On commuting operator exponentials

Let , be bounded operators on a Banach space with -congruence-free spectra such that . E. M. E. Wermuth has shown that . Ch. Schmoeger later established this result, using inner derivations and, in a second paper, has shown that: for in a complex unital Banach algebra, if the spectrum of is -congruence-free and , then (and thus, answering an open problem raised by E. M. E. Wermuth). In this paper we use the holomorphic functional calculus to give alternative simple proofs of both of these results. Moreover, we use the Borel functional calculus to give new proofs of recent results of Ch. Schmoeger concerning normal operator exponentials on a complex Hilbert space, under a weaker hypothesis on the spectra.

     

Remarks on commuting exponentials in Banach algebras

Suppose that and are elements of a complex unital Banach algebra such that the spectra of and are -congruence-free. E.M.E. Wermuth has shown that then

In this note we use two elementary facts concerning inner derivations on Banach algebras to give a very short proof of Wermuth's result.

     

Remarks on commuting exponentials in Banach algebras, II

Suppose that and are elements of a complex unital Banach algebra such that the spectrum of is -congruence-free and . We show that then is the sum of nilpotent elements. If denotes the spectral radius of , then we show that the additional assumption implies that

     

On normal operator exponentials

Suppose that and are bounded normal operators on a complex Hilbert space and that . In this paper some conditions implying are given.

     

On commuting operator exponentials, II

We prove that, under sufficient conditions on the spectra, $$ e^Me^N\subseteq e^Ne^M\Rightarrow MN\subseteq NM , $$ where N is an unbounded normal operator and M is a bounded normal operator in the Hilbert space.     

Modelling population growth vialaguerre-type exponentials

We use the Laguerre-type exponentials, i.e., eigenfunctions of the Laguerre-type derivatives, in order to construct new models for population growth. Relevant modifications of the classical exponential, logistic, and Volterra-Lotka models are investigated.     

A remark on the spectral multiplicity of normal extensions of commuting subnormal operator tuples

The authors gratefully acknowledge the support of the National Science Foundation.     

Approximating integrals of multivariate exponentials: A moment approach

We propose a method to calculate lower and upper bounds of some exponential multivariate integrals using moment relaxations and show that they asymptotically converge to the value of the integrals when the moment degree increases. We report computational results for integrals involving the normal distribution and exponential order statistic probabilities.     

The connectivity of commuting graphs

A necessary and sufficient condition is proven for the connectivity of commuting graphs C(G,X), where G is Sym(n), the symmetric group of degree n, and X is any G-conjugacy class.     

A remark on invariant operator ranges

Integral Equations and Operator Theory -     

Orthogonal exponentials on the generalized three-dimensional Sierpinski gasket

The self-affine measure μM,D corresponding to the expanding integer matrix

     

The cardinality of certain -orthogonal exponentials

The self-affine measures μ_M,D corresponding to the case (i) M=pI₃, D={0,e₁,e₂,e₃} in the space and the case (ii) M=pI₂, D={0,e₁,e₂,e₁+e₂} in the plane are non-spectral, where p>1 is odd, I_n is the n×n identity matrix, and e₁,…,e_n are the standard basis of unit column vectors in . One of the non-spectral problem on μ_M,D is to estimate the number of orthogonal exponentials in L²(μ_M,D) and to find them. In the present paper we show that, in both cases (i) and (ii), there are at most 4 mutually orthogonal exponentials in L²(μ_M,D) each, and the number 4 is the best.     

A strategy for analysing mixed and pooled exponentials

It is shown that a linear plot of the mean residual life on the failure rate characterizes the mixture of two exponentials. This plot is used to estimate the two components in the mixing distribution with the two largest mixing proportions. The EM algorithm is then used with these as initial values to obtain the MLE. Gradient plots are used to see if a higher-order fit is needed. A heuristic is given on how to use the gradient plots to identify components in the higher-order fit when this is the case. Graphs of an assignment function are then used to determine if the data are from a mixed model or simply the effect of pooling.     

Limit theorems for sums of random exponentials

We study limiting distributions of exponential sums as t→∞, N→∞, where (X_i) are i.i.d. random variables. Two cases are considered: (A) ess sup X_i = 0 and (B) ess sup X_i = ∞. We assume that the function h(x)= -log P{X_i>x} (case B) or h(x) = -log P {X_i>-1/x} (case A) is regularly varying at ∞ with index 1 < ϱ <∞ (case B) or 0 < ϱ < ∞ (case A). The appropriate growth scale of N relative to t is of the form , where the rate function H₀(t) is a certain asymptotic version of the function (case B) or (case A). We have found two critical points, λ₁<λ₂, below which the Law of Large Numbers and the Central Limit Theorem, respectively, break down. For 0 < λ < λ₂, under the slightly stronger condition of normalized regular variation of h we prove that the limit laws are stable, with characteristic exponent α = α (ϱ, λ) ∈ (0,2) and skewness parameter β ≡ 1.Research supported in part by the DFG grants 436 RUS 113/534 and 436 RUS 113/722.Mathematics Subject Classification (2000): 60G50, 60F05, 60E07     

Distance sets relating to orthogonal exponentials

The aim of this paper is to investigate the size properties of a planar set whose distance set has some prescribed arithmetic combinatorics. Such research is motivated by the conjecture that the disk has no more than 3 orthogonal exponentials. By proving a shifted version of Erdös-Solymosi’s theorem on the distance sets, we give some grounds on the conjecture. The results obtained here extend the corresponding results of Iosevich and Jaming in a simple manner.     

The single-valued extension property for sums and products of commuting operators

It is shown that the sum and the product of two commuting Banach space operators with Dunford's property (C) have the single-valued extension property.     

Fitting combinations of exponentials to probability distributions

Two techniques are described for approximating distributions on the positive half‐line by combinations of exponentials. One is based on Jacobi polynomial expansions, and the other on the logbeta distribution. The techniques are applied to some well‐known distributions (degenerate, uniform, Pareto, lognormal and others). In theory, the techniques yield sequences of combination of exponentials that always converge to the true distribution, but their numerical performance depends on the particular distribution being approximated. An error bound is given in the case the logbeta approximations. Copyright © 2006 John Wiley & Sons, Ltd.     

Sums of commuting square-zero transformations

We show that a linear transformation on a vector space is a sum of two commuting square-zero transformations if and only if it is a nilpotent transformation with index of nilpotency at most 3 and the codimension of in kerT is greater than or equal to the dimension of the space . We also characterize products of two commuting unipotent transformations with index 2.