Productively Fréchet Spaces

We solve the long standing problem of characterizing the class of strongly Fréchet spaces whose product with every strongly Fréchet space is also Fréchet.     

On Fredholm Operators Between Non-archimedean Fréchet Spaces

It is proved that the index of a Fredholm operator between non-Archimedean Fréchet spaces is preserved under compact perturbations. A similar result is shown for Fredholm operators between non-Archimedean polar regular LF-spaces.     

Fréchet-Valued Real Analytic Functions on Fréchet Spaces

The aim of this paper is to establish the equivalence between the real and topologically real analyticity of Fréchet-valued functions on Fréchet spaces. This is an extension of recent results of Bonet and Domanski [1, 2] to infinite dimension.Received February 19, 2002; in revised form June 25, 2002 Published online May 16, 2003     

Asymptotic Behavior of Gaussian Samples in Fréchet Spaces

Abstract

In this article we investigate unnormalized samples of Gaussian random elements in a separable Fréchet space 𝕄. First we describe a connection between shifts of a Gaussian measure μ in a separable Fréchet space and the infinite product of standard normal distributions in ?^∞, and on the basis of this result we derive the so‐called self‐sufficient expansion for Gaussian random elements in a Fréchet space. Moreover, we find lower bounds for the Gaussian measure μ of shifted balls in 𝕄 and estimate the metric entropy of balls in the Hilbert space ? ? 𝕄 which generates μ. Finally, applying the Brunn–Minkowski inequality we prove a kind of the logarithmic law of large numbers. The last result is an extension of the analogous theorem obtained by Goodman (Characteristics of normal samples. Ann. Probab. 1988, 16, 1281–1290), for a sequence of Gaussian random elements in a separable Banach space.     

On the Fréchet derivative of matrix functions

In 1956 Rinehart [4] discussed the derivatives of matrix functions by considering differences ƒ(A + E) − ƒ(A) for matrices E commuting with A. In that case the derivative turned out to be ƒ′(A). In this paper the case of noncommutative A and E is treated. This leads to the Fréchet derivative of the matrix function ƒ. An explicit integral representation is obtained. Using an approach that is similar to the one in [5], a finite sum in polynomials of A is obtained. The coefficients may be computed recursively. This is useful in computing the Fréchet derivative which is needed for Newton's method to solve nonlinear matrix equations.     

On Fréchet–Hilbert algebras

We consider Hilbert algebras with a supplementary Fréchet topology and get various extensions of the algebraic structure by using duality techniques. In particular we obtain optimal multiplier-type involutive algebras which in applications are large enough to be of significant practical use. The setting covers many situations arising from quantization rules, as those involving square-integrable families of bounded operators     

Spectra of Continuous Linear Operators on Separable Fréchet Spaces

Mathematical Notes -     

On Fréchet’s functional equation

Fréchet’s functional equation \(\Delta _{y_1,y_2,\dots ,y_{n+1}}f=0\) plays a key role in the theory of polynomial functions. A basic theorem of Djokovi? shows that under general conditions the functional equation \(\Delta _y^{n+1}f=0\) is equivalent to Fréchet’s equation. Here we give a short alternative proof for this result using spectral synthesis.     

Strongly Mixing Convolution Operators on Fréchet Spaces of Holomorphic Functions

A theorem of Godefroy and Shapiro states that non-trivial convolution operators on the space of entire functions on \({\mathbb{C}^n}\) are hypercyclic. Moreover, it was shown by Bonilla and Grosse-Erdmann that they have frequently hypercyclic functions of exponential growth. On the other hand, in the infinite dimensional setting, the Godefroy–Shapiro theorem has been extended to several spaces of entire functions defined on Banach spaces. We prove that on all these spaces, non-trivial convolution operators are strongly mixing with respect to a gaussian probability measure of full support. For the proof we combine the results previously mentioned and we use techniques recently developed by Bayart and Matheron. We also obtain the existence of frequently hypercyclic entire functions of exponential growth.     

Coorbit Spaces with Voice in a Fréchet Space

We set up a new general coorbit space theory for reproducing representations of a locally compact second countable group G that are not necessarily irreducible nor integrable. Our basic assumption is that the kernel associated with the voice transform belongs to a Fréchet space \(\mathcal T\) of functions on G, which generalizes the classical choice \(\mathcal T=L_w^1(G)\). Our basic example is \( \mathcal T=\bigcap _{p\in (1,+\infty )} L^p(G)\), or a weighted versions of it. By means of this choice it is possible to treat, for instance, Paley-Wiener spaces and coorbit spaces related to Shannon wavelets and Schrödingerlets.     

Hypercyclic Subspaces on Fréchet Spaces Without Continuous Norm

Known results about hypercyclic subspaces concern either Fréchet spaces with a continuous norm or the space ω. We fill the gap between these spaces by investigating Fréchet spaces without continuous norm. To this end, we divide hypercyclic subspaces into two types: the hypercyclic subspaces M for which there exists a continuous seminorm p such that ${M \cap {\rm ker} p = \{0\}}$ and the others. For each of these types of hypercyclic subspaces, we establish some criteria. This investigation permits us to generalize several results about hypercyclic subspaces on Fréchet spaces with a continuous norm and about hypercyclic subspaces on ω. In particular, we show that each infinite-dimensional separable Fréchet space supports a mixing operator with a hypercyclic subspace.     

On the regular variation of ratios of jointly Fréchet random variables

We provide a necessary and sufficient condition for the ratio of two jointly α-Fréchet random variables to be regularly varying. This condition is based on the spectral representation of the joint distribution and is easy to check in practice. Our result motivates the notion of the ratio tail index, which quantifies dependence features that are not characterized by the tail dependence index. As an application, we derive the asymptotic behavior of the quotient correlation coefficient proposed in Zhang (Ann Stat 36(2):1007–1030, 2008) in the dependent case. Our result also serves as an example of a new type of regular variation of products, different from the ones investigated by Maulik et al (J Appl Probab 39(4):671–699, 2002).     

Fréchet algebras of finite type

The main objects of study in this paper are Fréchet algebras having an Arens Michael representation in which every Banach algebra is finite dimensional. We shall classify these algebras using a theorem which ensures that the image of any continuous linear map of a Fréchet space of finite type (i.e., for which the defining seminorms have a finite dimensional cokernel) into any Fréchet space is in fact closed.This work is part of the research project of the European Research Training Network Analysis and Operators, contract HPRN-CT 2000 00116, funded by the European Commission.     

Random Coincidence Point Theorem in Fréchet Spaces with Applications

Abstract

We proved a random coincidence point theorem for a pair of commuting random operators in the setup of Fréchet spaces. As applications, we obtained random fixed point and best approximation results for *-nonexpansive multivalued maps. Our results are generalizations or stochastic versions of the corresponding results of Shahzad and Latif [Shahzad, N.; Latif, A. A random coincidence point theorem. J. Math. Anal. Appl. 2000, 245, 633–638], Khan and Hussain [Khan, A.R.; Hussain, N. Best approximation and fixed point results. Indian J. Pure Appl. Math. 2000, 31 (8), 983–987], Tan and Yaun [Tan, K.K.; Yaun, X.Z. Random fixed point theorems and approximation. Stoch. Anal. Appl. 1997, 15 (1), 103–123] and Xu [Xu, H.K. On weakly nonexpansive and *-nonexpansive multivalued mappings. Math. Japon. 1991, 36 (3), 441–445].     

Fréchet Embeddings of Negative Type Metrics

We show that every n-point metric of negative type (in particular, every n-point subset of L ₁) admits a Fréchet embedding into Euclidean space with distortion , a result which is tight up to the O(log log n) factor, even for Euclidean metrics. This strengthens our recent work on the Euclidean distortion of metrics of negative into Euclidean space. S. Arora supported by David and Lucile Packard Fellowship and NSF grant CCR-0205594. J.R. Lee supported by NSF grant CCR-0121555, NSF 0514993, NSF 0528414 and an NSF Graduate Research Fellowship.     

Smooth Fréchet globalizations of Harish-Chandra modules

We give an alternative proof of the Casselman-Wallach globalization theorem. The approach is based on lower bounds for matrix coefficients on a reductive group.     

Characterizations of nuclearity in Fréchet spaces

We extend the result of A. Bellow (Proc. Nat. Acad. Sci. USA73, No. 6 (1976), 1798–1799) on the characterization of finite-dimensional Banach spaces, to a characterization of nuclearity for Fréchet spaces. Those spaces are nuclear iff every Pettis-bounded and Pettis-uniformly integrable amart is mean convergent. Several other characterizations are given.