ON A CLASS OF POLYA PROPERTY PRESERVING OPERATORS

In this paper, we show that every continuous linear operator from H(OmegawXOmegaz) to H (OmegawxOmegaxi) has an integral representation with a kernel function M(z, w, xi). We give two sufficient conditions on M(z,w,() to ensure that its corresponding operator preserves Polya property. We also prove that a continuous linear operator from H(fl,,, x ) to H(! x S2() either preserves the Polya property for all functions with that property or does not preserve the Polya property for any function.

Observations on the Separable Quotient Problem for Banach Spaces

The longstanding Banach–Mazur separable quotient problem asks whether every infinite-dimensional Banach space has a quotient (Banach) space that is both infinite-dimensional and separable. Although it remains open in general, an affirmative answer is known in many special cases, including (1) reflexive Banach spaces, (2) weakly compactly generated (WCG) spaces, and (3) Banach spaces which are dual spaces. Obviously (1) is a special case of both (2) and (3), but neither (2) nor (3) is a special case of the other. A more general result proved here includes all three of these cases. More precisely, we call an infinite-dimensional Banach space X dual-like, if there is another Banach space E, a continuous linear operator T from the dual space E * onto a dense subspace of X, such that the closure of the kernel of T (in the relative weak* topology) has infinite codimension in E * . It is shown that every dual-like Banach space has an infinite-dimensional separable quotient.

Reiterative Distributional Chaos on Banach Spaces

If we change the upper and lower densities in the definition of distributional chaos of a continuous linear operator on a Banach space [Formula: see text] by the Banach upper and Banach lower densities, respectively, we obtain Li–Yorke chaos. Motivated by this, we introduce the notions of reiterative distributional chaos of types [Formula: see text], [Formula: see text] and [Formula: see text] for continuous linear operators on Banach spaces, which are characterized in terms of the existence of an irregular vector with additional properties. Moreover, we study its relations with other dynamical properties and present the conditions for the existence of a vector subspace [Formula: see text] of [Formula: see text], such that every nonzero vector in [Formula: see text] is both irregular for [Formula: see text] and distributionally near zero for [Formula: see text].

Some Expansion of Fuzzy Paranormal Operators

Let ℍ be a Fuzzy Hilbert space over the fields of ℝ/ℂ and FB(ℍ) is the set of all fuzzy continuous linear operator on ℍ .In this paper we introduce the expansion of different fuzzy paranormal operators like n- fuzzy paranormal operator, *- fuzzy paranormal operator and nth -fuzzy paranormal operator, which all are developed from paranormal operators and their characteristics. The study resulted the properties of an n- fuzzy paranormal operator, * fuzzy paranormal operator and nth -fuzzy paranormal operator and their relationship between them. To investigate the nature of these operators, all it needs the nature of the n- fuzzy paranormal operator.

Implicit Function Theorem. Part II

Summary In this article, we formalize differentiability of implicit function theorem in the Mizar system [3], [1]. In the first half section, properties of Lipschitz continuous linear operators are discussed. Some norm properties of a direct sum decomposition of Lipschitz continuous linear operator are mentioned here. In the last half section, differentiability of implicit function in implicit function theorem is formalized. The existence and uniqueness of implicit function in [6] is cited. We referred to [10], [11], and [2] in the formalization.

Construction of operators with prescribed orbits in Fréchet spaces with a continuous norm

Let $X$ be a separable, infinite dimensional real or complex Fréchet space admitting a continuous norm. Let $\{v_n:\ n\geq 1\}$ be a dense set of linearly independent vectors of $X$. We show that there exists a continuous linear operator $T$ on $X$ such that the orbit of $v_1$ under $T$ is exactly the set $\{v_n:\ n\geq 1\}$. Thus, we extend a result of Grivaux for Banach spaces to the setting of non-normable Fréchet spaces with a continuous norm. We also provide some consequences of the main result.

EXOTIC LAPLACIANS AND DERIVATIVES OF WHITE NOISE

In this paper, we give a relationship between the exotic Laplacians and the Lévy Laplacians in terms of the higher-order derivatives of white noise by introducing a bijective and continuous linear operator acting on white noise functionals. Moreover, we study a relationship between exotic Laplacians, acting on higher-order singular functionals, each other in terms of the constructed operator.

INVARIANT EXPECTATIONS AND VANISHING OF BOUNDED COHOMOLOGY FOR EXACT GROUPS

We study exactness of groups and establish a characterization of exact groups in terms of the existence of a continuous linear operator, called an invariant expectation, whose properties make it a weak counterpart of an invariant mean on a group. We apply this operator to show that exactness of a finitely generated group G implies the vanishing of the bounded cohomology of G with coefficients in a new class of modules, which are defined using the Hopf algebra structure of ℓ1(G).

Mixing operators on spaces with weak topology

We prove that a continuous linear operator