Stone's Theorem Research Articles

Toward a constructive theory of unbounded linear operators

AbstractWe show that the following results in the classical theory of unbounded linear operators on Hilbert spaces can be proved within the framework of Bishop's constructive mathematics: the Kato-Rellich theorem, the spectral theorem. Stone's theorem, and the self-adjointness of the most common quantum mechanical operators, including the Hamiltonians of electro-magnetic fields with some general forms of potentials.

A Stone-Weierstrass Type Theorem for an Unstructured Set - with Applications

A general version of the Stone–Weierstrass theorem is presented – one which involves no structure on the domain set of the real valued functions. This theorem is similar to the ‘Stone–Weierstrass theorem’ which appears in the book by Gillman and Jerison, but instead of involving the concept of stationary sets the one presented here involves stationary filters. As a corollary to our results we obtain Nel's theorem of Stone–Weierstrass type for an arbitrary topological space. Finally, an application is made to the setting of Cauchy spaces.

Unitary and Hermitian fractional operators and their relation to the fractional Fourier transform

Inspired by the recent popularity of the fractional Fourier transform (FRFT) and motivated by the use of Hermitian and unitary operator methods in signal analysis, we introduce a new unitary fractional operator associated with the FRFT. The new operator generalizes the unitary time-shift and frequency-shift operators by describing shifts at arbitrary orientations in the time-frequency (t-f) plane. We establish the connection with the FRFT by deriving two signal transformations, one invariant and one covariant, to the newly introduced unitary fractional operator. By using Stone's theorem and the duality concept, we derive what we call the Hermitian fractional operator which also generalizes the well-known Hermitian time and frequency operators.

Fuzzy ideals and congruences of lattices

General theory of algebraic fuzzy systems developed earlier by the authors is applied to extend the Stone's prime ideal theorem to fuzzy ideals of distributive lattices. A correspondence between fuzzy ideals and fuzzy congruences in a distributive lattice is obtained and certain important properties of these are discussed.

Regularized Quasi-Spectral Distributions

We introduce a generalization of a (regularized) spectral distribution,that does not contain any density assumptions; in particular, the momentumneed not be densely defined. We construct a C∞ (R) functional calculusfor the momentum. We prove analogues of Stone's theorem; that is,characterize the momenta of certain (regularized) quasi-spectraldistributions as generators of polynomially bounded once-integrated(regularized) groups.

Equiconvergence. A generalization of the Tamarkin—Stone theorem

Equiconvergence. A generalization of the Tamarkin—Stone theorem

Stone's representation theorem for Boolean hyperalgebras topological version

In this paper we present the topological version of the classical Stone's representation theorem for Boolean hyperalgebras.

Integral transforms covariant to unitary operators and their implications for joint signal representations

Fundamental to the theory of joint signal representations is the idea of associating a variable, such as time or frequency, with an operator, a concept borrowed from quantum mechanics. Each variable can be associated with a Hermitian operator, or equivalently and consistently, as we show, with a parameterized unitary operator. It is well known that the eigenfunctions of the unitary operator define a signal representation which is invariant to the effect of the unitary operator on the signal, and is hence useful when such changes in the signal are to be ignored. However, for detection or estimation of such changes, a signal representation covariant to them is needed. Using well-known results in functional analysis, we show that there always exists a translationally covariant representation; that is, an application of the operator produces a corresponding translation in the representation. This is a generalization of a recent result in which a transform covariant to dilations is presented. Using Stone's theorem, the "covariant" transform naturally leads to the definition of another, unique, dual parameterized unitary operator. This notion of duality, which we make precise, has important implications for joint distributions of arbitrary variables and their interpretation. In particular, joint distributions of dual variables are structurally equivalent to Cohen's class of time-frequency representations, and our development shows that, for two variables, the Hermitian and unitary operator correspondences can be used consistently and interchangeably if and only if the variables are dual.

Regularized spectral distributions

For C a bounded, injective operator with dense image, we define a C-regularized spectral distribution. This produces a functional calculus, f → f(B), from C∞(ℝ) into the space of closed densely defined operators, such that f(B)C is bounded when f has compact support. As an analogue of Stone's theorem, we characterize certain regularized spectral distributions as corresponding to generators of polynomially bounded C-regularized groups. We represent the regularized spectral distribution in terms of the regularized group and in terms of the C-resolvent. Applications include the Schrodinger equation with potential, and symmetric hyperbolic systems, all on Lp(ℝn) (1≤p<∞), Co(ℝn), BUC(ℝn), or any space of functions where translation is a bounded strongly continuous group.

Empirical quantum mechanics

Machida and Namiki developed a many-Hilbert-spaces formalism for dealing with the interaction between a quantum object and a measuring apparatus. Their mathematically rugged formalism was polished first by Araki from an operator-algebraic standpoint and then by Ozawa for Boolean quantum mechanics, which approaches a quantum system with a compatible family of continuous superselection rules from a notable and perspicacious viewpoint. On the other hand, Foulis and Randall set up a formal theory for the empirical foundation of all sciences, at the hub of which lies the notion of a manual of operations. They deem an operation as the set of possible outcomes and put down a manual of operations at a family of partially overlapping operations. Their notion of a manual of operations was incorporated into a category-theoretic standpoint into that of a manual of Boolean locales by Nishimura, who looked upon an operation as the complete Boolean algebra of observable events. Considering a family of Hilbert spaces not over a single Boolean locale but over a manual of Boolean locales as a whole, Ozawa's Boolean quantum mechanics is elevated into empirical quantum mechanics, which is, roughly speaking, the study of quantum systems with incompatible families of continuous superselection rules. To this end, we are obliged to develop empirical Hilbert space theory. In particular, empirical versions of the square root lemma for bounded positive operators, the spectral theorem for (possibly unbounded) self-adjoint operators, and Stone's theorem for one-parameter unitary groups are established.

WKL 0 and Stone's separation theorem for convex sets

The Stone's Separation Theorem (for countable convex sets of countable vector spaces over countable ordered fields) is equivalent to Weak König's Lemma (over RCA 0).

L- fuzzy version of Stone's representation theorem for distributive lattices

In this paper the category of L- fuzzy locales is introduced which is proved to play the same role with respect to stratified L- fuzzy topological spaces as the locales play for topological spaces. Secondly, Stone's representation theorem for distributive lattices is generalized to the L-fuzzy case, and the notion of L-fuzzy spectrum of distributive lattices is introduced and its properties are systematically studied; moreover, the harmony between the localic version of Stone's representation theorem and spectral spaces of distributive lattices is also valid in the L-fuzzy case.

Gelfand theorem implies Stone representation theorem of Boolean rings

Stone Theorem about representing a Boolean algebra in terms of open‐closed subsets of a topological space is a consequence of the Gelfand Theorem about representing a B∗‐ algebra as the algebra of continuous functions on a compact Hausdorff space.

An extension of the Erd?s-Stone theorem

LetK p(u1, ..., up) be the completep-partite graph whoseith vertex class hasu i vertices (l≤i≤p). We show that the theorem of Erdős and Stone can be extended as follows. There is an absolute constant α>0 such that, for allr≥1, 0<γ<1 and 0<e≤1/r, every graphG=G n of sufficiently large order |G|=n with at least $$\left( {1---\frac{1}{r} + \varepsilon } \right) \left( {_2^n } \right)$$ edges contains aK r+1(s,m,...,m,l), wherem=m(n)=[α(1−γ)(logn)/logr],s=s(n)=[α(1−γ)(logn)/rlog(1/e)], andl= l(n) ⌊αɛ1+γ/2 n γ ⌋. The above result strengthens a sharpening of the Erdős-Stone theorem due to Bollobas, Erdős, and Simonovits, which guaranteed the existence of aK r+1(s,...,s) inG. The strengthening in our result lies in the fact thatm above is independent of e andl can be demanded to be almost the first power ofn. A related conjecture extending the Chvatal-Szemeredi sharpening of the Erdős-Stone theorem is presented.

Spectral distributions and generalization of Stone's theorem

In this paper, we introduce the notion ofspectral distribution which is a generalization of the spectral measure. This notion is closely related to distribution semigroups and generalized scalar operators. The associated operator (called themomentum of the spectral distribution) has a functional calculus defined for infinitely differentiable functions on the real line. Our main result says thatA generating a smooth distribution group of orderk is equivalent to having ak-times integrated group that are O(¦t¦k) oriA being the momentum of a spectral distribution of degreek. We obtain the standard version of Stone's theorem as a special case of this result. The standard properties of a functional calculus together with spectral mapping theorem are derived. Finally, we show how the degree of a spectral distribution is related to the degree of the nilpotent operators which separate its momentum from its scalar part.

Some formal properties of operator polar decomposition

In this paper I consider the general mathematical problem of the polar decomposition of an operator in a linear space. Extending the space makes it possible to define a unitary operator related to the original nonhermitean one. By Stone's theorem this guarantees the existence of a phase operator in the extended space. The connection with supersymmetry is pointed out. Applying the general results to harmonic oscillator creation and annihilation operators we regain a phase description originally introduced by Newton. Projecting the phase operator from the extended space to the original one, we find a phase representation for the Boson operators. Introducing the conjugate rotation operator, one can describe the oscillator dynamics in the phase representation. The connection with the Barnett–Pegg phase operator is pointed out.

Remark on the theorem of stone

The Kakutani-Krein and Stone-Weierstrass theorems remain in force when the approximating family of functions satisfies more weak natural conditions. In the present note, these conditions are indicated.

Operator theory in the C∗-algebra framework

Properties of operators affiliated with a C ∗-algebra are studied. A functional calculus of normal elements is constructed. Representations of locally compact groups in a C ∗-algebra are considered. Generalizations of Stone and Nelson theorems are investigated. It is shown that the tensor product of affiliated elements is affiliated with the tensor product of corresponding algebras.

Hopfian and Co-Hopfian objects

The aim of the present paper is to study Hopfian and Co-Hopfian objects in categories like the category of rings, the module categories A-mod and mod-A for any ring A. Using Stone's representation theorem any Boolean ring can be regarded as the ring A of clopen subsets of compact Hausdorff totally disconnected space X. It turns out that the Boolean ring A will be Hopfian (resp. co-Hopfian) if and only if the space X is co-Hopfian (resp. Hopfian) in the category Top. For any compact Hausdorff space X let CR(X) (resp. CC(X)) denote the R (resp. C)-algebra of real (resp. complex) valued continuous functions on X. Using Gelfand's representation theorem we will prove that CR(X) (CC(X)) is Hopfian (respectively co-Hopfian) as an R(C) algebra if and only if X is co-Hopfian (respectively Hopfian) as an object of Top. We also study Hopfian and co-Hopfian compact topological manifolds.

An elementary nonstandard proof of Stone's representation theorem

A neat nonstandard proof of Stone's representation theorem is given. Improving on previous proofs (Loeb [5], Brunet [2]), it uses the remarkably simple fact that infinitesimal members of a filter on $X$, in any enlargement, are always compact for a natural topology on ${}^{\ast}X$.