On the Finite Dimensionality of Closed Subspaces in Lp(M,dμ)∩Lq(M,dν)

Finding effective finite-dimensional criteria for closed subspaces in Lp, endowed with some additional functional constraints, is a well-known and interesting problem. In this work, we are interested in some sufficient constraints on closed functional subspaces, Sp⊂Lp, whose finite dimensionality is not fixed a priori and can not be checked directly. This is often the case in diverse applications, when a closed subspace Sp⊂Lp is constructed by means of some additional conditions and constraints on Lp with no direct exemplification of the functional structure of its elements. We consider a closed topological subspace, Sp(q), of the functional Banach space, Lp(M,dμ), and, moreover, one assumes that additionally, Sp(q)⊂Lq(M,dν) is subject to a probability measure ν on M. Then, we show that closed subspaces of Lp(M,dμ)∩Lq(M,dν) for q>max{1,p},p>0 are finite dimensional. The finite dimensionality result concerning the case when q>p>0 is open and needs more sophisticated techniques, mainly based on analysis of the complementary subspaces to Lp(M,dμ)∩Lq(M,dν).

On the construction of the orthonormal wavelet in the Hardy space H2(ℝ)

We are concerned with the orthonormal wavelet [Formula: see text] in the Hardy space [Formula: see text] which is a closed subspace of [Formula: see text] without negative frequency components. It is well known that there does not exist an [Formula: see text]-wavelet such that [Formula: see text] is continuous on [Formula: see text] and satisfies [Formula: see text] for some [Formula: see text]. The aim of this paper is to find a critical decay rate in the existing [Formula: see text]-wavelet under the condition that [Formula: see text] is continuous on [Formula: see text]. Moreover, we also construct a concrete [Formula: see text]-wavelet having infinite vanishing moments.

An example regarding Kalton's paper ‘isomorphisms between spaces of vector-valued continuous functions’

The paper alluded to in the title contains the following striking result: Let $I$ be the unit interval and $\Delta$ the Cantor set. If $X$ is a quasi Banach space containing no copy of $c_{0}$ which is isomorphic to a closed subspace of a space with a basis and $C(I,\,X)$ is linearly homeomorphic to $C(\Delta ,\, X)$ , then $X$ is locally convex, i.e., a Banach space. We will show that Kalton result is sharp by exhibiting non-locally convex quasi Banach spaces $X$ with a basis for which $C(I,\,X)$ and $C(\Delta ,\, X)$ are isomorphic. Our examples are rather specific and actually, in all cases, $X$ is isomorphic to $C(K,\,X)$ if $K$ is a metric compactum of finite covering dimension.

Some Properties of Fuzzy Inner Product Space

Our goal in the present paper is to introduce a new type of fuzzy inner product space. After that, to illustrate this notion, some examples are introduced. Then we prove that that every fuzzy inner product space is a fuzzy normed space. We also prove that the cross product of two fuzzy inner spaces is again a fuzzy inner product space. Next, we prove that the fuzzy inner product is a non decreasing function. Finally, if U is a fuzzy complete fuzzy inner product space and D is a fuzzy closed subspace of U, then we prove that U can be written as a direct sum of D and the fuzzy orthogonal complement of D.

A Special One-Point Connectification that Characterizes $T_{0}$ Spaces

We show that every $T_{0}$ space $X$ has some $T_{0}$ "special" one-point connectification $X_{\infty}$, unique up to a homeomorphism, such that $X$ is a closed subspace of $X_{\infty}$ and a closed subset of $X$ is precisely a closed proper subset of $X_{\infty}$; moreover, having such a one-point connectification characterizes $T_{0}$ spaces. As an application, it is also shown that our one-point connectification of every given topological $n$-manifold is a space more general than but "close to" a topological $n$-manifold with boundary.

On Spectral Properties of Compact Toeplitz Operators on Bergman Space with Logarithmically Decaying Symbol and Applications to Banded Matrices

Let $L^2({{\mathbb{D}}})$ be the space of measurable square-summable functions on the unit disk. Let $L^2_a({{\mathbb{D}}})$ be the Bergman space, that is, the (closed) subspace of analytic functions in $L^2({{\mathbb{D}}})$. $P_+$ stays for the orthogonal projection going from $L^2({{\mathbb{D}}})$ to $L^2_a({{\mathbb{D}}})$. For a function $\varphi \in L^\infty ({{\mathbb{D}}})$, the Toeplitz operator $T_\varphi : L^2_a({{\mathbb{D}}})\to L^2_a({{\mathbb{D}}})$ is defined as $$\begin{align*} & T_\varphi f=P_+\varphi f, \quad f\in L^2_a({{\mathbb{D}}}). \end{align*}$$The main result of this article are spectral asymptotics for singular (or eigen-) values of compact Toeplitz operators with logarithmically decaying symbols, that is, $$\begin{align*} & \varphi(z)=\varphi_1(e^{i\theta})\, (1+\log(1/(1-r)))^{-\gamma},\quad \gamma>0, \end{align*}$$where $z=re^{i\theta }$ and $\varphi _1$ is a continuous (or piece-wise continuous) function on the unit circle. The result is applied to the spectral analysis of banded (including Jacobi) matrices.

A note on Banach fixed point property

In this short note we consider a sort of converse of the Banach fixed point theorem and prove that a metric space X is complete if and only if, for each closed subspace Y ⊆ X, any contraction f : Y → Y has a fixed point y ∈ Y.

Inscription on statistical convergence of order α

In this article we recall a remarkable result stated as "For a fixed ?, 0 < ? ? 1, the set of all bounded statistically convergent sequences of order ? is a closed linear subspace of m (m is the set of all bounded real sequences endowed with the sup norm)" by Bhunia et al. (Acta Math. Hungar. 130 (1-2) (2012), 153-161) and to develop the objective of this perception we demonstrate that the set of all bounded statistically convergent sequences of order ? may not form a closed subspace in other sequence spaces. Also we determine two different sequence spaces in which the set of all statistically convergent sequences of order ? (irrespective of boundedness) forms a closed set.

M-hypercyclicity of C 0-semigroup and Svep of its generator

Let 𝒯 = (Tt ) t ≥0 be a C 0-semigroup on a separable infinite dimensional Banach space X, with generator A. In this paper, we study the relationship between the single valued extension property of generator A, and the M-hypercyclicity of the C 0-semigroup. Specifically, we prove that if A does not have the single valued extension property at λ ∈ iℝ, then there exists a closed subspace M of X, such that the C 0-semigroup 𝒯 is M-hypercyclic. As a corollary, we get certain conditions of the generator A, for the C 0-semigroup to be M-hypercyclic.