The Radon-Nikodym theorem and the mean convergence of Banach space valued martingales

1969 ◽  
Vol 21 (1) ◽  
pp. 139-139 ◽  
Author(s):  
J. J. Uhl
Keyword(s):  
Banach Space ◽  
Mean Convergence ◽  
The Mean ◽  
Nikodym Theorem

scholarly journals Non-weak compactness of the integration map for vector measures

1993 ◽  
Vol 54 (3) ◽  
pp. 287-303 ◽  
Author(s):  
S. Okada ◽  
W. J. Ricker
Keyword(s):  
Banach Space ◽  
Volterra Operator ◽  
Continuous Operator ◽  
Weakly Compact ◽  
Vector Measures ◽  
Mean Convergence ◽  
Integrable Functions ◽  
The Mean ◽  
The Fourier Transform ◽  
Adjoint Operators

AbstractLet m be a vector measure with values in a Banach space X. If L1(m) denotes the space of all m integrable functions then, with respect to the mean convergence topology, L1(m) is a Banach space. A natural operator associated with m is its integration map Im which sends each f of L1(m) to the element ∫fdm (of X). Many properties of the (continuous) operator Im are closely related to the nature of the space L1(m). In general, it is difficult to identify L1(m). We aim to exhibit non-trivial examples of measures m in (non-reflexive) spaces X for which L1(m) can be explicitly computed and such that Im is not weakly compact. The examples include some well known operators from analysis (the Fourier transform on L1 ([−π, π]), the Volterra operator on L1 ([0, 1]), compact self-adjoint operators in a Hilbert space); such operators can be identified with integration maps Im (or their restrictions) for suitable measures m.

The Mean Convergence of Orthogonal Series

1950 ◽  
Vol 72 (4) ◽  
pp. 792 ◽  
Author(s):  
G. Milton Wing
Keyword(s):  
Orthogonal Series ◽  
Mean Convergence ◽  
The Mean

Some mean convergence theorems for weighted sums of Banach space valued random elements

Stochastics ◽  
2021 ◽  
pp. 1-19
Author(s):  
Pingyan Chen ◽  
Manuel Ordóñez Cabrera ◽  
Andrew Rosalsky ◽  
Andrei Volodin
Keyword(s):  
Banach Space ◽  
Weighted Sums ◽  
Convergence Theorems ◽  
Mean Convergence ◽  
Random Elements

CERTAIN ISOMETRIES RELATED TO THE BILATERAL LAPLACE TRANSFORM

2006 ◽  
Vol 11 (3) ◽  
pp. 331-346 ◽  
Author(s):  
S. B. Yakubovich
Keyword(s):  
Hilbert Space ◽  
Entire Function ◽  
Laplace Transform ◽  
Hilbert Spaces ◽  
Reproducing Kernel ◽  
Reproducing Kernel Hilbert Space ◽  
Inversion Formulas ◽  
Mean Convergence ◽  
The Laplace Transform ◽  
The Mean

We study certain isometries between Hilbert spaces, which are generated by the bilateral Laplace transform In particular, we construct an analog of the Bargmann‐Fock type reproducing kernel Hilbert space related to this transformation. It is shown that under some integra‐bility conditions on $ the Laplace transform FF(z), z = x + iy is an entire function belonging to this space. The corresponding isometrical identities, representations of norms, analogs of the Paley‐Wiener and Plancherel's theorems are established. As an application this approach drives us to a different type of real inversion formulas for the bilateral Laplace transform in the mean convergence sense.

On the mean convergence of Fourier–Jacobi series

2010 ◽  
Vol 62 (6) ◽  
pp. 943-960 ◽  
Author(s):  
V. P. Motornyi ◽  
S. V. Goncharov ◽  
P. K. Nitiema
Keyword(s):  
Jacobi Series ◽  
Mean Convergence ◽  
The Mean

scholarly journals UNCONDITIONAL CONVERGENCE IN THE STRONG OPERATOR TOPOLOGY AND ℓ∞

2011 ◽  
Vol 53 (3) ◽  
pp. 583-598 ◽  
Author(s):  
IOANA GHENCIU ◽  
PAUL LEWIS
Keyword(s):  
Banach Space ◽  
Vector Measure ◽  
Strong Operator Topology ◽  
Unconditional Convergence ◽  
Fundamental Result ◽  
Boundedness Theorem ◽  
Strong Operator ◽  
Spaces Of Operators ◽  
Funct Anal ◽  
Nikodym Theorem

AbstractIn this paper we study non-complemented spaces of operators and the embeddability of ℓ∞ in the spaces of operators L(X, Y), K(X, Y) and Kw*(X*, Y). Results of Bator and Lewis [2, 3] (Bull. Pol. Acad. Sci. Math.50(4) (2002), 413–416; Bull. Pol. Acad. Sci. Math.549(1) (2006), 63–73), Emmanuele [8–10] (J. Funct. Anal.99 (1991), 125–130; Math. Proc. Camb. Phil. Soc.111 (1992), 331–335; Atti. Sem. Mat. Fis. Univ. Modena42(1) (1994), 123–133), Feder [11] (Canad. Math. Bull.25 (1982), 78–81) and Kalton [16] (Math. Ann.208 (1974), 267–278), are generalised. A vector measure result is used to study the complementation of the spaces W(X, Y) and K(X, Y) in the space L(X, Y), as well as the complementation of K(X, Y) in W(X, Y). A fundamental result of Drewnowski [7] (Math. Proc. Camb. Phil. Soc. 108 (1990), 523–526) is used to establish a result for operator-valued measures, from which we obtain as corollaries the Vitali–Hahn–Saks–Nikodym theorem, the Nikodym Boundedness theorem and a Banach space version of the Phillips Lemma.

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