scholarly journals On the Fundamental Existence Theorem of Kishi

1963 ◽  
Vol 23 ◽  
pp. 189-198 ◽  
Author(s):  
Mitsuru Nakai
Keyword(s):  
Product Space ◽  
Borel Measure ◽  
Hausdorff Space ◽  
Lower Semicontinuous ◽  
Lower Semicontinuous Function ◽  
Semicontinuous Function ◽  
Locally Compact ◽  
Regular Borel Measure ◽  
Locally Compact Hausdorff Space ◽  
Semicontinuous Functions

Let Ω be a locally compact Hausdorff space and G(x, y) be a strictly positive lower semicontinuous function on the product space Ω×Ω of Ω. Such a function G(x, y) is called a kernel on Ω. The adjoint kernel Ğ(x, y) of G(x, y) is defined by Ğ(x, y) =G(y, x). Whenever we say a measure on Ω, we mean a positive regular Borel measure on Ω. The potential Gμ(x) and the adjoint potential Ğμ(x) of a measure μ relative to the kernel G(x, y) is defined byrespectively. These are also strictly positive lower semicontinuous functions on Ω provided μ≠0.

On the Ideal-Triangularizability of Semigroups of Quasinilpotent Positive Operators on C(K)

1998 ◽  
Vol 41 (3) ◽  
pp. 298-305
Author(s):  
M. T. Jahandideh
Keyword(s):  
Banach Lattice ◽  
Compact Hausdorff Space ◽  
Borel Measure ◽  
Hausdorff Space ◽  
Lower Semicontinuous ◽  
Integral Operators ◽  
Positive Operators ◽  
Locally Compact ◽  
Regular Borel Measure ◽  
The Ideal

AbstractIt is known that a semigroup of quasinilpotent integral operators, with positive lower semicontinuous kernels, on L2(X, μ), where X is a locally compact Hausdorff-Lindelöf space and μ is a σ-finite regular Borel measure on X, is triangularizable. In this article we use the Banach lattice version of triangularizability to establish the ideal-triangularizability of a semigroup of positive quasinilpotent integral operators on C(K) where K is a compact Hausdorff space.

scholarly journals An Existence Theorem in Potential Theory

1966 ◽  
Vol 27 (1) ◽  
pp. 133-137 ◽  
Author(s):  
Masanori Kishi
Keyword(s):  
Existence Theorem ◽  
Potential Theory ◽  
Compact Hausdorff Space ◽  
Hausdorff Space ◽  
Lower Semicontinuous ◽  
Locally Compact ◽  
Locally Compact Hausdorff Space

1. Concerning a positive lower semicontinuous kernel G on a locally compact Hausdorff space X the following existence theorem was obtained in [3].

Disjoint hypercyclic weighted translations on locally compact hausdorff spaces

2019 ◽  
Vol 69 (3) ◽  
pp. 647-664
Author(s):  
Ya Wang ◽  
Ze-Hua Zhou
Keyword(s):  
Borel Measure ◽  
Hausdorff Space ◽  
Weighted Space ◽  
Hausdorff Spaces ◽  
Locally Compact ◽  
Injective Mapping ◽  
Regular Borel Measure ◽  
Disjoint Hypercyclicity

Abstract Let G be a locally compact second countable Hausdorff space with a positive regular Borel measure λ, where λ is invariant under a continuous injective mapping φ : G → G. We characterize the disjoint hypercyclicity of finite weighted translations generated by φ acting on the weighted space Lp(G, ω) (1 ≤ p < ∞).

Measures Defined by Gages

1992 ◽  
Vol 44 (6) ◽  
pp. 1303-1316 ◽  
Author(s):  
Washek F. Pfeffer ◽  
Brian S. Thomson
Keyword(s):  
Set Theory ◽  
Measure Space ◽  
Hausdorff Space ◽  
Outer Measure ◽  
Riesz Representation Theorem ◽  
Locally Compact ◽  
Riemann Integral ◽  
Riesz Representation ◽  
Intuitive Concept ◽  
Locally Compact Hausdorff Space

AbstractUsing ideas of McShane ([4, Example 3]), a detailed development of the Riemann integral in a locally compact Hausdorff space X was presented in [1]. There the Riemann integral is derived from a finitely additive volume v defined on a suitable semiring of subsets of X. Vis-à-vis the Riesz representation theorem ([8, Theorem 2.141), the integral generates a Riesz measure v in X, whose relationship to the volume v was carefully investigated in [1, Section 7].In the present paper, we use the same setting as in [1] but produce the measure directly without introducing the Riemann integral. Specifically, we define an outer measure by means of gages and introduce a very intuitive concept of gage measurability that is different from the usual Carathéodory définition. We prove that if the outer measure is σ-finite, the resulting measure space is identical to that defined by means of the Carathéodory technique, and consequently to that of [1, Section 7]. If the outer measure is not σ-finite, we investigate the gage measurability of Carathéodory measurable sets that are σ-finite. Somewhat surprisingly, it turns out that this depends on the axioms of set theory.

scholarly journals On the theorem of Kishi for a continuous function-kernel

1974 ◽  
Vol 53 ◽  
pp. 127-135 ◽  
Author(s):  
Isao Higuchi ◽  
Masayuki Itô
Keyword(s):  
Continuous Function ◽  
Potential Theory ◽  
Compact Hausdorff Space ◽  
Symmetric Function ◽  
Hausdorff Space ◽  
Locally Compact ◽  
Continuity Principle ◽  
Lower Semi ◽  
Locally Compact Hausdorff Space

In the potential theory with respect to a non-symmetric function-kernel, the following theorem is obtained by M. Kishi ([3]).Let X be a locally compact Hausdorff space and G be a lower semi-continuous function-kernel on X. Assume that G(x, x)>0 for any x in X and that G and the adjoint kernel Ğ of G satisfy “the continuity principle”.

scholarly journals More On the countability index and the density index of S(X)

1990 ◽  
Vol 33 (1) ◽  
pp. 159-164
Author(s):  
K. D. Magill
Keyword(s):  
Finite Number ◽  
Topological Semigroup ◽  
Hausdorff Space ◽  
Extension Property ◽  
Dimensional Subspace ◽  
Countable Subset ◽  
Compact Open Topology ◽  
Locally Compact ◽  
Density Index ◽  
Locally Compact Hausdorff Space

The countability index, C(S), of a semigroup S is the smallest integer n, if it exists, such that every countable subset of S is contained in a subsemigroup with n generators. If no such integer exists, define C(S) = ∞. The density index, D(S), of a topological semigroup S is the smallest integer n, if it exists, such that S contains a dense subsemigroup with n generators. If no such integer exists, define D(S) = ∞. S(X) is the topological semigroup of all continuous selfmaps of the locally compact Hausdorff space X where S(X) is given the compact-open topology. Various results are obtained about C(S(X)) and D(S(X)). For example, if X consists of a finite number (< 1) of components, each of which is a compact N-dimensional subspace of Euclidean Nspace and has the internal extension property and X is not the two point discrete space. Then C(S(X)) exceeds two but is finite. There are additional results for C(S(X)) and similar results for D(S(X)).

A Characterization of Proximal Subgradient Set-Valued Mappings

1993 ◽  
Vol 36 (1) ◽  
pp. 116-122 ◽  
Author(s):  
R. A. Poliquin
Keyword(s):  
Lower Semicontinuous ◽  
Lower Semicontinuous Function ◽  
Semicontinuous Function ◽  
Set Valued Mapping ◽  
Selection Property ◽  
Set Valued Mappings ◽  
Bounded Below

AbstractIn this paper we tackle the problem of identifying set-valued mappings that are subgradient set-valued mappings. We show that a set-valued mapping is the proximal subgradient mapping of a lower semicontinuous function bounded below by a quadratic if and only if it satisfies a monotone selection property.

scholarly journals On projective lift and orbit spaces

1994 ◽  
Vol 50 (3) ◽  
pp. 445-449 ◽  
Author(s):  
T.K. Das
Keyword(s):  
Compact Hausdorff Space ◽  
Hausdorff Space ◽  
Covariant Functor ◽  
Hausdorff Spaces ◽  
Orbit Spaces ◽  
Locally Compact ◽  
Discontinuous Group ◽  
Coreflective Subcategory ◽  
Group Of Homeomorphisms ◽  
Locally Compact Hausdorff Space

By constructing the projective lift of a dp-epimorphism, we find a covariant functor E from the category Cd of regular Hausdorff spaces and continuous dp-epimorphisms to its coreflective subcategory εd consisting of projective objects of Cd We use E to show that E(X/G) is homeomorphic to EX/G whenever G is a properly discontinuous group of homeomorphisms of a locally compact Hausdorff space X and X/G is an object of Cd.

scholarly journals Multipliers of Modules of Continuous Vector-Valued Functions

2014 ◽  
Vol 2014 ◽  
pp. 1-6
Author(s):  
Liaqat Ali Khan ◽  
Saud M. Alsulami
Keyword(s):  
Compact Hausdorff Space ◽  
Hausdorff Space ◽  
Normed Spaces ◽  
Locally Compact ◽  
Continuous Vector ◽  
Vector Valued Functions ◽  
Vector Valued ◽  
Locally Compact Hausdorff Space ◽  
Locally Convex

In 1961, Wang showed that ifAis the commutativeC*-algebraC0(X)withXa locally compact Hausdorff space, thenM(C0(X))≅Cb(X). Later, this type of characterization of multipliers of spaces of continuous scalar-valued functions has also been generalized to algebras and modules of continuous vector-valued functions by several authors. In this paper, we obtain further extension of these results by showing thatHomC0(X,A)(C0(X,E),C0(X,F))≃Cs,b(X,HomA(E,F)),whereEandFarep-normed spaces which are also essential isometric leftA-modules withAbeing a certain commutativeF-algebra, not necessarily locally convex. Our results unify and extend several known results in the literature.

scholarly journals A Riemann integral in a locally compact Hausdorff space

1986 ◽  
Vol 41 (1) ◽  
pp. 115-137 ◽  
Author(s):  
S. I. Ahmed ◽  
W. F. Pfeffer
Keyword(s):  
Compact Hausdorff Space ◽  
Hausdorff Space ◽  
Sufficient Condition ◽  
Lebesgue Integral ◽  
Necessary And Sufficient Condition ◽  
Locally Compact ◽  
Riemann Integral ◽  
Variational Integrals ◽  
Necessary And Sufficient ◽  
Locally Compact Hausdorff Space

AbstractWe present a systematic and self-contained exposition of the generalized Riemann integral in a locally compact Hausdorff space, and we show that it is equivalent to the Perron and variational integrals. We also give a necessary and sufficient condition for its equivalence to the Lebesgue integral with respect to a suitably chosen measure.

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