On the Smirnov Class Defined by the Maximal Function

2002 ◽  
Vol 45 (2) ◽  
pp. 265-271 ◽  
Author(s):  
Marek Nawrocki
Keyword(s):  
Maximal Function ◽  
Linear Functional ◽  
Continuous Linear ◽  
Continuous Linear Functional ◽  
Smirnov Class ◽  
Convex Structure ◽  
Dual Spaces ◽  
Radial Maximal Function ◽  
Locally Convex

AbstractH. O. Kim has shown that contrary to the case of Hp-space, the Smirnov class M defined by the radial maximal function is essentially smaller than the classical Smirnov class of the disk. In the paper we show that these two classes have the same corresponding locally convex structure, i.e. they have the same dual spaces and the same Fréchet envelopes. We describe a general form of a continuous linear functional on M and multiplier from M into Hp, 0 < p ≤ ∞.

scholarly journals Extreme points and support points of conformal mappings

2019 ◽  
Vol 17 (1) ◽  
pp. 23-31
Author(s):  
Ronen Peretz
Keyword(s):  
Remainder Term ◽  
Linear Functional ◽  
Convex Combination ◽  
Extreme Points ◽  
Holomorphic Functions ◽  
Support Point ◽  
Continuous Linear ◽  
Continuous Linear Functional ◽  
The Third ◽  
Support Points

Abstract There are three types of results in this paper. The first, extending a representation theorem on a conformal mapping that omits two values of equal modulus. This was due to Brickman and Wilken. They constructed a representation as a convex combination with two terms. Our representation constructs convex combinations with unlimited number of terms. In the limit one can think of it as an integration over a probability space with the uniform distribution. The second result determines the sign of ℜ L(z0(f(z))2) up to a remainder term which is expressed using a certain integral that involves the Löwner chain induced by f(z), for a support point f(z) which maximizes ℜ L. Here L is a continuous linear functional on H(U), the topological vector space of the holomorphic functions in the unit disk U = {z ∈ ℂ | |z| < 1}. Such a support point is known to be a slit mapping and f(z0) is the tip of the slit ℂ − f(U). The third demonstrates some properties of support points of the subspace Sn of S. Sn contains all the polynomials in S of degree n or less. For instance such a support point p(z) has a zero of its derivative p′(z) on ∂U.

scholarly journals Maximal ideals in some F-algebras of holomorphic functions

Filomat ◽  
2015 ◽  
Vol 29 (1) ◽  
pp. 1-5 ◽  
Author(s):  
Romeo Mestrovic
Keyword(s):  
Linear Functional ◽  
Holomorphic Functions ◽  
Complete Characterization ◽  
Open Unit ◽  
Continuous Linear ◽  
Linear Functionals ◽  
Continuous Linear Functional ◽  
Maximal Ideals ◽  
Algebras Of Holomorphic Functions

For 1 < p < ?, the Privalov class Np consists of all holomorphic functions f on the open unit disk D of the complex plane C such that sup 0?r<1?2?0 (log+ |f(rei?)j|p d?/2? < + ? M. Stoll [16] showed that the space Np with the topology given by the metric dp defined as dp(f,g) = (?2?0 (log(1 + |f*(ei?) - g*(ei?)|))p d?/2?)1/p, f,g ? Np; becomes an F-algebra. Since the map f ? dp(f,0) (f ? Np) is not a norm, Np is not a Banach algebra. Here we investigate the structure of maximal ideals of the algebras Np (1 < p < ?). We also give a complete characterization of multiplicative linear functionals on the spaces Np. As an application, we show that there exists a maximal ideal of Np which is not the kernel of a multiplicative continuous linear functional on Np.

scholarly journals The states of a Banach algebra generate the dual

1971 ◽  
Vol 17 (4) ◽  
pp. 341-344 ◽  
Author(s):  
Allan M. Sinclair
Keyword(s):  
Banach Space ◽  
Banach Algebra ◽  
Linear Combination ◽  
Linear Functional ◽  
Measure Theory ◽  
Complex Field ◽  
Continuous Linear ◽  
Continuous Linear Functional ◽  
Unital Banach Algebra ◽  
Dual Banach Space

In this paper we prove that the states of a unital Banach algebra generate the dual Banach space as a linear space (Theorem 2). This is a result of R. T. Moore (4, Theorem 1(a)) who uses a decomposition of measures in his proof. In the proof given here the measure theory is replaced by a Hahn-Banach separation argument. We shall let A denote a unital Banach algebra over the complex field, and D(1) denote {f ∈ A′: ‖f‖ = f(1) = 1} where A′ is the dual of A. The motivation of Moore's results is the theorem that in a C*-algebra every continuous linear functional is a linear combination of four states (the states are the elements of D(1)) (see (2, 2.6.4, 2.1.9, 1.1.10)).

On Linear Functionals and Summability Factors for Strong Summability II

1981 ◽  
Vol 33 (4) ◽  
pp. 769-781
Author(s):  
W. Balser ◽  
W. B. Jurkat ◽  
A. Peyerimhoff
Keyword(s):  
Linear Functional ◽  
Strong Summability ◽  
Summability Factors ◽  
Continuous Linear ◽  
Linear Functionals ◽  
Continuous Linear Functional ◽  
Satisfactory Answer ◽  
Two Parameters

The first part of this paper, which will be referred to by I, appeared in Volume 30 of this journal. The present paper will use the same bibliography as I.Theorem 1 in I shows that the knowledge of all continuous linear functionals in o[A]p is essential in determining convergence and summability factors for strong summability. So far, Theorem 7 in I was for a general A the only tool in deciding whether a given sequence ∈ generates such a functional. We mentioned in a remark following Theorem 7 the difficulties in verifying the conditions of this theorem (two parameters are involved). In the present paper, we study continuous linear functionals in o[A]1 in more detail, and we obtain in a corollary to Theorem 22 a condition which appears to be a more satisfactory answer to the question, whether a given sequence ∈ generates a continuous linear functional in o[A]1.

Properties (V) and (w V) on C(Ω X)

1995 ◽  
Vol 117 (3) ◽  
pp. 469-477 ◽  
Author(s):  
Elizabeth M. Bator ◽  
Paul W. Lewis
Keyword(s):  
Banach Space ◽  
Continuous Function ◽  
Linear Functional ◽  
Formal Series ◽  
Continuous Linear ◽  
Continuous Linear Functional ◽  
Weakly Compact ◽  
Image Position ◽  
Continuous Function Space ◽  
Fundamental Paper

A formal series Σxn in a Banach space X is said to be weakly unconditionally converging, or alternatively weakly unconditionally Cauchy (wuc) if Σ|x*(xn)| < ∞ for every continuous linear functional x* ∈ X*. A subset K of X* is called a V-subset of X* iffor each wuc series Σxn in X. Further, the Banach space X is said to have property (V) if the V-subsets of X* coincide with the relatively weakly compact subsets of X*. In a fundamental paper in 1962, Pelczynski [10] showed that the Banach space X has property (V) if and only if every unconditionally converging operator with domain X is weakly compact. In this same paper, Pelczynski also showed that all C(Ω) spaces have property (V), and asked if the abstract continuous function space C(Ω, X) has property (F) whenever X has property (F).

scholarly journals A noncommutative theory of Bade functionals

1991 ◽  
Vol 33 (1) ◽  
pp. 73-81 ◽  
Author(s):  
Don Hadwin ◽  
Mehmet Orhon
Keyword(s):  
Banach Space ◽  
Linear Functional ◽  
Hausdorff Space ◽  
Boolean Algebras ◽  
Representation Space ◽  
Continuous Linear ◽  
Continuous Linear Functional ◽  
Banach Modules ◽  
Key Concepts ◽  
Stone Representation

Since the pioneering work of W. G. Bade [3, 4] a great deal of work has been done on bounded Boolean algebras of projections on a Banach space ([11, XVII.3.XVIII.3], [21, V.3], [16], [6], [12], [13], [14], ]17], [18], [23], [24]). Via the Stone representation space of the Boolean algebra, the theory can be studied through Banach modules over C(K), where K is a compact Hausdorff space. One of the key concepts in the theory is the notion of Bade functionals. If X is a Banach C(K)-module and x ε X, then a Bade functional of x with respect to C(K) is a continuous linear functional α on X such that, for each a in C(K) with a ≥ 0, we have(i) α (ax) ≥0,(ii) if α (ax) = 0, then ax = 0.

On order structure and operators in L ∞(μ)

2009 ◽  
Vol 7 (4) ◽  
Author(s):  
Irina Krasikova ◽  
Miguel Martín ◽  
Javier Merí ◽  
Vladimir Mykhaylyuk ◽  
Mikhail Popov
Keyword(s):  
Banach Space ◽  
Compact Operator ◽  
Linear Functional ◽  
Continuous Operator ◽  
The Other ◽  
Order Structure ◽  
Continuous Linear ◽  
Continuous Linear Functional ◽  
Other Hand ◽  
Continuous Operators

AbstractIt is known that there is a continuous linear functional on L ∞ which is not narrow. On the other hand, every order-to-norm continuous AM-compact operator from L ∞(μ) to a Banach space is narrow. We study order-to-norm continuous operators acting from L ∞(μ) with a finite atomless measure μ to a Banach space. One of our main results asserts that every order-to-norm continuous operator from L ∞(μ) to c 0(Γ) is narrow while not every such an operator is AM-compact.

scholarly journals On linear continuous operators on C_p-spaces

2021 ◽  
Vol 21 (1) ◽  
pp. 45-50
Author(s):  
A.P. Devyatkov ◽  
◽  
S.D. Shalaginov ◽  
Keyword(s):  
Linear Functional ◽  
Pointwise Convergence ◽  
Continuous Operator ◽  
Continuous Functions ◽  
Linear Continuous Operator ◽  
Continuous Linear ◽  
Space Of Continuous Functions ◽  
Continuous Linear Functional ◽  
Topology Of Pointwise Convergence ◽  
Continuous Operators

The paper describes the structure of a linear continuous operator on the space of continuous functions in the topology of pointwise convergence. The corresponding theorem is a generalization of A.V.Arkhangel'skii's theorem on the general form of a continuous linear functional on such spaces.

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