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A Caltech Library Repository Feedhttp://www.rssboard.org/rss-specificationpython-feedgenenThu, 30 Nov 2023 19:39:18 +0000Fourier Transforms of Certain Classes of Integrable Functions
https://resolver.caltech.edu/CaltechETD:etd-06152006-085338
Authors: Ryan, Robert Dean
Year: 1960
DOI: 10.7907/GHJR-RD61
NOTE: Text or symbols not renderable in plain ASCII are indicated by [...]. Abstract is included in .pdf document.
Let G be a locally compact Abelian group with character group [...]. M(G) will denote the class of all bounded Radon measures on G and P(G) will denote the class of all continuous positive definite functions on G. For [...] we write [...] = [...] and for [...] we write [...] = [...]. [...] will denote the linear space spanned by [...]. We find necessary and sufficient conditions on [...] in order that [...] for [...]. Theorem 5, Chapter II: [...] for [...] if and only if there exists a constant K > 0 such that [...] for all [...] where [...]. Theorem 6, Chapter II: [...] for [...] if and only if [...] for all [...]. Theorems 3 and 4, Chapter III: [...] if and only if there exists some p, [...], such that for each [...] > 0 there exists a [...] > 0 with the property that [...] whenever [...] and [...]. By taking G to be the unit circle and p = 2 in Theorems 3 and 4, Chapter III, we obtain a generalization of a theorem by R. Salem (Comptes Rendus Vol. 192 (1931)). Taking G to be the additive group of reals and p = 1 gives a generalization of a theorem by A. Berry (Annals of Math. (2) Vol. 32 (1931)).https://thesis.library.caltech.edu/id/eprint/2609The Egoroff property and related properties in the theory of Riesz spaces
https://resolver.caltech.edu/CaltechETD:etd-01142003-101748
Authors: Holbrook, John A. R.
Year: 1965
DOI: 10.7907/BN4B-9460
NOTE: Text or symbols not renderable in plain ASCII are indicated by [...]. Abstract is included in .pdf document.
A Riesz space L is said to be Egoroff, if, whenever [...] and [...], there is a sequence [...] in L such that [...] and, for each n,m, there exists an index k(n,m) such that [...]. This notion was introduced, in rather a different form, by Nakano. Banach function spaces are Egoroff, and Lorentz showed that, for any function seminorm [...], the maximal seminorm [...] among those which are dominated by [...] and which are [...] (a monotone seminorm [...] is [...] if [...]) is precisely the "Lorentz seminorm" [...], where [...]. In this thesis the extent to which [...] holds in general Riesz spaces is determined. In fact, [...] for every monotone seminorm [...] on a Riesz space L if, and only if, L is "almost-Egoroff". The almost-Egoroff property is closely related to the Egoroff property and, indeed, coincides with it in the case of Archimedean spaces. Analogous theorems for Boolean algebras are discussed. The almost-Egoroff property is shown to yield a number of results which ensure that, under certain conditions, a monotone seminorm is [...] when restricted to an appropriate super order dense ideal. Riesz spaces L possessing an integral, Riesz norm [...](i.e., a Riesz norm such that [...] are considered also, since in many cases these are known to be Egoroff. In particular if [...] is normal on L (i.e., [...] a directed system, [...] ), then L is Egoroff. In this connection, a pathological space, possessing an integral Riesz norm which is nowhere normal, is constructed.https://thesis.library.caltech.edu/id/eprint/157Multiplication in Riesz spaces
https://resolver.caltech.edu/CaltechTHESIS:10192015-111930391
Authors: Rice, Norman Molesworth
Year: 1966
DOI: 10.7907/PZMJ-B369
<p>A.G. Vulih has shown how an essentially unique intrinsic multiplication can be defined in certain types of Riesz spaces (vector lattices) L. In general, the multiplication is not universally defined in L, but L can always be imbedded in a large space L<sup>#</sup> in which multiplication is universally defined.</p>
<p>If ф is a normal integral in L, then ф can be extended to a normal integral on a large space L<sub>1</sub>(ф) in L<sup>#</sup>, and L<sub>1</sub>(ф) may be regarded as an abstract integral space. A very general form of the Radon-Nikodym theorem can be proved in L<sub>1</sub>(ф), and this can be used to give a relatively simple proof of a theorem of Segal giving a necessary and sufficient condition that the Radon-Nikodym theorem hold in a measure space.</p>
<p>In another application, the multiplication is used to give a representation of certain Riesz spaces as rings of operators on a Hilbert space. </p>
https://thesis.library.caltech.edu/id/eprint/9227Locally convex Riesz spaces and Archimedean quotient spaces
https://resolver.caltech.edu/CaltechTHESIS:10052015-142308267
Authors: Moore, Lawrence Carlton
Year: 1966
DOI: 10.7907/15GP-R248
<p>A Riesz space with a Hausdorff, locally convex topology determined by Riesz seminorms is called a <u>locally</u> <u>convex</u> <u>Riesz</u> <u>space</u>. A sequence {x<sub>n</sub>} in a locally convex Riesz space L is said to <u>converge</u> <u>locally</u> to x ϵ L if for some topologically bounded set B and every real r ˃ 0 there exists N (r) and n ≥ N (r) implies x – x<sub>n</sub> ϵ r<sup>b</sup>. Local Cauchy sequences are defined analogously, and L is said to be locally complete if every local Cauchy sequence converges locally. Then L is locally complete if and only if every monotone local Cauchy sequence has a least upper bound. This is a somewhat more general form of the completeness criterion for Riesz – normed Riesz spaces given by Luxemburg and Zaanen. Locally complete, bound, locally convex Riesz spaces are barrelled. If the space is metrizable, local completeness and topological completeness are equivalent.</p>
<p>Two measures of the non-archimedean character of a non-archimedean Riesz space L are the smallest ideal A<sub>o</sub> (L) such that quotient space is Archimedean and the ideal I (L) = { x ϵ L: for some 0 ≤ v ϵ L, n |x| ≤ v for n = 1, 2, …}. In general A<sub>o</sub> (L) ᴝ I (L). If L is itself a quotient space, a necessary and sufficient condition that A<sub>o</sub> (L) = I (L) is given. There is an example where A<sub>o</sub> (L) ≠ I (L). </p>
<p>A necessary and sufficient condition that a Riesz space L have every quotient space Archimedean is that for every 0 ≤ u, v ϵ L there exist u<sub>1</sub> = sup (inf (n v, u): n = 1, 2, …), and real numbers m<sub>1</sub> and m<sub>2</sub> such that m<sub>1</sub> u<sub>1</sub> ≥ v<sub>1</sub> and m<sub>2</sub> v<sub>1</sub> ≥ u<sub>1</sub>. If, in addition, L is Dedekind σ – complete, then L may be represented as the space of all functions which vanish off finite subsets of some non-empty set. </p>
https://thesis.library.caltech.edu/id/eprint/9200Invariant subspaces in Hilbert and normed spaces
https://resolver.caltech.edu/CaltechETD:etd-10042002-144336
Authors: Taylor, Richard Forsythe
Year: 1968
DOI: 10.7907/1453-JV44
NOTE: Text or symbols not renderable in plain ASCII are indicated by [...]. Abstract is included in .pdf document.
This dissertation concerns itself with the following question: Suppose T is a bounded linear operator from an infinite dimensional Hilbert Space into itself. What are sufficient conditions to imply the existence of a nonzero, proper subspace M of H such that T(M)[...]M? The methodology used to approach the question is in line with the methods developed by Aronzajn and Smith [1] and Bernstein and Robinson [3]. The entire thesis is exposited within the framework of nonstandard analysis as developed by Robinson [9].
Chapter 1 of the dissertation develops the necessary theory involved, and presents a necessary and sufficient condition for T to have a proper invariant subspace. The conditions involve assumptions on certain finite dimensional approximations of T.
Chapter 2 demonstrates two situations under which the conditions presented in Chapter 1 come about. The first of these, which was announced by Feldman [5] and has been published in preprint form by Gillespie [6], was proved independently by the author under more relaxed conditions. For simplicity, we state here the Feldman result.
Theorem: If T is quasi-nilpotent and if the algebra generated by T has a nonzero compact operator in its uniform closure, then T has an invariant subspace.
It is still an open question whether or not the condition "T commutes with a compact operator" implies the desired result. By insisting that C be "very compact" (to be defined) the following result is demonstrated.
Theorem: If C is a nonzero "very compact" operator, and if TC=CT, then T has an invariant subspace.https://thesis.library.caltech.edu/id/eprint/3899Measures in topological spaces
https://resolver.caltech.edu/CaltechETD:etd-09252002-093739
Authors: Kirk, Ronald Brian
Year: 1968
DOI: 10.7907/00H3-6K91
NOTE: Text or symbols not renderable in plain ASCII are indicated by [...].
Abstract is included in .pdf document.
Let X be a completely-regular topological space and let C*(X) denote the space of all bounded, real-valued continuous functions on X. For a positive linear functional [...] on C*(X), consider the following two continuity conditions. [...] is said to be a B-integral if whenever [...] and [...] for all [...], then [...]. [...] is said to be B-normal if whenever [...] is a directed system with [...] for all [...], then [...]. It is obvious that a B-normal functional is always a B-integral. The main concern of this paper is what can be said in the converse direction.
Methods are developed for discussing this question. Of particular importance is the representation of C*(X) as a space [...] of finitely-additive set functions on a certain algebra of subsets of X. This result was first announced by A. D. Alexandrov, but his proof was obscure. Since there seem to be no proofs readily available in the literature, a complete proof is given here. Supports of functionals are discussed and a relatively simple proof is given of the fact that every B-integral is B-normal if and only if every B-integral has a support.
The space X is said to be B-compact if every B-integral is B-normal. It is shown that B-compactness is a topological invariant and various topological properties of B-compact spaces are investigated. For instance, it is shown that B-compactness is permanent on the closed sets and the co-zero sets of a B-compact space. In the case that the spaces involved are locally-compact, it is shown that countable products and finite intersections of B-compact spaces are B-compact.
Also B-compactness is studied with reference to the classical compactness conditions. For instance, it is shown that if X is B-compact, then X is realcompact. Or that if X is paracompact and if the continuum hypothesis holds, then X is B-compact if and only if X is realcompact.
Finally, the methods and results developed in the paper are applied to formulate and prove a very general version of the classical Kolmogorov consistency theorem of probability theory. The result is as follows. If X is a locally-compact, B-compact space and if S is an abstract set, then a necessary and sufficient condition that a finitely-additive set function defined on the Baire (or the Borel) cylinder sets of X[superscript S] be a measure is that its projection on each of the finite coordinate spaces be Baire (or regular Borel) measures.
https://thesis.library.caltech.edu/id/eprint/3748The Egoroff property and its relation to the order topology in the theory of Riesz spaces
https://resolver.caltech.edu/CaltechETD:etd-10072002-143502
Authors: Chow, Theresa Kee Yu
Year: 1969
DOI: 10.7907/T2KV-BF37
NOTE: Text or symbols not renderable in plain ASCII are indicated by [...]. Abstract is included in .pdf document.
A sequence(f[subscript]n : n = 1, 2, ...) in a Riesz space L is order convergent to an element [...] whenever there exists a sequence [...] in L such that [...] holds for all n. Sequential order convergence defines the order topology on L. The closure of a subset S in this topology is denoted by cl(S). The pseudo order closure S' of a subset S is the set of all [...] such that there exists a sequence in S which is order convergent to f. If S' = cl(S) for every convex subset S, then S' = cl(S) for every subset S. L has the Egoroff property if and only if S' = cl(S) for every order bounded subset S of L. A necessary and sufficient condition for L to have the property that S' = cl(S) for every subset S of L is that L has the strong Egoroff property.
A sequence(f[subscript]n : n = 1, 2, ...) in a Riesz space L is ru-convergent to an element [...] whenever there exists a real sequence [...] and an element [...] such that [...] holds for all n. Sequential ru-convergence defines the ru-topology on L. The closure of a subset S in this topology is denoted by [...]. The pseudo ru-closure S'[subscript ru] of a subset S is the set of all [...] such that there exists a sequence in S which is ru-convergent to f. If L is Archimedean, then [...] for every convex subset S implies that [...] for every subset S. A characterization of those Archimedean Riesz spaces L with the property that [...] for every subset S of L is obtained.
If [...] is a monotone seminorm on a Riesz space L, then a necessary and sufficient condition for [...] in L implies [...] is that the set [...] is order closed. For every monotone seminorm [...] on L, the largest [...]-Fatou monotone serninorm bounded by [...] is the Minkowski functional of the order closure of [...].
A monotone seminorm p on a Riesz space L is called strong Fatou whenever [...]. A characterization of those Riesz spaces L which have the following property is given: "For every monotone seminorm [...], the largest strong Fatou monotone seminorm bounded by [...] : [...]." A similar characterization for Boolean algebras is also obtained.
https://thesis.library.caltech.edu/id/eprint/3955The Riesz space structure of an Abelian W*-algebra
https://resolver.caltech.edu/CaltechTHESIS:02222016-142556483
Authors: Dodds, Peter Gerard
Year: 1969
DOI: 10.7907/MBWY-0552
<p>Let M be an Abelian W*-algebra of operators on a Hilbert space <i>H</i>. Let M<sub>0</sub> be the set of all linear, closed, densely defined transformations in <i>H</i> which commute with every unitary operator in the commutant M’ of M. A well known result of R. Pallu de Barriere states that if ɸ is a normal positive linear functional on M, then ɸ is of the form T → (Tx, x) for some x in <i>H</i>, where T is in M. An elementary proof of this result is given, using only those properties which are consequences of the fact that ReM is a Dedekind complete Riesz space with plenty of normal integrals. The techniques used lead to a natural construction of the class M<sub>0</sub>, and an elementary proof is given of the fact that a positive self-adjoint transformation in M<sub>0</sub> has a unique positive square root in M<sub>0</sub>. It is then shown that when the algebraic operations are suitably defined, then M<sub>0</sub> becomes a commutative algebra. If ReM<sub>0</sub> denotes the set of all self-adjoint elements of M<sub>0</sub>, then it is proved that ReM<sub>0</sub> is Dedekind complete, universally complete Riesz spaces which contains ReM as an order dense ideal. A generalization of the result of R. Pallu de la Barriere is obtained for the Riesz space ReM<sub>0</sub> which characterizes the normal integrals on the order dense ideals of ReM<sub>0</sub>. It is then shown that ReM<sub>0</sub> may be identified with the extended order dual of ReM, and that ReM<sub>0</sub> is perfect in the extended sense. </p>
<p>Some secondary questions related to the Riesz space ReM are also studied. In particular it is shown that ReM is a perfect Riesz space, and that every integral is normal under the assumption that every decomposition of the identity operator has non-measurable cardinal. The presence of atoms in ReM is examined briefly, and it is shown that ReM is finite dimensional if and only if every order bounded linear functional on ReM is a normal integral.</p>
https://thesis.library.caltech.edu/id/eprint/9578Rearrangements of Measurable Functions
https://resolver.caltech.edu/CaltechTHESIS:10292010-132102259
Authors: Day, Peter William
Year: 1970
DOI: 10.7907/6V2Z-F375
<p>Let (X, Λ, μ) be a measure space and let M(X, μ) denote the set of all extended real valued measurable functions on X. If (X<sub>1</sub>, Λ<sub>1</sub>, μ<sub>1</sub>) is also a measure space and f ϵ M(X, μ) and g ϵ M(X<sub>1</sub>, μ<sub>1</sub>), then f and g are said to be equimeasurable (written f ~ g) iff μ (f<sup>-1</sup>[r, s]) = μ<sub>1</sub>(g<sup>-1</sup>[r, s]) whenever [r, s] is a bounded interval of the real numbers or [r, s] = {+ ∝} or = {- ∝}. Equimeasurability is investigated systematically and in detail.</p>
<p>If (X, Λ, μ) is a finite measure space (i. e. μ (X) < ∝) then for each f ϵ M(X, μ) the decreasing rearrangement δ<sub>f</sub> of f is defined by</p>
<p>δ<sub>f</sub>(t) = inf {s: μ ({f > s}) ≤ t} 0 ≤ t ≤ μ(X).</p>
<p>Then δ<sub>f</sub> is the unique decreasing right continuous function on [0, μ(X)] such that δ<sub>f</sub> ~ f. If (X, Λ, μ) is non-atomic, then there is a measure preserving map σ: X → [0, μ(X)] such that δ<sub>f</sub>(σ) = f μ-a.e.</p>
<p>If (X, Λ, μ) is an arbitrary measure space and f ϵ M(X, μ) then f is said to have a decreasing rearrangement iff there is an interval J of the real numbers and a decreasing function δ on J such that f ~ δ. The set D(X, μ} of functions having decreasing rearrangements is characterized, and a particular decreasing rearrangement δ<sub>f</sub> is defined for each f ϵ D. If ess. inf f ≤ 0 < ess. sup f, then δ<sub>f</sub> is obtained as the right inverse of a distribution function of f. If ess. inf f < 0 < ess. sup f then formulas relating (δ<sub>f</sub>)<sup>+</sup> to δ<sub>f+</sub>, (δ<sub>f</sub>)<sup>-</sup> to δ<sub>f-</sub> and δ<sub>-f</sub> to δ<sub>f</sub> are given. If (X, Λ, μ) is non-atomic and σ-finite and δ is a decreasing rearrangement of f on J, then there is a measure preserving map σ: X → J such that δ(σ) = f μ-a.e.</p>
<p>If (X, Λ, μ) and (X<sub>1</sub>, Λ<sub>1</sub>, μ<sub>1</sub>) are finite measure spaces such that a = μ(X) = μ<sub>1</sub>(X<sub>1</sub>), if f, g ϵ M(X, μ) ∪ M(X<sub>1</sub>, μ<sub>1</sub>), and if ∫<sub>o</sub><sup>a</sup> δ<sub>f+</sub> and ∫<sub>o</sub><sup>a</sup> δ<sub>g+</sub> are finite, then g < < f means ∫<sub>o</sub><sup>t</sup> δ<sub>g</sub> ≤ ∫<sub>o</sub><sup>t</sup> δ<sub>f</sub> for all 0 ≤ t ≤ a, and g < f means g < < f and ∫<sub>o</sub><sup>a</sup> δ<sub>f</sub> = ∫<sub>o</sub><sup>a</sup> δ<sub>g</sub>. The preorder relations < and < < are investigated in detail.</p>
<p>If f ϵ L<sup>1</sup>(X, μ), let Ω(f) = {g ϵ L<sup>1</sup>(X, μ): g < f} and Δ(f) = {g ϵ L<sup>1</sup>(X, μ): g ~ f}. Suppose ρ is a saturated Fatou norm on M(X, μ) such that L<sup>ρ</sup> is universally rearrangement invariant and L<sup>∝</sup> ⊂ L<sup>ρ</sup> ⊂ L<sup>1</sup>. If f ϵL<sup>ρ</sup> then Ω(f) ⊂ L<sup>ρ</sup> and Ω(f) is convex and σ(L<sup>ρ</sup>, L<sup>ρ'</sup>)-compact. If ξ is a locally convex topology on L<sup>ρ</sup> in which the dual of L<sup>ρ</sup> is L<sup>ρ'</sup>, then Ω(f) is the ξ-closed convex hull of Δ(f) for all f ϵ L<sup>ρ</sup> iff (X, Λ, μ) is adequate. More generally, if f ϵ L<sup>1</sup>(X<sub>1</sub>, μ<sub>1</sub>) let Ω<sub>f</sub>(X, μ) = {g ϵ L<sup>1</sup>(X, μ): g < f} and Δ<sub>f</sub>(X, μ) = {g ϵ L<sup>1</sup>(X, μ): g ~ f}. Theorems for Ω(f) and Δ(f) are generalized to Ω<sub>f</sub> and Δ<sub>f</sub>, and a norm ρ<sub>1</sub> on M(X<sub>1</sub>, μ<sub>1</sub>) is given such that Ω<sub>|f|</sub> ⊂ L<sup>ρ</sup> iff f ϵ L<sup>ρ</sup>1.</p>
<p>A linear map T: L<sup>1</sup>(X<sub>1</sub>, μ<sub>1</sub>) → L<sup>1</sup>(X, μ) is said to be doubly stochastic iff Tf < f for all f ϵ L<sup>1</sup>(X<sub>1</sub>, μ<sub>1</sub>). It is shown that g < f iff there is a doubly stochastic T such that g = Tf.</p>
<p>If f ϵ L<sup>1</sup> then the members of Δ(f) are always extreme in Ω(f). If (X, Λ, μ) is non-atomic, then Δ(f) is the set of extreme points and the set of exposed points of Ω(f).</p>
<p>A mapping Φ: Λ<sub>1</sub> → Λ is called a homomorphism if (i) μ(Φ(A)) = μ<sub>1</sub>(A) for all A ϵ Λ<sub>1</sub>; (ii) Φ(A ∪ B) = Φ(A) ∪ Φ(B) [μ] whenever A ∩ B = Ø [μ<sub>1</sub>]; and (iii) Φ(A ∩ B) = Φ(A) ∩ Φ(B)[μ] for all A, B ϵ Λ<sub>1</sub>, where A = B [μ] means C<sub>A</sub> = C<sub>B</sub> μ-a.e. If Φ: Λ<sub>1</sub> → Λ is a homomorphism, then there is a unique doubly stochastic operator T<sub>Φ</sub>: L<sup>1</sup>(X<sub>1</sub>, μ<sub>1</sub>) → L<sup>1</sup> (X, μ) such that T<sub>Φ</sub>C<sub>E</sub> = C<sub>Φ(E)</sub> for all E. If T: L<sup>1</sup> (X<sub>1</sub>, μ<sub>1</sub>) → L<sup>1</sup>(X, μ) is linear then Tf ~ f for all f ϵ L<sup>1</sup>(X<sub>1</sub>, μ<sub>1</sub>) iff T = T<sub>Φ</sub> for some homomorphism Φ.</p>
<p>Let X<sub>o</sub> be the non-atomic part of X and let A be the union of the atoms of X. If f ϵ L<sup>1</sup>(X, μ) then the σ(L<sup>1</sup>, L<sup>∝</sup>)-closure of Δ(f) is shown to be {g ϵ L<sup>1</sup>: there is an h ~ f such that g|X<sub>o</sub> < h|X<sub>o</sub> and g|A = h|A} whenever either (i) X consists only of atoms; (ii) X has only finitely many atoms; or (iii) X is separable.</p>https://thesis.library.caltech.edu/id/eprint/6164Applications of Model Theory to Complex Analysis
https://resolver.caltech.edu/CaltechTHESIS:04112018-095018367
Authors: Stroyan, Keith Duncan
Year: 1971
DOI: 10.7907/69R3-SY38
<p>We use a nonstandard model of analysis to study two main topics
in complex analysis.</p>
<p>UNIFORM CONTINUITY AND RATES OF GROWTH OF MEROMORPHIC FUNCTIONS
is a unified nonstandard approach to several
theories; the Julia-Milloux theorem and Julia exceptional functions,
Yosida's class (A), normal meromorphic functions, and Gavrilov's
W<sub>p</sub> classes. All of these theories are reduced to the study of uniform
continuity in an appropriate metric by means of S-continuity in the
nonstandard model (which was introduced by A. Robinson).</p>
<p>The connection with the classical Picard theorem is made
through a generalization of a result of A. Robinson on S-continuous
*-holomorphic functions.</p>
<p>S-continuity offers considerable simplifications over the standard
sequential approach and permits a new characterization of these growth
requirements.</p>
<p>BOUNDED ANALYTIC FUNCTIONS AS THE DUAL OF A
BANACH SPACE is a nonstandard approach to the pre-dual Banach
space for H<sup>∞</sup>(D) which was introduced by Rubel and Shields.</p>
<p>A new characterization of the pre-dual by means of the
nonstandard hull of a space of contour integrals infinitesimally near the
boundary of an arbitrary region is given.</p>
<p>A new characterization of the strict topology is given in terms
of the infinitesimal relation: "h b k provided ||h-k|| is finite and
h(z) ≈ k(z) for z∈(*D)".</p>
<p>A new proof of the noncoincidence of the strict and Mackey
topologies is given in the case of a smooth finitely connected region.
The idea of the proof is that the infinitesimal relation: "h γ k provided
||h-k|| is finite and h(z) ≈ k(z) on nearly all of the boundary", gives
rise to a compatible topology finer than the strict topology.</p>https://thesis.library.caltech.edu/id/eprint/10798On Order and Topological Properties of Riesz Spaces
https://resolver.caltech.edu/CaltechTHESIS:09192018-073917103
Authors: Aliprantis, Charalambos Dionisios
Year: 1973
DOI: 10.7907/HPNF-KH28
<p>Chapter 1 contains a summary of results on Riesz spaces frequently used in this thesis.</p>
<p>Chapter 2 considers the real linear space L<sub>b</sub>(L, M) of all order bounded linear transformations from a Riesz space L into a Dedekind complete Riesz space M. The order structure of the Dedekind complete Riesz space L<sub>b</sub>(L, M) is studied in some detail. Dual formulas for T(f<sup>+</sup>), T(f<sup>-</sup>) and T(|f|) are proved. The linear space of all extendable operators from the ideal A of L into M is denoted by L<sup>e</sup> <sub>b</sub>(A, M). Two theorems are proved:</p>
<p>(i) If θ ≦ T is extendable, then T has a smallest positive extension T<sub>m</sub><sub>'</sub> given by T<sub>m</sub>(u) = sup {T(v): v ∈ A; θ ≦ v ≦ u} for all u in L<sup>+</sup>.</p>
<p>(ii) The mapping T →(T<sup>+</sup>)<sub>m</sub> - (T<sup>-</sup>)<sub>m</sub> from L<sup>e</sup><sub>b</sub>(A, M) into L<sub>b</sub>(L, M) is a Riesz isomorphism.</p>
<p>Chapter 3 studies integral and normal integral transformations. Some of the theorems included in this chapter are:</p>
<p>(i) If T ∈ L<sup>e</sup> <sub>b</sub>(A,M) is a normal integral, then so is T<sub>m</sub>.</p>
<p>(ii) If L is σ-Dedekind complete and M is super Dedekind complete, then T in L<sub>b</sub>(L,M) is a normal integral if and only if N<sub>T</sub> = {u ∈ L: |T |(|u|) = θ} is a band of L.</p>
<p>(iii) If L is σ-Dedekind complete and M is super Dedekind complete and if there exists a strictly positive operator for L into M, then L is super Dedekind complete.</p>
<p>(iv) If M admits a strictly positive linear functional which is normal then the normal component T<sub>n</sub> of the operator θ ≦ T ∈ L<sub>b</sub>(L,M) is given by T<sub>n</sub>(u) = inf {sup <sub>α</sub>T(u<sub>α</sub>): θ ≦ u<sub>α</sub> ↑ u} for all u in L<sup>+</sup>.</p>
<p>Chapter 4 studies ordered topological vector spaces (E,τ) with particular emphasis on locally solid linear topological Riesz spaces. Order continuity and topological continuity are considered by introducing the properties (A,o), (A,i), (A,ii), (A,iii) and (A,iv). Some results from this chapter are:</p>
<p>(i) If (L, τ) is a locally solid Riesz space, then (L,τ) satisfies (A,i) if every τ-closed ideal is a σ-ideal, and (L, τ) satisfies (A,ii) if every τ-closed ideal is a band.</p>
<p>(ii) If (L,τ) is a metrizable locally solid Riesz space with (A,ii), then L satisfies the Egoroff property.</p>
<p>(iii) If (L,τ) is a metrizable locally solid Riesz space, then both (A,i) and (A,iii) hold if (A,ii) holds. A counter example shows that this is not true for non-metrizable locally solid Riesz spaces.</p>
<p>The fifth and final chapter considers Hausdorff locally solid Riesz spaces (L, τ). The topological completion of (L, τ) is denoted by (L^, τ^). Some results from this chapter are:</p>
<p>(i) (L^,τ^) is a Hausdorff locally solid Riesz space with cone L^<sup>+</sup> = L<sup>+</sup> = the τ^-closure of L + in L^, containing L as a Riesz subspace.</p>
<p>(ii) (L^,τ^) satisfies the (A,iii) property, if (L, τ) does.</p>
<p>(iii) (L^,τ^) satisfies the (A,ii) property, if (L, τ) does.</p>
<p>(iv) If τ is metrizable, then (L^,τ^) satisfies the (A,i) property if (L, τ) does.</p>
<p>(v) If L<sub>ρ</sub> is a normed Riesz space with the (sequential) Fatou property, then L^<sub>ρ^</sub> has the (sequential) Fatou property.</p>https://thesis.library.caltech.edu/id/eprint/11189Generalized multipliers on locally compact Abelian groups
https://resolver.caltech.edu/CaltechETD:etd-10122005-082659
Authors: Ford, Lawrence Charles
Year: 1974
DOI: 10.7907/NB31-1Y34
NOTE: Text or symbols not renderable in plain ASCII are indicated by [...]. Abstract is included in .pdf document.
Let G be a locally compact Abelian group with dual [...], [...], and [...] supp [...] is compact}. Then for [...], the containments are proper if G is noncompact, and [...] is a dense, translation invariant subspace of [...] for [...]. Let [...] be a complex valued function defined on [...], and [...] = [...]. Suppose [...]. Define the operator, [...] by the equation [...] for each [...]. Then [...] is a module over M(G), [...] is a module homomorphism, and [...] is (p, q) closed. We call [...] a generalized (p, q) multiplier.
The main results include:
(1) Suppose T is an operator satisfying: (a) The domain D(T) is a translation invariant subspace of [...], and the range R(T) [...]; (b) D(T) [...]; (c) T is (p, q) closed, linear, and commutes with all translations; (d) C X T(C) is dense in [...]. Then T = [...] for some [...].
(2) The set of all generalized (p, q) multipliers, denoted [...], is a linear space, and the set of all generalized (p, p) multipliers, denoted [...], is an algebra containing [...] and contained in [...].
(3) If [...], then [...] is locally the transform of a bounded (p, q) multiplier.
Further sections include a deeper study of [...], [...], and special results obtainable for compact G.https://thesis.library.caltech.edu/id/eprint/4041A Counterexample in the Theory of Fourier Transforms in the Complex Domain
https://resolver.caltech.edu/CaltechTHESIS:08312021-161900257
Authors: Delaney, William Kenneth
Year: 1975
DOI: 10.7907/8s0b-ck03
The Borel transform of an entire function of exponential type is
defined outside a closed bounded convex set D. Paley and Wiener have
given a necessary and sufficient condition on the entire function F(z)
such that φ(w), the Borel transform of F(z), is contained in E<sup>2</sup>(ℂ\D)
for the case when D is a line segment. Kacnel'son has shown that the
natural extension of this result provides a necessary condition for a
general closed bounded convex set D. Here, by counterexample, we
show that the natural extension does not provide a sufficient condition.https://thesis.library.caltech.edu/id/eprint/14348Aspects of the Theory of Normed Spaces
https://resolver.caltech.edu/CaltechTHESIS:03092017-112801788
Authors: Wiid, Frans Gerhardus Johannes
Year: 1982
DOI: 10.7907/rytj-9p73
<p>The dissertation will be divided into two parts. The first part will, in essence, be a study of weak compactness in a variety of families of normed spaces. Included in this study will be general characterizations of weak compactness in spaces of vector measures and tensor products that contain all known results of this nature as special cases (in particular, we do not need to restrict attention to only those range spaces with strong geometric properties such as, for example, the Radon-Nikodym property). The methods of Nonstandard Analysis constitute a fundamental tool in these investigations.</p>
<p>The second part of the dissertation will contain a discussion and a study of Model theoretic aspects of categories of normed spaces. We will introduce multi-sorted formal languages that enable us to view various subcategories of the category of normed spaces as being equivalent to categories of set-valued models of coherent theories in these languages. We see, in particular, that the category of real normed spaces is equivalent to the category of set-valued models of a lim-theory, and that, for instance, the category of L-spaces is equivalent to the category of set-valued models of a coherent extension of this lim-theory. These considerations allow for proofs of existence of 2-adjoints to inclusion functors from some 2-categories into the
2-category of Topos-valued normed spaces, and the study of the elementary properties of these adjoints.</p>
<p>The coherent theory of Hilbert spaces gives rise to interesting spatial Toposes when the appropriate "adjoint functor theorems" are proved. The sites of these toposes are spectral spaces (in the sense of Algebraic geometry) with interesting cohomological properties.</p>
https://thesis.library.caltech.edu/id/eprint/10092Disjointness Preserving Operators
https://resolver.caltech.edu/CaltechTHESIS:03232017-113645091
Authors: Hart, Dean Robert
Year: 1983
DOI: 10.7907/88fv-2p60
<p>Let E and F be Archimedian Riesz spaces. A linear operator T : E → F is called disjointness preserving if |f| ∧ |g| = 0 in E implies |Tf| ∧ |Tg| = 0 in F. An order continuous disjointness preserving operator T : E → E is called bi-disjointness preserving if the order closure of |T|E is an ideal in E. If the order dual of E separates the points of E, then every order continuous disjointness preserving operator whose adjoint is disjointness preserving is bi-disjointness preserving. If E is in addition Dedekind complete, then the converse holds.</p>
<p>DEFINITION. Let T : E → E be a bi-disjointness preserving operator. We say that T is:</p>
<p>(i) quasi-invertible if T is injective and {TE}<sup>dd</sup> = E.</p>
<p>(ii) of forward shift type if T is injective and <sub>n=1</sub>∩<sup>∞</sup>{T<sup>n</sup>E}<sup>dd</sup> = {0}.</p>
<p>(iii) of backward shift type if <sub>n=1</sub>∨<sup>∞</sup> Ker T<sup>n</sup> = E and{TE}<sup>dd</sup> = E.</p>
<p>(iv) hypernilpotent if <sub>n=1</sub>∨<sup>∞</sup> Ker T<sup>n</sup> = E and <sub>n=1</sub>∩<sup>∞</sup> {T<sup>n</sup>E}<sup>dd</sup> = {0}.</p>
<p>The supremum in (iii) and (iv) is taken in the Boolean algebra of bands.</p>
<p>The following decomposition theorem is proved.</p>
<p>THEOREM. Let T : E → E be a bi-disjointness preserving operator on a Dedekind complete Riesz space E. Then there exist T-reducing bands E<sub>i</sub> (i = 1,2,3,4) such that <sub>i=1</sub>⊕<sup>4</sup> E<sub>i</sub> = E and the restriction of T to E<sub>i</sub> satisfies the ith property listed in the preceding definition.</p>
<p>Quasi-invertible operators can be decomposed further in the following way. Set 0rth(E) :={T ∈ ℒ<sub>b</sub>(E) : TB ⊂ B for every band B}. We say that a quasi-invertible operator T has strict period n (n ∈ℕ) if T<sup>n</sup> ∈ 0rth(E) and for every non-zero band B ⊂ E, there exists a band A s.t. {0} ≠ A ⊂ B and A, {TA}<sup>dd</sup>, ... , {T<sup>n-1</sup>A}<sup>dd</sup> are mutually disjoint. A quasi-invertible operator is called aperiodic if for every n ∈ℕ and every non-zero band B ⊂ E, there exists a band A s.t. {0} ≠ A ⊂ B and A, {TA}<sup>dd</sup> , ... , {T<sup>n</sup>A}<sup>dd</sup> are mutually disjoint.</p>
<p>THEOREM. Let T : E → E be a quasi-invertible operator on a Dedekind complete Riesz space E. Then there exist T-reducing bands E<sub>n</sub> (n ∈ ℕ ⋃ {∞}) such that the restriction of T to E<sub>n</sub> (n ∈ ℕ) has strict period n, the restriction of T to E<sub>∞</sub> is aperiodic and E = <sub> n∈ℕ ⋃ {∞}</sub>⊕ E<sub>n</sub>.</p>
<p>Finally, the spectrum of bi-disjointness preserving operators is considered.</p>
<p>THEOREM. Let E be a Banach lattice which is either Dedekind complete or has a weak Fatou norm. Let T : E → E be a bi-disjointness preserving operator. If T is either of forward shift type, of backward shift type, hypernilpotent or aperiodic quasi-invertible, then the spectrum of T is rotationally invariant. If T is quasi-invertible with strict period n, then λ ∈ σ(T) implies λα ∈ σ(T) for any nth root of unity α.</p>
<p>The above theorems can be combined to deduce results concerning the spectrum of arbitrary bi-disjointness preserving operators. One such result is given below.</p>
<p>THEOREM. Let T : E → E be a bi-disjointness preserving operator on a Dedekind complete Banach lattice E. Suppose, for each 0 < r ∈ ℝ, {z ∈ ℂ : |z| = r} ⋂ σ(T) lies in an open half plane. Then there exists T-reducing bands E<sub>1</sub> and E<sub>2</sub> such that E = E<sub>1</sub>⊕ E<sub>2</sub> , T|<sub>E<sub>1</sub></sub> is an abstract multiplication operator (i.e. is in the center of E) and T|<sub>E<sub>2</sub></sub> is quasi-nilpotent.</p>https://thesis.library.caltech.edu/id/eprint/10101Boundaries of Smooth Sets and Singular Sets of Blaschke Products in the Little Bloch Class
https://resolver.caltech.edu/CaltechTHESIS:10232009-113530661
Authors: Hungerford, Gregory Jude
Year: 1988
DOI: 10.7907/ehgq-c421
<p>A subset of R is called smooth if the integral of its characteristic function is smooth in the sense defined by Zygmund. It is shown that such a set is either trivial or its boundary has Hausdorff dimension 1. Sets are constructed here which are as close to smooth as one likes but whose boundaries do not have dimension 1.</p>
<p>It was conjectured by T. Wolff that if B is Blaschke product in the Little Bloch class, its zeroes accumulate to a set of dimension 1. This conjecture is proven here.</p>
https://thesis.library.caltech.edu/id/eprint/5326Generic Differentiability of Convex Functions and Monotone Operators
https://resolver.caltech.edu/CaltechTHESIS:08232013-082402986
Authors: Verona, Maria Elena
Year: 1989
DOI: 10.7907/be7m-vv03
<p>The aim of this paper is to investigate to what extent the known theory of subdifferentiability and generic differentiability of convex functions defined on open sets can be carried out in the context of convex functions defined on not necessarily open sets. Among the main results obtained I would like to mention a Kenderov type theorem (the subdifferential at a generic point is contained in a sphere), a generic Gâteaux differentiability result in Banach spaces of class S and a generic Fréchet differentiability result in Asplund spaces. At least two methods can be used to prove these results: first, a direct one, and second, a more general one, based on the theory of monotone operators. Since this last theory was previously developed essentially for monotone operators defined on open sets, it was necessary to extend it to the context of monotone operators defined on a larger class of sets, our "quasi open" sets. This is done in Chapter III. As a matter of fact, most of these results have an even more general nature and have roots in the theory of minimal usco maps, as shown in Chapter II.</p>https://thesis.library.caltech.edu/id/eprint/7933On spectral properties of positive operators
https://resolver.caltech.edu/CaltechTHESIS:04112011-131850601
Authors: Zhang, Xiao-Dong
Year: 1991
DOI: 10.7907/r9x6-7m39
This thesis deals with the spectral behavior of positive operators and related ones
on Banach lattices. We first study the spectral properties of those positive operators
that satisfy the so-called condition (c). A bounded linear operator T on a Banach
space is said to satisfy the condition (c) if it is invertible and if the number 0 is in
the unbounded connected component of its resolvent set p(T). By using techniques
in complex analysis and in operator theory, we prove that if T is a positive operator
satisfying the condition (c) on a Banach lattice E then there exists a positive number a
and a positive integer k such that T^k ≥ a•I, where I is the identity operator on E. As
consequences of this result, we deduce some theorems concerning the behavior of the
peripheral spectrum of positive operators satisfying the condition (c). In particular,
we prove that if T is a positive operator with its spectrum contained in the unit circle
Γ then either σ(T) = Γ or σ(T) is finite and cyclic and consists of k-th roots of unity
for some k. We also prove that under certain additional conditions a positive operator
with its spectrum contained in the unit circle will become an isometry. Another main
result of this thesis is the decomposition theorem for disjointness preserving operators.
We prove that under some natural conditions if T is a disjointness preserving operator
on an order complete Banach lattice E such that its adjoint T' is also a disjointness
preserving operator then there exists a family of T-reducing bands {E_n : ≥ 1} U
{E_∞} of E such that T|E_n has strict period n and that T|E_∞ is aperiodic. We also
prove that any disjointness preserving operator with its spectrum contained in a
sector of angle less than π can be decomposed into a sum of a central operator and a
quasi-nilpotent operator. Among other things we give some conditions under which
an operator T lies in the center of the Banach lattice. Also discussed in this thesis
are certain conditions under which a positive operator T with σ(T) = {1} is greater
than or equal to the identity operator I.
https://thesis.library.caltech.edu/id/eprint/6294Evolution equations and semigroups of operators with the disjoint support property
https://resolver.caltech.edu/CaltechETD:etd-09052007-110700
Authors: Biyanov, Andrey Y.
Year: 1995
DOI: 10.7907/k7nd-5671
NOTE: Text or symbols not renderable in plain ASCII are indicated by [...]. Abstract is included in .pdf document.
Let [...], [...] be locally compact Hausdorff spaces, [...], [...] Banach spaces.
Theorem. T is an operator in [...], [...] with the disjoint support property if and only if [...] open, [...] such that:
(1) [...].
(2) [...] compact, [...] compact, [...] with the following property: [...].
(3) [...]
[...].
Let X be a locally compact Hausdorff space, E a Banach space.
Theorem. [...] is a [...]-group on [...](X,E) with the disjoint support property if and only if [...] a continuous flow, [...] a continuous cocycle of [...] such that [...].
There is a corresponding result about [...]-semigroups on [...](X,E) with the disjoint support property, where semiflows and semicocycles play the roles of flows and cocycles respectively.
Suppose [...], X is either (a,b) or [a,b], where by [[...],b] we mean ([...],b], and by [a,[...]] we mean [a,[...]).
Theorem. Let [...] be a [...]-group on [...](X) with the disjoint support property. Then [...] is the union of pairwise disjoint intervals [...], [...], where I is either finite or countable and [...]: [...] such that [...] = [...] : [...] is a homeomorphism and the corresponding group dual
[...].
The above theorem generalizes the well-known result of A. Plessner that if [...] and [...], then f is absolutely continuous if and only if [...].
The following theorem generalizes the result of N. Wiener and R. C. Young about the behavior of measures on [...] under translation.
Theorem. Let [...] be a [...]-group on [...](X) with the disjoint support property. Then [...]
lim sup[...],
where [...] is the component of in [...]. Moreover, if lim sup[...] = 1, then the last inequality becomes an equality.https://thesis.library.caltech.edu/id/eprint/3339Sun-Dual Characterizations of the Translation Group of ℝ
https://resolver.caltech.edu/CaltechTHESIS:11212019-103601328
Authors: Jackson, Frances Yvonne
Year: 1998
DOI: 10.7907/3y5x-kg66
<p>Let <i>E</i> be a Banach space. The mapping <i>t</i> → <i>T</i> (<i>t</i>) of ℝ (real numbers) into <i>L<sub>b</sub></i>(<i>E</i>), the Banach algebra of all bounded linear operators on <i>E</i>, is called a <i>strongly continuous group</i> or a <i>C₀</i>-group, if <i>G</i> = {<i>T</i>(<i>t</i>) : <i>t</i> ∈ ℝ} defines a group representation of (ℝ, +) into the multiplicative group of <i>L<sub>b</sub></i>(<i>E</i>), and if ∀<i>f</i> ∈ <i>E</i>,</p>
<p>[equation; see abstract in scanned thesis for details].</p>
<p>For example, if <i>E</i> = <i>C₀</i>(ℝ), the function space which consists of all continuous, complex functions that vanish at infinity, then (∀<i>t</i> ∈ ℝ) (∀<i>f</i> ∈ <i>C₀</i>(ℝ)), the function <i>T</i>(<i>t</i>)<i>f</i>(<i>x</i>) = <i>f</i>(<i>x</i> + <i>t</i>), <i>x</i> ∈ ℝ, defines a strongly continuous group, since each <i>f</i> ∈ <i>E</i> is uniformly continuous; this group is called the <i>translation group</i>. If we now consider <i>E</i> = <i>B</i>(ℝ), the space of bounded, continuous complex functions on ℝ, then although the translation group on <i>E</i> is not strongly continuous, it is strongly continuous on the subspace <i>BUC</i>(ℝ) of <i>E</i>, which consists of bounded, uniformly continuous functions. <i>BUC</i>(ℝ) is the largest subspace of <i>E</i> on which the translation group is strongly continuous.</p>
<p>The <i>adjoint family</i> of a <i>C₀</i>-group defined on a Banach space <i>E</i>, need not be strongly continuous on the Banach dual <i>E*</i> of <i>E</i>. Let <i>E</i><sup>⊙</sup> (pronounced <i>E</i>-sun) be the largest linear subspace of <i>E*</i> relative to which the adjoint family is a <i>C₀</i>-group:</p>
<p>[equation; see abstract in scanned thesis for details].</p>
<p><i>E</i><sup>⊙</sup> is called the <i>sun-dual</i> or <i>sun-space</i> of <i>E</i>. If <i>E</i> = <i>C₀</i>(ℝ), then it follows from a well-known result of A. Plessner that <i>E</i><sup>⊙</sup> = <i>L</i>¹(ℝ) ([Ple]). This research paper contains a characterization of the sun-dual of <i>BUC</i>(ℝ) and of the subspace <i>W</i> <i>AP</i>(ℝ) of <i>BUC</i>(ℝ), which consists of weakly almost periodic functions on ℝ.</p>https://thesis.library.caltech.edu/id/eprint/13588Dade's Ordinary Conjecture for the Finite Unitary Groups in the Defining Characteristic
https://resolver.caltech.edu/CaltechTHESIS:11212019-153036477
Authors: Ku, Chao
Year: 1999
DOI: 10.7907/xhe3-q841
<p>There has been rising interest in the study of Dade's conjectures, which not only generalize Alperin's weight conjecture, but unify some other major conjectures in (modular) representation theory, such as Brauer's height conjecture in abelian blocks and McKay’s conjecture. In this thesis we verify Dade's ordinary conjecture for the finite unitary groups in the defining characteristic. Dade's conjectures involve proving the vanishing of the alternating sum of certain <i>G</i>-stable function over the <i>p</i>-group complex of a finite group <i>G</i>. We develop some machinery to treat alternating sums which we hope will serve as part of a general approach to such problems. This includes extending some of the existing techniques in a functorial way. We also show how to make use of the topological properties of <i>p</i>-group complexes to reduce the alternating sums. While this work is mainly intended for the unitary groups, it should also apply to other groups of Lie type, and part of the work can be generalized to treat a much wider class of groups. Among other things, we also obtain a formula which expresses the McKay's numbers of the finite unitary groups in term s of partitions of integers.</p>https://thesis.library.caltech.edu/id/eprint/13591Regularization of the Amended Potential around a Symmetric Configuration
https://resolver.caltech.edu/CaltechETD:etd-12122003-115521
Authors: Garduño, Antonio Hernández
Year: 2002
DOI: 10.7907/NBWA-EQ57
Relative equilibria are periodic trajectories that, in a dynamical system with continuous symmetry, correspond to fixed points in the projected dynamics to the quotient space. In Hamiltonian systems with symmetry, it is of interest to understand the structure of relative equilibria near symmetric states. In this context, we give a method that in some cases of simple mechanical systems with compact symmetry group gives information about the relative equilibria bifurcating from a set of relative equilibria with isotropy subgroup isomorphic to S^1. This method is based on the blowing-up of the amended potential.https://thesis.library.caltech.edu/id/eprint/4960