Subspace Theorem Research Articles

The relative interiors of convex fuzzy sets under induced fuzzy topology

By introducing affine hulls of fuzzy sets and using Lowen's Representation Theorem for fuzzy subspaces (also for affine fuzzy sets), we go somewhat deeper into the fuzzy topological properties of fuzzy sets and convex fuzzy sets, respectively. Finally we establish the theory of relative interiors of convex fuzzy sets.

A subspace theorem for ordinary linear differential equations

AbstractThe study of the S-unit equation for algebraic numbers rests very heavily on Schmidt's Subspace Theorem. Here we prove an effective subspace theorem for the differential function field case, which should be valuable in the proof of results concerning the S-unit equation for function fields. Theorem 1 states that either has a given upper bound where are linearly independent linear forms in the polynomials with coefficients that are formal power series solutions about x = 0 of non-zero differential equations and where Orda denotes the order of vanishing about a regular (finite) point of functions ƒk, i: (k = 1, n; i = 1, n) or lies inside one of a finite number of proper subspaces of (K(x))n. The proof of the theorem is based on the wroskian methods and graded sub-rings of Picard-Vessiot extensions developed by D. V. Chudnovsky and G. V. Chudnovsky in their function field analogues of the Roth and Schmidt theorems. A brief discussion concerning the possibility of a subspace theorem for a product of valuations including the infinite one is also included.

The number of subspaces occurring in the p-adic subspace theorem in diophantine approximation.

The number of subspaces occurring in the p-adic subspace theorem in diophantine approximation.

Notes on invariant subspaces

The main purpose of this article is to give an approach to the recent invariant subspace theorem of Brown, Chevreau and Pearcy: Every contraction on a Hubert space, whose spectrum contains the unit circle has nontrivial invariant subspaces. Our proof incorporates several of the recent ideas tying together function theory and operator theory. 1. I N T R O D U C T I O N The Jordan structure theorem for finite matrices has been known now for over one hundred years, and its usefulness can hardly be overstated. It says that every square matrix A over the complex numbers C is similar to another matrix B (i.e., B = XAX~ for some invertible matrix X) which is a direct sum of Jordan cells. That is, B can be written in block form \BX 0 ••• 0 ] B=\° * ' ° [o 0 ... Bk and each Bt has the form [A, 0 0

A Refinement of Schmidt's Subspace Theorem

A Refinement of Schmidt's Subspace Theorem

The spectrum of orthogonal sums of subnormal pairs

This note provides yet another example of the difficulties that arise when one wants to extend the spectral theory of subnormal operators to subnormal tuples. Several basic properties of a subnormal operator Y remain true for tuples; e.g. the existence and uniqueness of its minimal normal extension N, the spectral inclusion σ(N)⊂ σ(Y)-proved for n-tuples in [4] and generalized to infinite tuples in [5]. However, neither the invariant subspace theorem nor the spectral mapping theorem in the “strong form” as in [3] is known so far for subnormal tuples.

Galois generators and the subspace theorem

In an earlier paper [2], a preliminary attempt was made to investigate the divisibility properties of the set of normal integral generators in a tame, prime extension ofQ. The result suggested that the contribution from an arbitrary, fixed, finite set of primes is very small. In this paper we are going to show (1) that it is about as small as it could be and (2) that the results hold in general for tame, abelian extensions (with only a mild technical condition). The approach is twofold, depending upon (a) a good understanding of the geometry of the units in abelian group rings, and (b) the generalisation, due to Schlickewei, of the beautiful and powerful subspace theorem of W. Schmidt. In previous papers, attempts to use diophantine approximation for these problems have been frustrated by the unwieldy nature of the theorems involved. Here we go to the heart of the matter by making an appeal directly to the subspace theorem.

An example of a domain of parreau-widom type

Several theorems on the Hardy classes on the open unit disk have been extended to the Hardy classes on Riemann surfaces of Parreau–Widom type. However, some of the theorems, for instance, Beurling's invariant subspace theorem ([3,4, 7,9]) and the F. and M. Riesz theorem ([6]), are generalized under an additional condition. In this note we show that this additional condition cannot be dropped. Namely, we give an example of a plane domain which is of Parreau–Widom type and yet does not allow these theorems.

A Choquet theorem for general subspaces of vector-valued functions

Let X be a compact Hausdorff space, let E be a (real or complex) Banach space, and let C(X, E) stand for the Banach space of all continuous E-valued functions defined on X under the supremum norm. If A is an arbitrary linear subspace of C(X, E), then it is shown that each bounded linear functional l on A can be represented by a boundary E*-valued vector measure μ on X that has the same norm as l. This result constitutes an extension to vector-valued functions of the so-called analytic version of Choquet's integral representation theorem.

Recursive solution of the covariance equations for linear prediction

An efficient recursive algorithm is presented to solve the “covariance” equations of the linear prediction modeling procedure. This algorithm is based on the conjugate direction optimization procedure and the expanding subspace theorem, and we show it is a natural extension as well as a geometric interpretation of the Levinson algorithm. The developed algorithm is simple to implement computationally, and can be extended to the multidimensional case.

Separadores de puntos y dimensión

We define the K-dimension K(X) of a topological space X by means of separating-points. The zero K-dimensional spaces are those that your quasicomponents are the point sets. We establishes the theorem of topological invariance, the subspace theorem, the sum theorem in a restricted case and the cartesian product theorem. The relation with the classical dimension functions are also studied.

Beurling-Lax representations using classical Lie groups with many applications II: GL(n,�) and Wiener-Hopf factorization

The classical Beurling-Lax-invariant subspace theorem characterizes the full range simply invariant subspacesM of L n 2 as being of the formM=ΘH n 2 where Θ∈L n×n ∞ is a phase function. Here L n 2 is the Hilbert space of measurable ℂn-valued functions on the unit circle {eit|0≤t≤2π} which are square-integrable in norm, H n 2 is the subspace of functions in L n 2 with analytic continuation to the interior of the disk {z‖z|<1}, L n×n 2 is the space of measurable essentially bounded n×n matrix functions on the unit circle, and a phase function Θ is one whose values Θ(eit) are unitary for a.e. t (i.e., Θ(eit) is in the Lie group U(n) a.e.). Halmos extended this to L ∞ 2 . A subspace M⊂L n 2 is said to beinvariant if eit M⊂M,simply invariant if in addition $$\mathop \cap \limits_{k \geqslant 0} $$ eikt M=(0), andfull range if $$\mathop \cup \limits_{N > 0} $$ e−iNt M is dense in L n 2 . In the Beurling-Lax representationM=ΘH n 2 ,M uniquely determines Θ up to a unitary constant factor on the right if one insists that Θ(eit)∈U(n). If one demands only that Θ(eit) ∈ GL(n,ℂ) (the group of invertible n×n complex matrices), however, there is considerably more freedom; in fact ΘH n 2 =Θ1H n 2 where Θ1 ΘF and F∈L n×n ∞ is outer with inverse F−1∈L n×n ∞ . More generally, we have ΘH n 2 =[Θ1H n ∞ ]− whenever Θ1=ΘF and F is outer with F and F−1 in L n×n 2 . (An F∈L n×n 2 will be said to beouter if FH n ∞ is a dense subset of H n 2 .) In particular one can use this freedom to obtain representationsM=[ΘH n ∞ ]− where the representor Θ has values Θ(eit) in other matrix Lie groups. This program was carried out in accompanying work of the authors [B-H1-4] for the classical simple Lie groups U(m,n), O(p,q), O*(2n), Sp(n,C), Sp(n,R), Sp(p,q), O(n,C), GL(n,R), U*(2n), GL(n,R) and SL(n,C) and many applications were given. In this paper we give a natural theorem for GL(n,ℂ), by introducing the extra structure of preassigning the spaceM x=[ΘH n ∞ ]− as well asM=[ΘH n ∞ ]−. The theorems in [B-H1-4] can be derived by specializing our main result here for GL(n,ℂ) to the various subgroups which we listed.

The signal subspace approach for multiple wide-band emitter location

The rational vector space generalization of the signal subspace approach is presented and applied to the estimation of multiple wide-band emitter locations from the signals received at multiple sensors. The signal subspace and array manifold concepts first introduced by Schmidt are generalized to rational vector space. These concepts are used to develop the rational signal subspace theory and prove the signal subspace theorem, on which signal subspace algorithms are based. The theory is applied in discrete time to derive a class of rational signal subspace algorithms for source location and spectral estimation using unit circle eigendecomposition of multivariate rational models of sensor outputs. Simulation results are presented for an algorithm in this class, including sample statistics from Monte Carlo trials and comparisons with the Cramer-Rao bound.

Invariant Subspaces on Riemann Surfaces of Parreau-Widom Type

In this paper we generalize Beurling’s invariant subspace theorem to the Hardy classes on a Riemann surface with infinite handles. The problem is to classify all closed ($\text {weak}^{\ast }$ closed, if $p = \infty$) ${H^\infty }(d\chi )$-submodules, say $\mathfrak {m}$, of ${L^p}(d\chi )$, $1 \leqslant p \leqslant \infty$, where $d\chi$ is the harmonic measure on the Martin boundary of a Riemann surface $R$, and ${H^\infty }(d\chi )$ is the set of boundary functions of all bounded analytic functions on $R$. Our main result is stated roughly as follows. Let $R$ be of Parreau-Widom type, that is, the space ${H^\infty }(R,\gamma )$ of bounded analytic sections contains a nonzero element for every complex flat line bundle $\gamma \in \pi {(R)^{\ast }}$. We may assume, without loss of generality, that the Green’s function of $R$ vanishes at the infinity. Set ${m^\infty }(\gamma ) = \sup \{ |f({\mathbf {O}})|:f \in {H^\infty }(R,\gamma ),|f| \leqslant 1\}$ for a fixed point ${\mathbf {O}}$ of $R$. Then, a necessary and sufficient condition in order that every such an $\mathfrak {m}$ takes either the form $\mathfrak {m} = {C_E}{L^p}(d\chi )$, where ${C_E}$ is the characteristic function of a set $E$, or the form $\mathfrak {m} = q{H^p}(d\chi ,\gamma )$, where $|q| = 1$ a.e. and $\gamma$ is some element of $\pi {(R)^{\ast }}$ is that ${m^\infty }(\gamma )$ is continuous for the variable $\gamma \in \pi {(R)^{\ast }}$.

Invariant subspaces on Riemann surfaces of Parreau-Widom type

In this paper we generalize Beurling’s invariant subspace theorem to the Hardy classes on a Riemann surface with infinite handles. The problem is to classify all closed ( weak ∗ \text {weak}^{\ast } closed, if p = ∞ p = \infty ) H ∞ ( d χ ) {H^\infty }(d\chi ) -submodules, say m \mathfrak {m} , of L p ( d χ ) {L^p}(d\chi ) , 1 ⩽ p ⩽ ∞ 1 \leqslant p \leqslant \infty , where d χ d\chi is the harmonic measure on the Martin boundary of a Riemann surface R R , and H ∞ ( d χ ) {H^\infty }(d\chi ) is the set of boundary functions of all bounded analytic functions on R R . Our main result is stated roughly as follows. Let R R be of Parreau-Widom type, that is, the space H ∞ ( R , γ ) {H^\infty }(R,\gamma ) of bounded analytic sections contains a nonzero element for every complex flat line bundle γ ∈ π ( R ) ∗ \gamma \in \pi {(R)^{\ast }} . We may assume, without loss of generality, that the Green’s function of R R vanishes at the infinity. Set m ∞ ( γ ) = sup { | f ( O ) | : f ∈ H ∞ ( R , γ ) , | f | ⩽ 1 } {m^\infty }(\gamma ) = \sup \{ |f({\mathbf {O}})|:f \in {H^\infty }(R,\gamma ),|f| \leqslant 1\} for a fixed point O {\mathbf {O}} of R R . Then, a necessary and sufficient condition in order that every such an m \mathfrak {m} takes either the form m = C E L p ( d χ ) \mathfrak {m} = {C_E}{L^p}(d\chi ) , where C E {C_E} is the characteristic function of a set E E , or the form m = q H p ( d χ , γ ) \mathfrak {m} = q{H^p}(d\chi ,\gamma ) , where | q | = 1 |q| = 1 a.e. and γ \gamma is some element of π ( R ) ∗ \pi {(R)^{\ast }} is that m ∞ ( γ ) {m^\infty }(\gamma ) is continuous for the variable γ ∈ π ( R ) ∗ \gamma \in \pi {(R)^{\ast }} .

An invariant subspace theorem in the abstract Hardy algebra theory

An invariant subspace theorem in the abstract Hardy algebra theory

An operator not satisfying Lomonosov's hypothesis

An example is presented of a Hilbert space operator such that no non-scalar operator that commutes with it commutes with a non-zero compact operator. This shows that Lomonosov's invariant subspace theorem does not apply to every operator.

Convex fuzzy sets

This paper gathers some elementary known results about convex fuzzy sets and completes the theory, introducing the necessary concepts. Using a representation theorem for fuzzy subspaces it gives separation theorems for convex fuzzy sets in the proper setting.

The Thue-Siegel-Roth-Schmidt theorem for algebraic functions

The theorem of the title on simultaneous rational approximation to algebraic numbers is carried over to simultaneous approximation by rational functions to algebraic functions. More generally, Schmidt's Subspace Theorem is proved in the context of functions.

Hilden's simple proof of Lomonosov's invariant subspace theorem

Perhaps the best-known unsolved problemin functional analysis is theinvariant subspace problem: Does every bounded operator on (separable, infinite-dimensional, complex) Hilbert space have a nontrivial invariant subspace ? (P. Enflo has shown, in work that has not yet been published, that there is an operator on a nonreflexive Banach space with only the trivial invariant subspaces.) Many special cases of this problem have been solved; see [Z]. One of the earliest and most elegant invariant subspace theorems is the result of von Neumann, Aronszajn, and Smith that compact operators have invariant subspaces. In the years 19% 1973, a number of authors worked on extending this result and significant progress was made; (see [2, pp. 93-941). In 1973 operator theorists were stunned by the generalization achieved by Lomonosov [l]. The theorem Lomonosov obtained was a more general result than anyone had ever hoped to be able to prove; moreover, the proof was considerably simpler than even the original von Neumann-Aronszajn-Smith proof! Lomonosov treated the problem as a nonlinear one. He ingeniously constructed a certain function to which Schauder’s fixed point theorem applied; the fixed points of this function produced invariant subspaces. L. J. Wallen of the University of Hawaii told his colleague H. M. Hilden about Lomonosov’s proof. Wallen suggested to Hilden that it would be desirable if Schauder’s theorem could be replaced by Banach’s contraction mapping theorem to yield a proof that could be presented in introductory functional analysis courses. Within a few days Hilden found a proof that does not require any fixed point theorem, nor any other result beyond the most elementary notions of functional analysis. This proof appears in [2, p. 1581, but there it is surrounded by more technical results. Hopefully this exposition will make the theorem more accessible to nonspecialists, Many operator theorists are aware of a preprint (never published) which appeared to imply that some of Lomonosov’s ideas were due to the authors of the preprint. No such implication should be drawn from the present note. The ideas are all Lomonosov’s and Hilden’s. Even the exposition of the proof is derivative; it is largely based on Wallen’s 1973 Wabash conference lecture.