Nuclear operators on Banach function spaces

Let X be a Banach space and E be a perfect Banach function space over a finite measure space $$(\Omega ,\Sigma ,\lambda )$$ ( Ω , Σ , λ ) such that $$L^\infty \subset E\subset L^1$$ L ∞ ⊂ E ⊂ L 1 . Let $$E'$$ E ′ denote the Köthe dual of E and $$\tau (E,E')$$ τ ( E , E ′ ) stand for the natural Mackey topology on E. It is shown that every nuclear operator $$T:E\rightarrow X$$ T : E → X between the locally convex space $$(E,\tau (E,E'))$$ ( E , τ ( E , E ′ ) ) and a Banach space X is Bochner representable. In particular, we obtain that a linear operator $$T:L^\infty \rightarrow X$$ T : L ∞ → X between the locally convex space $$(L^\infty ,\tau (L^\infty ,L^1))$$ ( L ∞ , τ ( L ∞ , L 1 ) ) and a Banach space X is nuclear if and only if its representing measure $$m_T:\Sigma \rightarrow X$$ m T : Σ → X has the Radon-Nikodym property and $$|m_T|(\Omega )=\Vert T\Vert _{nuc}$$ | m T | ( Ω ) = ‖ T ‖ nuc (= the nuclear norm of T). As an application, it is shown that some natural kernel operators on $$L^\infty $$ L ∞ are nuclear. Moreover, it is shown that every nuclear operator $$T:L^\infty \rightarrow X$$ T : L ∞ → X admits a factorization through some Orlicz space $$L^\varphi $$ L φ , that is, $$T=S\circ i_\infty $$ T = S ∘ i ∞ , where $$S:L^\varphi \rightarrow X$$ S : L φ → X is a Bochner representable and compact operator and $$i_\infty :L^\infty \rightarrow L^\varphi $$ i ∞ : L ∞ → L φ is the inclusion map.

The complex moment problem: determinacy and extendibility

Complex moment sequences are exactly those which admit positive definite extensions on the integer lattice points of the upper diagonal half-plane. Here we prove that the aforesaid extension is unique provided the complex moment sequence is determinate and its only representing measure has no atom at $0$. The question of converting the relation is posed as an open problem. A partial solution to this problem is established when at least one of representing measures is supported in a plane algebraic curve whose intersection with every straight line passing through $0$ is at most one point set. Further study concerns representing measures whose supports are Zariski dense in $\mathbb{C} $ as well as complex moment sequences which are constant on a family of parallel “Diophantine lines”. All this is supported by a bunch of illustrative examples.

An Inductive Julia-Carathéodory Theorem for Pick Functions in Two Variables

Classically, Nevanlinna showed that functions from the complex upper half plane into itself which satisfy nice asymptotic conditions are parametrized by finite measures on the real line. Furthermore, the higher order asymptotic behaviour at infinity of a map from the complex upper half plane into itself is governed by the existence of moments of its representing measure, which was the key to his solution of the Hamburger moment problem. Agler and McCarthy showed that an analogue of the above correspondence holds between a Pick function f of two variables, an analytic function which maps the product of two upper half planes into the upper half plane, and moment-like quantities arising from an operator theoretic representation for f. We apply their ‘moment’ theory to show that there is a fine hierarchy of levels of regularity at infinity for Pick functions in two variables, given by the Löwner classes and intermediate Löwner classes of order N, which can be exhibited in terms of certain formulae akin to the Julia quotient.

Control of Shareholders’ Wealth Maximization in Nigeria

This research focuses on who controls shareholder’s wealth maximization and how does this affect firm’s performance in publicly quoted non-financial companies in Nigeria. The shareholder fund was the dependent while explanatory variables were firm size (proxied by log of turnover), retained earning (representing management control) and dividend payment (representing measure of shareholders control). The data used for this study were obtained from the Nigerian Stock Exchange [NSE] fact book and the annual reports of the six sampled companies from Food/ Beverages and tobacco sub-sector for twenty years (1986-2005) to constitute pooled data of 120 observations. Using auto-regression technique for correcting serial auto-correlation in time series data, the results converge at ten iterations. Results showed that all the independent variables provide good explanation for the model. It was observed that firm size (log of turnover) and retained earnings had positive relationships and statistically significant impacts on the shareholders fund while dividend payment had negative relationship. The results show that turnover and retained earnings are of more significance in the control of shareholders wealth than the dividend payment. Thus, we can conclude that the management of the organizations under the present study is in major control of shareholders wealth maximization objective and impact on the firm performance. Implication is that selecting high quality management for the organizations would help in achieving shareholders wealth maximization objective in organizations.

UNIQUENESS IN MOMENT — PROBLEMS OVER NUCLEAR SPACES AND WEAK CONVERGENCE OF PROBABILITY MEASURES

Based on results from the theory of ordered (topological) vector spaces and on the theory of Fourier transforms of Radon probability measures (Bochner–Minlos–Schwartz) we present a solution of infinite dimensional moment problems over real nuclear spaces E. Both moment and truncated moment problems are treated simultaneously. In both cases uniqueness of the representing measure is characterized in terms of conditions on the set of moments directly. Concentration of the representing measures is expressed through continuity properties of the second moment. This is finally applied to characterize weak convergence of sequences of measures in terms of pointwise convergence of the associated sequence of moment functionals on the tensor algebra over E.

Representation of p-Lattice Summing Operators

In this paper we study some aspects of the behaviour of p-lattice summing operators. We prove first that an operator T from a Banach space E to a Banach lattice X is p-lattice summing if and only if its bitranspose is. Using this theorem we prove a characterization for 1 -lattice summing operators defined on a C(K) space by means of the representing measure, which shows that in this case 1 -lattice and ∞-lattice summing operators coincide. We present also some results for the case 1 ≤ p < ∞ on C(K,E).

Strictly singular and strictly cosingular operators on spaces of continuous functions

In this paper we will be concerned with studying operators T: C(K, X) → Y defined on Banach spaces of continuous functions. We will be particularly interested in studying the classes of strictly singular and strictly cosingular operators. In the process, we obtain answers to certain questions recently raised by Bombal and Porras in [5]. Specifically, we study Banach space X and Y for which an operator T: C(K, X) → Y with representing measure m is strictly singular (strictly cosingular) whenever m is strongly bounded and m(A) is strictly singular (strictly cosingular) for each Borel subset A of K. Along the way we establish several results dealing with non-compact operators on continuous function spaces, and we consolidate numerous results concerning extension theorems for operators defined on these same spaces. Also, we join Saab and Saab [25] in demonstrating that if l1 does not embed in X* then the adjoint T* of a strongly bounded map must be weakly precompact, thereby presenting an alternative solution to a question raised in [2].

Nuclear operators on spaces of continuous vector-valued functions

Let Ω: be a compact Hausdorff space, let E be a Banach space, and let C(Ω, E) stand for the Banach space of continuous E-valued functions on Ω under supnorm. It is well known [3, p. 182] that if F is a Banach space then any bounded linear operator T:C(Ω, E)→ F has a finitely additive vector measure G defined on the σ-field of Borel subsets of Ω with values in the space ℒ(E, F**) of bounded linear operators from E to the second dual F** of F. The measure G is said to represent T. The purpose of this note is to study the interplay between certain properties of the operator T and properties of the representing measure G. Precisely, one of our goals is to study when one can characterize nuclear operators in terms of their representing measures. This is of course motivated by a well-known theorem of L. Schwartz [5] (see also [3, p. 173]) concerning nuclear operators on spaces C(Ω) of continuous scalar-valued functions. The study of nuclear operators on spaces C(Ω, E) of continuous vector-valued functions was initiated in [1], where the author extended Schwartz's result in case E* has the Radon-Nikodym property. In this paper, we will show that the condition on E* to have the Radon-Nikodym property is necessary to have a Schwartz's type theorem. This leads to a new characterization of dual spaces E* with the Radon-Nikodym property. In [2], it was shown that if T:C(Ω, E)→ F is nuclear than its representing measure G takes its values in the space (E, F) of nuclear operators from E to F. One of the results of this paper is that if T:C(Ω, E)→ F is nuclear then its representing measure G is countably additive and of bounded variation as a vector measure taking its values in (E, F) equipped with the nuclear norm. Finally, we show by easy examples that the above mentioned conditions on the representing measure G do not characterize nuclear operators on C(Ω, E) spaces, and we also look at cases where nuclear operators are indeed characterized by the above two conditions. For all undefined notions and terminologies, we refer the reader to [3].

Commuting Dilations and Uniform Algebras

Let X be a compact Hausdorff space, let C(X) be the algebra of complex-valued continuous functions on X, and let A be a uniform algebra on X. Fix a nonzero complex homomorphism τ on A and a representing measure m for τ on X. The abstract Hardy space Hp = Hp(m), 1 ≤ p ≤ ∞, determined by A is defined to the closure of Lp = Lp(m) when p is finite and to be the weak*-closure of A in L∞ = L∞(m) p = ∞.