Some tilings of the plane whose singular points are uncountable and unbounded

Let I be a tiling of the plane such that for every tile T of I there correspond a tile T of I (not necessarily unique) and an integer k(T, T) (depending on T and T), k(T, T)>2, such that T meets T in k(T, T) connected components. Tiles T and T satisfying this condition are called associated tiles in I. Various properties concerning I and its singular points are obtained. First, it is not possible that every tile in I have a unique associated tile. In fact, there exist infinite families of tiles {F} {F _n:n1} such that F is the unique associated tile for every F _n. Next, if x is a singular point of I, then every neighborhood of x contains uncountably many singular points of I. Finally, the set of singular points of I is unbounded.     

Cyclicity results for Jordan and compressed Toeplitz operators

A vectorx in a Hilbert spaceH iscyclic for a bounded linear operatorTHH if the closed linear span of the orbit {T ⁿ xn0} ofx underT is all ofH. Operators which have a cyclic vector are said to be cyclic.Jordan operators are the infinite direct sums of Jordan cells acting on finite- dimensional Hilbert spaces. Necessary and sufficient conditions for a Jordan operator to be cyclic are given (see Corollary 6). In this case, a dense set of cyclic vectors is exhibited (see Corollary 4). Sufficient conditions for uncountable collections of cyclic Jordan operators to have a common cyclic vector are given and, in this case, a dense set of common cyclic vectors is exhibited (see Corollary 9).Analogues of these cyclicity results for Jordan operators are obtained for compressions of analytic Toeplitz operatorsT _A FAF on the Hardy spaceH ² to subspaces (BH ²) invariant for the backward shiftT _z ^* whereB is a Blaschke product by showing that such compressions are quasisimilar to Jordan operators.     

Weight functions on the Kneser graph and the solution of an intersection problem of Sali

LetX, Y be finite sets and suppose thatF is a collection of pairs of sets (F, G),FX,GY satisfying |FF|s, |GG|t and |FF|+|GG|s+t+1 for all (F, G),F, GF. Extending a result of Sali, we determine the maximum ofF.     

On invariant subspaces of Volterra-type operators

The lattice of all the closed, invariant subspaces of the Volterra integration operator onL ²[0, 1] is equal to {B(a):a[0, 1]}, whereB(a)={fL ²[0, 1]:f=0 a.e. on [0,a]}. In order to extend this result to Banach function spaces we study the Volterra-type operatorV that was introduced in [7] for the case ofL ^p -spaces. Our main result characterizesL-closed subspaces of a Banach function spaceL that are invariant underV, whereL denotes the associate space ofL. In particular, if the norm ofL is order continuous and ifV is injective, then all the closed, invariant subspaces ofV are determined.This work was supported by the Research Ministry of Slovenia.     

Small compact perturbation of strongly irreducible operators

An operatorT onH is called strongly irreducible ifT is not similar to any reducible operators. In this paper, we shall say yes to answer the following question raised by D. A. Herrero.Given an operatorT with connected spectrum (T) and a positive number , can we find a compact operatorK with K < such thatT+K is strongly irreducible?Supported by National Natural Science Foundation of China(19901011), Mathematical Center of State Education Commission of China and 973 Project of China     

Orthomodular lattices from 3-dimensional quadratic spaces

If E is a vector space over a field K, then any regular symmetric bilinear form on E induces a polarity on the lattice of all subspaces of E. In the particular case where E is 3-dimensional, the set of all subspaces M of E such that both M and are not N-subspaces (which, in most cases, is equivalent to saying that M is nonisotropic), ordered by inclusion and endowed with the restriction of the above polarity, is an orthomodular lattice T(E, ). We show that if K is a proper subfield of K, with K F₂, and E a 3-dimensional K -subspace of E such that the restriction of to E × E is, up to multiplicative constant, a bilinear form on the K -space E , then T(E , ) is isomorphic to an irreducible 3-homogeneous proper subalgebra of T(E, ). Our main result is a structure theorem stating that, when K is not of characteristic 3, the converse is true, i.e., any irreducible 3-homogeneous proper subalgebra of T(E, ) is of this form. As a corollary, we construct infinitely many finite orthomodular lattices which are minimal in the sense that all their proper subalgebras are modular. In fact, this last result was our initial aim in this paper.Received June 4, 2003; accepted in final form May 18, 2004.     

Square of ω-hyponormal operators

In this paper, we show that ifT is a -hyponormal operator, thenT ² is also -hyponormal.     

On the boundedness of Cauchy singular operator from the spaceL p toL q,p>q>-1

It is proved that for a Cauchy type singular operator, given by equality (1), to be bounded from the Lebesgue spaceL _p () tol _q (), as = _n=1 ^Ȟ _n , _n ={z:|z|=r _n }, it is necessary and sufficient that either condition (4) or (5) be fulfilled.     

Factorization of a selfadjoint nonanalytic operator function II. Single spectral zone

We consider a selfadjoint and smooth enough operator-valued functionL() on the segment [a, b]. LetL(a)0,L(b)0, and there exist two positive numbers and such that the inequality |(L()f, f)|< ([a, b] f=1) implies the inequality (L'()f, f)>. Then the functionL() admits a factorizationL()=M()(I-Z) whereM() is a continuous and invertible on [a, b] operator-valued function, and operatorZ is similar to a selfadjoint one. This result was obtained in the first part of the present paper [10] under a stronge conditionL()0 ( [a,b]). For analytic functionL() the result of this paper was obtained in [13].     

Isolated point of spectrum ofP-hyponormal, log-hyponormal operators

LetT B(H) be a bounded linear operator on a complex Hilbert spaceH. Let ₀ (T) be an isolated point of (T) and let be the Riesz idempotent for ₀. In this paper, we prove that ifT isp-hyponormal or log-hyponormal, thenE is self-adjoint andE H=ker(H–₀)=ker(H–₀ ^*.This research was supported by Grant-in-Aid Research 1 No. 12640187.     

Isometries of symmetric operator spaces associated with AFD factors of typeII and symmetric vector-valued spaces

If is a surjective isometry of the separable symmetric operator spaceE(M, ) associated with the approximately finite-dimensional semifinite factorM and if · _E(M,) is not proportional to · _{L 2}, then there exist a unitary operatorUM and a Jordan automorphismJ ofM such that(x)=UJ(x) for allxME(M, ). We characterize also surjective isometries of vector-valued symmetric spacesF((0, 1), E(M, )).Research supported by the Australian Research Council     

Pseudo-RiemannianT-duals of compact Riemannian homogeneous spaces

The first part of this paper describes the construction of pseudo-Riemannian homogeneous spaces with special curvature properties such as Einstein spaces, using corresponding known compact Riemannian ones. This construction is based on the notion of a certain duality between compact and non-compact homogeneous spaces. In the second part we apply this method to obtain pseudo-Riemannian homogeneous manifolds with real Killing spinors. We will prove that under a certain additional condition a dual pseudo-Riemannian space (G/H, g) of a compact Riemannian homogeneous space (G/H, g) with homogeneousSpin-structure admits a homogeneousSpin ⁺-structure and theG_invariant Killing spinors on (G/H, g) correspond toG-invariant Killing spinors on (G/H, g). We can ensure that in most cases the hypothesis onG-invariance is satisfied.     

Joint torsion of Toeplitz operators with H∞ symbols

We give a new² index theorem for the basic example of Toeplitz operators on the circle. The joint torsion, a non zero complex valued analytic index, of a pair of Fredholm Toeplitz operatorsT andT withH symbols is computed by residues in the disk, and is determined by a monodromy integral which specifies the isomorphism class of a flat line bundle on the circle. When the symbols and are rational a product of joint torsions identifies the isomorphism class of the bundle inH ¹ (S ¹,C ^*), and the identification extends by rational approximation to the case of smooth symbols defined on the circle.Partially supported by National Science Foundation grants to both authors.     

Extensions and duality of finite geometric closure operators

If P and P are finite partially ordered sets such that P=P–e for some maximal element e P, all geometric closure operators on P are determined whose restriction to P equals a given closure operator on P.The classes Q(P) and Q(P^*) of all geometric closure operators on P and on its order dual P^* are shown to be anti-isomorphic partially ordered sets.     

On the asymptotic equivalence ofL pmetrics for convergence to normality

Summary This paper is concerned with the rate of convergence to zero of theL _pmetrics _np1p, constructed out of differences between distribution functions, for departure from normality for normed sums of independent and identically distributed random variables with zero mean and unit variance. It is shown that the _np are, under broad conditions, asymptotically equivalent in the strong sense that, for 1p, p, _np/_np is universally bounded away from zero and infinity asn.     

Ergodicity and weak-mixing of homogeneous extensions of measure-preserving transformations with applications to Markov shifts

Given a homogeneous extensionS of a measure-preserving transformationT, we provide necessary and sufficient conditions for the ergodicity and weak-mixing ofS in terms of functional equations. We then apply our findings to the case whenT is a Markov shift and the associated skewing function ofS depends on a finite number of coordinates. In this case, we obtain a simplification to the appropriate functional equations.     

Multiple wiener integrals and nonlinear functionals of a nuclear space valued Wiener process

In this work we develop techniques for the study of nonlinear functionals of a -valued Wiener processW _t, where is the dual of a countably Hilbert nuclear space. We construct stochastic integrals and multiple Wiener integrals of operator-valued processes with respect toW _t. The Wiener decomposition of the space of -valued nonlinear functionals ofW _t is established. We also obtain multiple stochastic integral expansions and representations of -valued nonlinear functionals ofW _t as operator-valued stochastic integrals of Itô type.This research was partially supported by CONACYT grants 22537 and PCEXCNA-040651, and Air Force Office of Scientific Research No. F49620 85 C 0144.Presently at CIMAT, A.P. 402 Guanajuato 36000, GTO, México.     

On nonconvex perturbations of maximal monotone differential inclusions

The paper is concerned with the evolution inclusionxAx+F(t,x), whereA generates a contractive semigroup andF is a lower semicontinuous multifunction. Constructing a suitable directionally continuous selection fromF, we prove the existence of solutions on a closed domain and the connectedness of the set of trajectories.     

Non-semibounded sesquilinear forms and left-indefinite Sturm-Liouville problems

By Kato's First and Second Representation Theorem a closed densely defined semibounded hermitian sesquilinear form [·,·] in a Hilbert spaceH can be represented by a selfadjoint operatorT inH and if, in particular, the formt[·,·] is nonnegative thenD(t)=D(T ^1/2 ). In the present note this result is generalized to non-semibounded forms by means of Krein space methods. An application to the form where the functionp changes its sign leads to an expansion theorem for the Sturm-Liouville problem –(pu)=u,u(a)=u(b)=0 with respect to the weighted Sobolev norm .     

On perturbations of linear m-accretive operators on reflexive Banach spaces

In this note we prove some results on the m-accretivity of sums and products of linear operators. In particular we obtain the following theorem: LetA, B be two m-accretive operators on a reflexive Banach space. IfA is invertible and (A)^–1 B is accretive thenBA ^–1 andA+B are m-accretive.