Compact covariance operators

A group G G is representable in a Banach space X X if G G is isomorphic to the group of isometries on X X in some equivalent norm. We prove that a countable group G G is representable in a separable real Banach space X X in several general cases, including when G ≃ { − 1 , 1 } × H G \simeq \{-1,1\} \times H , H H finite and dim ⁡ X ≥ | H | \dim X \geq |H| , or when G G contains a normal subgroup with two elements and X X is of the form c 0 ( Y ) c_0(Y) or ℓ p ( Y ) \ell _p(Y) , 1 ≤ p > + ∞ 1 \leq p >+\infty . This is a consequence of a result inspired by methods of S. Bellenot (1986) and stating that under rather general conditions on a separable real Banach space X X and a countable bounded group G G of isomorphisms on X X containing − I d -Id , there exists an equivalent norm on X X for which G G is equal to the group of isometries on X X . We also extend methods of K. Jarosz (1988) to prove that any complex Banach space of dimension at least 2 2 may be renormed with an equivalent complex norm to admit only trivial real isometries, and that any complexification of a Banach space may be renormed with an equivalent complex norm to admit only trivial and conjugation real isometries. It follows that every real Banach space of dimension at least 4 4 and with a complex structure may be renormed to admit exactly two complex structures up to isometry, and that every real Cartesian square may be renormed to admit a unique complex structure up to isometry.

Let $B$ be a real separable Banach space and $R:{B^*} \to B$ a covariance operator. All representations of $R$ in the form $\sum {e_n} \otimes {e_n}$, $\{ {e_n},n \geqslant 1\} \subset B$, are characterized. Necessary and sufficient conditions for $R$ to be compact are obtained, including a generalization of Mercer’s theorem. An application to characteristic functions is given.

Geometric duality theory of cones in dual pairs of vector spaces

Let $X$ be a real reflexive, smooth and separable Banach space having the Kadeč-Klee property and compactly imbedded in the real Banach space $Y$ and let $G:Y\rightarrow \mathbb{R} $ be a differentiable functional. By using the "fountain theorem" (Bartsch [3]), we will study the multiplicity of solutions for the operator equation \[ J_{\varphi}u=G^{\prime}(u)\text{,} \] where $J_{\varphi}$ is the duality mapping on $X$, corresponding to the gauge function $\varphi$. Equations having the above form with $J_{\varphi}$ a duality mapping on Orlicz-Sobolev spaces are considered as applications. As particular cases of the latter results, some multiplicity results concerning duality mappings on Sobolev spaces are derived.

Let $X$ be a real reflexive and separable Banach space having the Kadeč-Klee property, compactly imbedded in the real Banach space $V$ and let $G\colon V\rightarrow {\mathbb R} $ be a differentiable functional. By using ``fountain theorem'' and ``dual fountain theorem'' (Bartsch [< i> Infinitely many solutions of a symmetric Dirichlet problem< /i> , Nonlinear Anal. < b> 20< /b> (1993), 1205–1216] and Bartsch-Willem [< i> On an elliptic equation with concave and convex nonlinearities< /i> , Proc. Amer. Math. Soc. < b> 123< /b> (1995), 3555–3561], respectively), we will study the multiplicity of solutions for operator equation $$ J_{\varphi}u=G^{\prime}(u), $$ where $J_{\varphi}$ is the duality mapping on $X$, corresponding to the gauge function $\varphi$. Equations having the above form with $J_{\varphi}$ a duality mapping on Orlicz-Sobolev spaces are considered as applications. As particular cases of the latter results, some multiplicity results concerning duality mappings on Sobolev spaces are derived.

Rao has firstly introduced the Riemannian structure associated with the Fisher information matrix over a finite dimensional parametrized statistical model. He proposed the Riemannian distance as a measure of dissimilality between two probability measures, (cf. [2], for example.) In [1], Amari introduced a pair of dual affine connections with respect to the metric and discussed of the differential geometry of the space of a finite dimensional parametrized statistical model. It provides a differential geometrical meaning to statistical inference. In the present paper, we realize the above idea for a family of equivalent (i.e.,mutually absolute continuous) Gaussian measures on a Banach space. Our main result is as follows. Let 5 b e a real separable Banach space and P be a centered gaussian measure on the topological dual of B (cf. [6]). The covariance of P naturally determines the Hubert space H. i.e., for arbitrary x1 and x2eB, let

In this paper, we will prove the following : Let E be a real separable Banach space. Then every probability measure on E has a Hilbertian support if and only if E is isomorphic to a Hilbert space. In the case of lp (1 ≤ p < 2) we will give an explicit construction of probability measures without Hilbertian support.

Strassen's version of the law of the iterated logarithm is proved for Brownian motion in a real separable Banach space. We apply this result to obtain the law of the iterated logarithm for a sequence of independent Gaussian random variables with values in a Banach space and to obtain Strassen's result. Introduction. Let H denote a real separable Hilbert space with norm 11*IH and assume llIIB is a measurable norm on H in the sense of (2). Then there exists a constant M > 0 such that IIXIIB 0, let m, denote the canonical Gaussian cylinder set measure on H with variance parameter t and let t1t (t > 0) denote the Borel probability measure on B induced by m, (t > 0). We call , the Wiener measure on B generated by H with variance parameter t. Let SIB denote the space of continuous functions w from (0, om) into B such that w(0) = 0, and let 9 be the a-field of 2B generated by the functions w -* w(t).

In this correspondence, complete convergence theorems are established for weighted row sums from arrays of random elements taking values in real separable Rademacher type Banach spaces as well as real separable martingale type Banach spaces. It is assumed that A version of the Rademacher type p complete convergence theorem is also established with random variable weights. Illustrative examples are included.

The aim of this paper is to study new results of an approximate orthogonality of Birkhoff-James techniques in real Banach space , namely Chiemelinski orthogonality (even there is no ambiguity between the concepts symbolized by orthogonality) and provide some new geometric characterizations which is considered as the basis of our main definitions. Also, we explore relation between two different types of orthogonalities. First of them orthogonality in a real Banach space and the other orthogonality in the space of bounded linear operator . We obtain a complete characterizations of these two orthogonalities in some types of Banach spaces such as strictly convex space, smooth space and reflexive space. The study is designed to give different results about the concept symmetry of Chmielinski-orthogonality for a compact linear operator on a reflexive, strictly convex Banach space having Kadets-Klee property by exploring a new type of a generalized some results with Birkhoff James orthogonality in the space of bounded linear operators. We also exhibit a smooth compact linear operator with a spectral value that is defined on a reflexive, strictly convex Banach space having Kadets-Klee property either having zero nullity or not -right-symmetric.

Cylindrical Wiener processes in real separable Banach spaces are defined, and an approximation theorem involving scalar Wiener processes is given for such processes. A weak stochastic integral for Banach spaces involving a cylindrical Wiener process as integrator and an operator-valued stochastic process as integrand is defined. Basic properties of this integral are stated and proved. A class of linear, time-invariant, stochastic differential equations in real, separable, reflexive Banach spaces is formulated in such fashion that a solution of the equation is a cylindrical process. An existence and uniqueness theorem is proved. A stochastic version of the problem of heat conduction in a ring provides an example.

For a double array of independent random elements {V mn ,m ≥ 1, n ≥ 1} in a real separable Banach space, conditions are provided under which the weak and strong laws of large numbers for the double sums Σ i=1 m Σ j=1 n V ij , m ≥ 1, n ≥ 1 are equivalent. Both the identically distributed and the nonidentically distributed cases are treated. In the main theorems, no assumptions are made concerning the geometry of the underlying Banach space. These theorems are applied to obtain Kolmogorov, Brunk-Chung, and Marcinkiewicz-Zygmund type strong laws of large numbers for double sums in Rademacher type p (1 ≤ p ≤ 2) Banach spaces.

1. Introduction. Let {X n n=1 ∞} be a sequence of independent symmetric random variables defined on the probability space (Ω,ℑ, P) and taking values in the real separable Banach space (B,∥ • ∥). In this note, we will investigate the validity of the bounded law of the iterated logarithm, in both the usual (BLIL) and self-normalized (SNBLIL) forms, for such variables, where the normalizer will often depend on the type of the Banach space in question. One of our goals will be to drop the standard boundedness assumption that has been made in such a situation ever since Kolmogorov's fundamental LIL was proved; see, for example, (1977) and (1991). This will not be possible without self-normalization in the absence of an additional real-valued almost sure boundedness condition (Corollaries 1 and 2). Our main result (Theorem 2) also yields a characterization of a class of Banach spaces in a manner similar to that in (1976), (1975-76): A Banach space B is of "(LLn)p-1 — type p" if and only if each independent symmetric B-valued sequence X n n-1 ∞ satisfying a mild real-valued almost sure boundedness condition verifies a SNBLIL.

On Cantrell–Rosalsky’s strong laws of large numbers

The mixing and ergodicity of Gaussian measures have been characterized in terms of their covariances, first for random sequences, and then in the framework of linear dynamics on Banach spaces using covariance operators. In this thesis, we extend the latter results to the setting of infinitely divisible measures on Banach spaces, by deriving necessary and sufficient conditions for the strong and weak mixing of linear operators. Our results are specialized in explicit form to stable and tempered stable measures, with examples of linear operators satisfying the required measure invariance conditions. We also derive rates of convergence for the mixing of linear operators in the infinite-dimensional framework. Explicit mixing rates are obtained for weighted shifts under compound Poisson, α-stable, and tempered α-stable measures. Our approach relies on characterizations of mixing for infinitely divisible stochastic processes, and replaces the use of using covariance operators with codifference functionals and control measures on Banach spaces. We also investigate mixing in the setting of non-Frechet spaces and show that weak*-discontinuity is a necessary condition for the mixing of linear operators on duals of real countably Hilbert nuclear spaces equipped with a non-Gaussian Mittag-Leffler or Gamma-grey probability measure. For the Gaussian measure, we derive a partial characterization for weak*-continuous linear operators in terms of covariance operators.