improvements concerning the Baker--Campbell--Hausdorff expansion are also made. In this Part IE, we consider the case of finite dimensional Banach algebras. We show that Magnus expansion is convergent (and works in logarithmic sense) if the cumulative norm $< 2+\varepsilon_{n}$, where $\varepsilon_{n}$ is a positive number depending on the dimension $n$ of the Banach algebra. We also show concrete finite-dimensional counterexamples of multiple Baker--Campbell--Hausdorff type for any cumulative norm $>2$ (necessarily of possibly great dimension).
Assume that $\mathfrak A$ is a real Banach space of finite dimension $n\geq2$. Consider any Borel probability measure $\nu$ supported on the unit ball $K$ of $\mathfrak A$. We show that \[\Delta(\nu)=\int_{x \in K}\int_{ y\in K}|x-y|_{\mathfrak A} \,\,\,\nu(x)\,\nu(y)\leq 2(1-2^{-n}f(n)),\] where $f:\mathbb N\setminus \{0,1\}\rightarrow (0,1]$ is a concrete universal function such that $f(n)\sim \frac{2}{\mathrm e n^2\log n}$. It is hoped that in the estimate`$f(n)$' can be replaced by `$1$'.
We approach the convergence of the Magnus, Wilcox, and symmetric Wilcox expansions by a non-commutative heat equation derived from the Maurer-Cartan equation.
In the first part of the paper, some extensions of the classical Dynkin-Specht-Wever lemma are developed. In the second part, we extend Burgunder's splitting construction, and relate back to the Kashiwara-Vergne problem.
The objective of this paper is to give alternative proofs for the symmetric Poincar\'e-Birkhoff-Witt theorem utilizing the Magnus recursion formulae or Dynkin's non-commutative polynomial comparison method and simple universal algebraic principles. As an application of these principles, a theorem of Nouaz\'e-Revoy type is also obtained.
Considering commutator monomials of the non-commutative associative variables $X_1,\ldots,X_n$; we determine the maximal possible number of alternating associative monomials in their noncommutative polynomial expansions. This is achieved by replacing generating functions with polytope sequences, which turn out to be finitely generated in some sense.
We review and provide simplified proofs related to the Magnus expansion, and improve convergence estimates. Observations and improvements concerning the Baker--Campbell--Hausdorff expansion are also made. In this Part IA, we consider uniform convexity. Notions of uniformly convex algebras are discussed, and uniform convexity is shown to improve convergence estimates.
We review and provide simplified proofs related to the Magnus expansion, and improve convergence estimates. Observations and improvements concerning the Baker--Campbell--Hausdorff expansion are also made. In this Part IIA, we investigate the case of $2\times2$ matrices with respect to the operator norm. We consider norm estimates and minimal presentations in terms of the Magnus and BCH expansions. Some results are obtained in the complex case, but a more complete picture is obtained in the real case.
A short proof of the elliptical range theorem concerning the numerical range of $2\times2$ complex matrices is given.
We consider the elliptical range theorems for the Davis--Wielandt shell, the numerical range, and the conformal range in terms of and related to their quadratic representations. The emphasis is on exposing a variety of elementary approaches.
We review and provide simplified proofs related to the Magnus expansion, and improve convergence estimates. Observations and improvements concerning the Baker--Campbell--Hausdorff expansion are also made. In this Part II, we consider the case of $C^*$-algebras, i. e. essentially the case of operators on Hilbert spaces. We present the spectral approach to the Magnus expansion in the context of the conformal range (which is a projection of the Davis--Wielandt shell), allowing a more effective approach. This makes possible to clarify certain convergence properties of the BCH expansion related to the critical cumulative norm $\pi$. In particular, we prove that for finite dimensional matrices $A,B$, the norm condition $\|A\|_2+\|B\|_2\leq\pi$ implies that the BCH expansion of $A$ and $B$ is convergent. Several counterexamples regarding convergence of the Magnus and BCH expansions are presented. In the rest, we prove growth estimates for the Magnus expansion in the setting of Hilbert space operators, both in terms of the overall sum and the individuals terms.
We review and provide simplified proofs related to the Magnus expansion, and improve convergence estimates. Observations and improvements concerning the Baker--Campbell--Hausdorff expansion are also made. In this Part III, we consider the Banach--Lie algebraic setting. We show how to improve the "standard" convergence bound $\boldsymbol\delta= 2.1737374\ldots$ using the customary generating function / ODE methods. Then we discuss how to achieve better convergence bounds using the resolvent method. The emphasis here is on the variety of the methods, rather than pushing them to the extreme. Nevertheless, regarding the cumulative convergence radii, we show how to establish $2.427<\mathrm C_\infty^{\mathrm{Lie}}\leq 4$ for the Magnus expansion, and $2.93<\mathrm C^{\mathrm{Lie}}_2 \leq 2\boldsymbol v_{\mathrm{Mi}}=5.4028570\ldots$ for the BCH expansion.
This expository paper focuses on free Lie $K$-algebras and the basic PBW theorem. We argue in various ways that the basic PBW theorem is a quite close consequence of the Magnus-Witt theorems concerning free Lie algebras.
We review and provide simplified proofs related to the Magnus expansion, and improve convergence estimates. Observations and improvements concerning the Baker--Campbell--Hausdorff expansion are also made. In this Part I, we consider the general Banach algebraic setting. We show that the (cumulative) convergence radius of the Magnus expansion is $2$; and of the Baker--Campbell--Hausdorff series is $\mathrm C_2=2.89847930\ldots$. More generally, the resolvent method is developed in the analytic setting.
The objective of this series of papers is to recover information regarding the behaviour of FQ operations in the case $n=2$, and FQ conform-operations in the case $n=3$. In this first part we study how the basic invariance properties of FQ operations ($n=2$) are reflected in their formal power series expansions.
The objective of this series of papers is to recover information regarding the behaviour of FQ operations in the case $n=2$, and FQ conform-operations in the case $n=3$. In this second part we show some arithmetically constructible examples of FQ operations ($n=2$), concentrating on monoaxiality, related extensions, and (hyper)scaling.