Academic literature on the topic 'Spectral spaces'

In the paper [14] Hochster gave a topological characterisation of those spaces X which arise as the prime spectrum of a commutative ring: they are the spectral spaces, defined as those topological spaces which are T_0, quasi-compact and sober, whose quasi-compact and open subsets form a basis for the topology and are closed under finite intersections. It is well known that the prime spectrum of a ring is always spectral; Hochster proved the converse by describing a construction which, starting from such a space X, builds a ring having the desired prime spectrum; however the construction given is (in Hochster's own words) very intricate, and has not been further exploited in the literature (a passing exception perhaps being the use of [14] Theorem 4 in the example on page 272 of [24]). In the finite setting, alternative constructions of a ring having a given spectrum are provided by Lewis [17] and Ershov [7], which, particularly in light of the work of Fontana in [8], appear to be more tractable, and at least more readily understood. This insight into the methods of constructing rings with a given spectrum is used to prove a result about which spaces may arise as the prime spectrum of a Noetherian ring: it is shown that every 1-dimensional Noetherian spectral space may be realised as the prime spectrum of a Noetherian ring. A close analysis of the two finite constructions considered here reveals considerable similarities between their underlying operation, despite their radically different presentations. Furthermore, we generalise the framework of Ershov's construction beyond the finite setting, finding that the ring we thus define on a space X contains the ring defined by Hochster's construction on X as a subring. We find that in certain examples these rings coincide, but that in general the containment is proper, and that the spectrum of the ring provided by our generalised construction is not necessarily homeomorphic to our original space. We then offer an additional condition on our ring which may (-and indeed in certain examples does) serve to repair this disparity. It is hoped that the analysis of the constructions presented herein, and the demonstration of the heretofore unrecognised connections between the disparate ring constructions proposed by various authors, will facilitate further investigation into the prime ideal structure of commutative rings.

The principal objects of study in this thesis are arbitrary spectral families, E, on a complex Banach space X. The central theme is the relationship between the geometry of X and the p-variation of E. We show that provided X is super- reflexive, then given any E, there exists a value 1 · p < 1, depending only on E and X, such that var p(E) < 1. If X is uniformly smooth we provide an explicit range of such values p, which depends only on E and the modulus of convexity of X*, delta X*(.).

Spectral theory is the study of Mark Kac's famous question [K], "can one hear the shape of a drum?" That is, can we determine the geometrical or topological properties of a manifold by using its Laplace Spectrum? In recent years, the problem has been extended to include the study of Riemannian orbifolds within the same context. In this thesis, on the one hand, we answer Kac's question in the negative for orbifolds that are spherical space forms of dimension higher than eight. On the other hand, for the three-dimensional and four-dimensional cases, we answer Kac's question in the affirmative for orbifold lens spaces, which are spherical space forms with cyclic fundamental groups. We also show that the isotropy types and the topology of the singularities of Riemannian orbifolds are not determined by the Laplace spectrum. This is done in a joint work with E. Stanhope and D. Webb by using P. Berard's generalization of T. Sunada's theorem to obtain isospectral orbifolds. Finally, we construct a technique to get examples of orbifold lens spaces that are not isospectral, but have the same asymptotic expansion of the heat kernel. There are several examples of such pairs in the manifold setting, but to the author's knowledge, the examples developed in this thesis are among the first such examples in the orbifold setting.

Spectral multiplicity theory solves the problem of unitary equivalence of normal operate on a Hilbert space ${ cal H}$ by associating with each normal operator N a multiplicity function, such that two operators are unitarily equivalent if and only if their multiplicity functions are equal. This problem was first solved in the classical case in which ${ cal H}$ is separable by Hellinger in 1907, and in the general case in which ${ cal H}$ is nonseparable by Wecken in 1939. This thesis develops the later versions of multiplicity theory in the nonseparable case given by Halmos and Brown, and gives the simplification of Brown's version to the classical theory. Then the versions of Halmos and Brown are shown directly to be equivalent. Also, the multiplicity function of Brown is expressed in terms of the multiplicity function of Halmos.

In this thesis we consider a connected locally finite graph G that possesses the Cheeger isoperimetric property. We define a decreasing one parameter family of Hardy-type spaces associated with the standard nearest neighbour Laplacian on G. We show that the space with parameter ½ is the space of all integrable functions whose Riesz transform is integrable. We show that if G has bounded geometry and the parameter is an integer, the corresponding Hardy-type space admits an atomic decomposition. We also show that if G is a homogeneous tree and the parameter is not an integer, the corresponding Hardy-type space does not admit an atomic decomposition. Furthermore, we consider the Hardy-type spaces defined in terms of the heat and Poisson maximal operators, and we analyse their relationships with the family of spaces defined previously. We also show that the space associated with the heat maximal operator is properly contained in the one associated with the heat maximal operator, a phenomenon which has no counterpart in the euclidean setting. Applications to the purely imaginary powers of the Laplacian are also given. Finally, we characterise, for every p, the class of spherical multipliers on the p-integrable functions on homogeneous trees in terms of Fourier multipliers on the torus. Furthermore we give a sharp sufficient condition on spherical multipliers on the product of homogeneous trees.

Thesis (Ph.D.) - University of Glasgow, 2000.<br>Ph.D. thesis submitted to the Department of Mathematics, University of Glasgow, 2000. Includes bibliographical references. Print version also available.