Some varieties and convexities generated by fractal lattices

We introduce definitions of semifractal, 0–1-fractal, quasifractal and fractal lattices. A variety generated by a fractal lattice is called fractal generated, with analogous terminology for the other variants. We show that a semifractal generated nondistributive lattice variety cannot be of residually finite length. This easily implies that there are exactly continuously many lattice varieties which are not semifractal generated. On the other hand, for each prime field F, the variety generated by all subspace lattices of vector spaces over F is shown to be fractal generated. These countably many varieties and the class of all distributive lattices are the only known fractal generated lattice varieties at present. Four distinct countable distributive fractal lattices are given each of which generates . After showing that each lattice can be embedded in a quasifractal, continuously many quasifractals are given each of which has cardinality and generates the variety of all lattices. Semifractal considerations are applied to construct examples of convexities that include no minimal convexity, thus answering a question of Jakubík. (A convexity is a class of lattices closed under taking homomorphic images, convex sublattices and direct products, a notion due to Ervin Fried.) This research was partially supported by the NFSR of Hungary (OTKA), grant no. T 049433 and K 60148.     

Embedding operators into strongly continuous semigroups

We study linear operators T on Banach spaces for which there exists a C₀-semigroup (T(t))_t≥0 such that T = T(1). We present a necessary condition in terms of the spectral value 0 and give classes of examples for which such a C₀-semigroup does or does not exist. Received: 22 December 2008, Revised: 7 April 2009     

Additive maps preserving the minimum and surjectivity moduli of operators

Let be a complex Hilbert space and let be the algebra of all bounded linear operators on . We characterize additive maps from onto itself preserving different spectral quantities such as the minimum modulus, the surjectivity modulus, and the maximum modulus of operators. Received: 15 July 2008     

The Schur property for subgroup lattices of groups

A classical theorem of Schur states that if the centre of a group G has finite index, then the commutator subgroup G′ of G is finite. A lattice analogue of this result is proved in this paper: if a group G contains a modularly embedded subgroup of finite index, then there exists a finite normal subgroup N of G such that G/N has modular subgroup lattice. Here a subgroup M of a group G is said to be modularly embedded in G if the lattice is modular for each element x of G. Some consequences of this theorem are also obtained; in particular, the behaviour of groups covered by finitely many subgroups with modular subgroup lattice is described. Received: 16 October 2007, Final version received: 22 February 2008     

Semidirect products of lattices

The semidirect product of lattices is a lattice analogue of the semidirect product of groups. In this article we introduce this construction, show some basic facts and study a class of lattices closed under semidirect products. We also generalise this notion presenting the semidirect product of semilattices. Received February 22, 2005; accepted in final form August 29, 2006.     

On the Cowen-Douglas Class for Banach Space Operators

In [3], M. J. Cowen and R. G. Douglas prove that the adjoint of a Hilbert space operator T is in the class if and only if T is unitarily equivalent with the operator M _z on a Hilbert space -valued analytic functions, where M _z denotes the operator of multiplication by the independent variable. The proof involves holomorphic vector bundles and Grauert’s theorem. In this paper we use a theorem by I. Gohberg and L. Rodman [4] to give a more elementary proof of this fact, which also works for Banach space operators.      

Properties of generalized Toeplitz operators

An operatorX: is said to be a generalized Toeplitz operator with respect to given contractionsT ₁ andT ₂ ifX=T ₂XT₁ ^*. The purpose of this line of research, started by Douglas, Sz.-Nagy and Foia, and Pták and Vrbová, is to study which properties of classical Toeplitz operators depend on their characteristic relation. Following this spirit, we give appropriate extensions of a number of results about Toeplitz operators. Namely, Wintner's theorem of invertibility of analytic Toeplitz operators, Widom and Devinatz's invertibility criteria for Toeplitz operators with unitary symbols, Hartman and Wintner's theorem about Toeplitz operator having a Fredholm symbol, Hartman and Wintner's estimate of the norm of a compactly perturbed Toeplitz operator, and the non-existence of compact classical Toeplitz operators due to Brown and Halmos.Dedicated to our friend Cora Sadosky on the occasion of her sixtieth birthday     

Measure theory based on lattices and transfinite recursion

In this methodological study we develop the foundations of measure theory using lattices as prime structures instead of rings. Topological as well as abstract regularity is incorporated into this approach from the outset. The use of inner and outer measures is replaced by transfinite constructions. Basic extension steps are transfinitely iterated to yield generalizations of Carathéodory’s theorem which are optimal with respect to inner and outer approximations. Received: 5 May 2008     

Bicommutants of Toeplitz operators

In this paper we discuss an unusual phenomenon in the context of Toeplitz operators in the Bergman space on the unit disc: If two Toeplitz operators commute with a quasihomogeneous Toeplitz operator, then they commute with each other. In the Bourbaki terminology, this result can be stated as follows: The commutant of a quasihomogeneous Toeplitz operator is equal to its bicommutant. Received: 11 March 2008     

On operators with closed analytic core

As shown by Mbekhta [9] and [10], the analytic core and the quasi-nilpotent part of an operator play a significant role in the local spectral and Fredholm theory of operators on Banach spaces. It is a basic fact that the analytic core is closed whenever 0 is an isolated point of the spectrum. In this note, we explore the extent to which the converse is true, based on the concept of support points. Our results are exemplified in the case of decomposable operators, Riesz operators, convolution operators, and semi-shifts.     

On the Orbit of an m-Isometry

A bounded linear operator T on a Hilbert space H is called an m-isometry for a positive integer m if . We prove some properties concerning the behaviour of the orbit of an m-isometry. For example, every orbit of an m-isometry is eventually norm increasing and some m-isometries can not be N-supercyclic, that is, there does not exist an N-dimensional subspace E_N such that the orbit of T at E_N is dense in H.

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To Professor José Rodríguez Expósito on his 60th birthday     

Orthocomplemented lattices in ordered groups

On a partially ordered set G the orthogonality relation is defined by incomparability and is a complete orthocomplemented lattice of double orthoclosed sets. We will prove that the atom space of the lattice has the same order structure as G. Thus if G is a partially ordered set (an ordered group, or an ordered vector space), then is a canonically partially ordered set (an ordered quotient group, or an ordered quotient vector space, respectively). We will also prove: if G is an ordered group with a positive cone P, then the lattice has the covering property iff , where g is an element of G and M is the intersection of all maximal subgroups contained in . Received August 1, 2006; accepted in final form May 29, 2007.     

A lattice approach to narrow operators

It is known that if a rearrangement invariant function space E on [0,1] has an unconditional basis then each linear continuous operator on E is a sum of two narrow operators. On the other hand, the sum of two narrow operators in L₁ is narrow. To find a general approach to these results, we extend the notion of a narrow operator to the case when the domain space is a vector lattice. Our main result asserts that the set N_r(E, F) of all narrow regular operators is a band in the vector lattice L_r(E, F) of all regular operators from a non-atomic order continuous Banach lattice E to an order continuous Banach lattice F. The band generated by the disjointness preserving operators is the orthogonal complement to N_r(E, F) in L_r(E, F). As a consequence we obtain the following generalization of the Kalton-Rosenthal theorem: every regular operator T : E → F from a non-atomic Banach lattice E to an order continuous Banach lattice F has a unique representation as T = T_D + T_N where T_D is a sum of an order absolutely summable family of disjointness preserving operators and T_N is narrow. Supported by Ukr. Derzh. Tema N 0103Y001103.     

Existence and extensions of positive linear operators

In this paper we develop some unified methods, based on the technique of the auxiliary sublinear operator, for obtaining extensions of positive linear operators. In the first part, a version of the Mazur-Orlicz theorem for ordered vector spaces is presented and then this theorem is used in diverse applications: decomposition theorems for operators and functionals, minimax theory and extensions of positive linear operators. In the second part, we give a general sufficient condition (an implication between two inequalities) for the existence of a monotone sublinear operator and of a positive linear operator. Some particular cases in which this condition becomes necessary are also studied. Dedicated to Prof. Romulus Cristescu on his 80th birthday     

Hyponormality of Toeplitz Operators with Polynomial and Symmetric-Type Symbols

Suppose that is a trigonometric polynomial of the form (z) = ^N_n=-N a_n zⁿ. It is well-known that T is normal if and only if | a_{– N}| =  | a_N| and the Fourier coefficients of satisfy the following symmetry condition:

Wiener-Hopf operators on L_{\omega}^{2}(\mathbb{R}^{+})

Let be a weighted space with weight . In this paper we show that for every Wiener-Hopf operator T on and for every a I, there exists a function such that

The complex Busemann-Petty problem for arbitrary measures

The complex Busemann-Petty problem asks whether origin symmetric convex bodies in with smaller central hyperplane sections necessarily have smaller volume. The answer is affirmative if and negative if . In this article we show that the answer remains the same if the volume is replaced by an “almost” arbitrary measure. This result is the complex analogue of Zvavitch’s generalization to arbitrary measures of the original real Busemann-Petty problem. Received: 6 May 2008     

On the Wu metric in unbounded domains

We discuss the properties of the Wu pseudometric and present counterexamples for its upper semicontinuity that answers the question posed by Jarnicki and Pflug. We also give formulae for the Wu pseudometric in elementary Reinhardt domains. Received: 12 September 2007     

On approximate solutions to the first initial boundary value problem for the m-Hessian evolution equations

We adapt to degenerate m-Hessian evolution equations the notion of m-approximate solutions introduced by N. Trudinger for m-Hessian elliptic equations, and we present close to necessary and sufficient conditions guaranteeing the existence and uniqueness of such solutions for the first initial boundary value problem. Dedicated to Professor Felix Browder     

On two theorems of Flavell

We extend two theorems due to P. Flavell [6] to arbitrary fusion systems. Received: 18 June 2008