scholarly journals Levi Problem: Complement of a closed subspace in a Stein space and its applications

2021 ◽  
Vol 359 (8) ◽  
pp. 1023-1046
Author(s):  
Jing Zhang
Keyword(s):  
Closed Subspace ◽  
Stein Space ◽  
Levi Problem

scholarly journals In what spaces is every closed normal cone regular?

1970 ◽  
Vol 17 (2) ◽  
pp. 121-125 ◽  
Author(s):  
C. W. McArthur
Keyword(s):  
Banach Space ◽  
Convex Space ◽  
Unconditional Basis ◽  
Normal Cone ◽  
Closed Subspace ◽  
Locally Convex Space ◽  
Regular Part ◽  
Fréchet Space ◽  
Sequential Completeness ◽  
Locally Convex

It is known (13, p. 92) that each closed normal cone in a weakly sequentially complete locally convex space is regular and fully regular. Part of the main theorem of this paper shows that a certain amount of weak sequential completeness is necessary in order that each closed normal cone be regular. Specifically, it is shown that each closed normal cone in a Fréchet space is regular if and only if each closed subspace with an unconditional basis is weakly sequentially complete. If E is a strongly separable conjugate of a Banach space it is shown that each closed normal cone in E is fully regular. If E is a Banach space with an unconditional basis it is shown that each closed normal cone in E is fully regular if and only if E is the conjugate of a Banach space.

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

2021 ◽  
Author(s):  
Mahamet Koïta ◽  
Stanislas Kupin ◽  
Sergey Naboko ◽  
Belco Touré
Keyword(s):  
Spectral Analysis ◽  
Continuous Function ◽  
Bergman Space ◽  
Orthogonal Projection ◽  
Spectral Properties ◽  
Toeplitz Operators ◽  
Closed Subspace ◽  
Jacobi Matrices ◽  
Banded Matrices ◽  
Eigen Values

Abstract 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.

scholarly journals Some properties of Grauert type surfaces

2017 ◽  
Vol 28 (08) ◽  
pp. 1750063 ◽  
Author(s):  
Samuele Mongodi ◽  
Zbigniew Slodkowski ◽  
Giuseppe Tomassini
Keyword(s):  
Convex Space ◽  
Level Sets ◽  
Double Cover ◽  
Classification Theorem ◽  
Stein Space ◽  
Pluriharmonic Function ◽  
Exhaustion Function ◽  
Real Analytic ◽  
Complete Surfaces

In a previous work, we classified weakly complete surfaces which admit a real analytic plurisubharmonic exhaustion function; we showed that, if they are not proper over a Stein space, then they admit a pluriharmonic function, with compact Levi-flat level sets foliated with dense complex leaves. We called these Grauert type surfaces. In this note, we investigate some properties of these surfaces. Namely, we prove that the only compact curves that can be contained in them are negative in the sense of Grauert and that the level sets of the pluriharmonic function are connected, thus completing the analogy with the Cartan–Remmert reduction of a holomorphically convex space. Moreover, in our classification theorem, we had to pass to a double cover to produce the pluriharmonic function; the last part of the present paper is devoted to the construction of an example where passing to a double cover cannot be avoided.

scholarly journals Best Simultaneous Approximation in Orlicz Spaces

2007 ◽  
Vol 2007 ◽  
pp. 1-7 ◽  
Author(s):  
M. Khandaqji ◽  
Sh. Al-Sharif
Keyword(s):  
Banach Space ◽  
Simultaneous Approximation ◽  
Closed Subspace ◽  
Orlicz Spaces ◽  
Unit Interval ◽  
Luxemburg Norm ◽  
Integrable Functions ◽  
Best Simultaneous Approximation ◽  
Distance Formula

LetXbe a Banach space and letLΦ(I,X)denote the space of OrliczX-valued integrable functions on the unit intervalIequipped with the Luxemburg norm. In this paper, we present a distance formula dist(f1,f2,LΦ(I,G))Φ, whereGis a closed subspace ofX, andf1,f2∈LΦ(I,X). Moreover, some related results concerning best simultaneous approximation inLΦ(I,X)are presented.

scholarly journals Characterizing Complete Erdős Space

2009 ◽  
Vol 61 (1) ◽  
pp. 124-140 ◽  
Author(s):  
Jan J. Dijkstra ◽  
Jan van Mill
Keyword(s):  
Hilbert Space ◽  
Banach Spaces ◽  
Closed Subspace ◽  
Convergent Sequence ◽  
Cantor Set

Abstract. The space now known as complete Erdős space was introduced by Paul Erdős in 1940 as the closed subspace of the Hilbert space ℓ2 consisting of all vectors such that every coordinate is in the convergent sequence ﹛0﹜ ∪ ﹛1/n : n ∈ℕ﹜. In a solution to a problem posed by Lex G. Oversteegen we present simple and useful topological characterizations of . As an application we determine the class of factors of . In another application we determine precisely which of the spaces that can be constructed in the Banach spaces ℓp according to the ‘Erdős method’ are homeomorphic to . A novel application states that if I is a Polishable Fσ-ideal on ω, then I with the Polish topology is homeomorphic to either ℤ, the Cantor set 2ω, ℤ × 2ω, or . This last result answers a question that was asked by Stevo Todorčević.

scholarly journals Almost Surjective Epsilon-Isometry in The Reflexive Banach Spaces

CAUCHY ◽  
2017 ◽  
Vol 4 (4) ◽  
pp. 167
Author(s):  
Minanur Rohman
Keyword(s):  
Banach Space ◽  
Linear Operator ◽  
Banach Spaces ◽  
Lorentz Space ◽  
Bounded Linear Operator ◽  
Closed Subspace ◽  
Reflexive Banach Space ◽  
Reflexive Banach Spaces ◽  
Bounded Linear ◽  
Star Topology

<p class="AbstractCxSpFirst">In this paper, we will discuss some applications of almost surjective epsilon-isometry mapping, one of them is in Lorentz space ( L_(p,q)-space). Furthermore, using some classical theorems of w star-topology and concept of closed subspace -complemented, for every almost surjective epsilon-isometry mapping  <em>f </em>: <em>X to</em><em> Y</em>, where <em>Y</em> is a reflexive Banach space, then there exists a bounded linear operator   <em>T</em> : <em>Y to</em><em> X</em>  with  such that</p><p class="AbstractCxSpMiddle">  </p><p class="AbstractCxSpLast">for every x in X.</p>

scholarly journals On a Closed Subspace of L^(p(.)(Ω))

2020 ◽  
Vol 9 (2) ◽  
pp. 682-688
Author(s):  
Yasin Kaya
Keyword(s):  
Closed Subspace

Das Levi-Problem

2017 ◽  
pp. 147-172
Author(s):  
Jörg Eschmeier
Keyword(s):  
Levi Problem
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