Unbounded quasi-integrals

Let be a locally compact Hausdorff space. We define a quasi-measure in , a quasi-integral on , and a quasi-integral on . We show that all quasi-integrals on are bounded, continuity properties of the quasi-integral on , representation of quasi-integrals on in terms of quasi-measures, and unique extension of quasi-integrals on to .

     

Unbounded derivations

In this note we discuss various extensions of a normal 1 derivation of a uniformly hyperfinite C¹-algebra. Various approximation theorems are employed to show when said extensions generate automorphism groups of the C¹-algebra. We characterize the “maximal” extension of Sakai and Powers as a graph limit and show when this extension is the closure of the given derivation. We also discuss an identity obeyed by the resolvent of a derivation.     

Unbounded symmetrizable idempotents

The relationship between closed unbounded idempotents and dense decompositions of a Hilbert space is explored by extending the notion of compatibility between closed subspaces and positive bounded operators.     

Unbounded semidistributive lattices

We consider two properties which are close to being lower bounded in the class of finite join semidistributive lattices. An example is constructed in which a finite join semidistributive lattice has both these two properties, but it is not lower bounded. Translated fromAlgebra i Logika, Vol. 39, No. 1, pp. 87–92, January–February, 2000.     

Unbounded Toeplitz Operators

This partly expository article develops the basic theory of unbounded Toeplitz operators on the Hardy space H ², with emphasis on operators whose symbols are not square integrable. Unbounded truncated Toeplitz operators on coinvariant subspaces of H ² are also studied. In memory of Paul R. Halmos     

Unbounded Quasinormal Operators Revisited

Various characterizations of unbounded closed densely defined operators commuting with the spectral measures of their moduli are established. In particular, Kaufman’s definition of an unbounded quasinormal operator is shown to coincide with that given by the third-named author and Szafraniec. Examples demonstrating the sharpness of results are constructed.     

Unbounded domains of normality

Let be a transcendental entire function of order less than 1/2. We introduce the method of ``self-sustaining spread' to study the components of the set of normality of such a function. We give a new proof of the fact that any preperiodic or periodic component of the set of normality of is bounded. We obtain the same conclusion for a wandering domain if the growth rate of is never too small.

     

Unbounded strictly singular operators

Let D(T)⊂X→Y be an unbounded linear operator where X and Y are normed spaces. It is shown that if Y is complete then T is strictly singular if and only if T is the sum of a continuous strictly singular operator and an unbounded finite rank operator. A counterexample is constructed for the case in which Y is not complete.     

Quasi-Affinity for Unbounded Representations

Quasi-affinity for unbounded representations are introduced and it is shown that, if a self-adjoint representation π is quasi-affine to a *-representation ?, then ? is also self-adjoint and π and ? are unitarily equivalent.     

Unbounded derivations in operator algebras

Unbounded derivations in uniformly hyperfinite C¹-algebras will be studied. Various conditions, under which normal ¹-derivations in the C¹-algebras can be extended to the infinitesimal generators of the approximately inner strongly continuous one-parameter subgroups of ¹-automorphisms on the algebras, will be given.     

Cotorsion Dimension of Unbounded Complexes

We extend the cotorsion dimension of R-modules to unbounded R-complexes by applying the flat model structure on Ch(R) proposed by J. Gillespie. This is not natural because there has been no sufficiently general result available for the existence of proper “cotorsion” resolutions of unbounded complexes, for which one would be able to define the derived functors. The global cotorsion dimension of ring is discussed in our present framework, and the relations between it and other dimensions are investigated as well. Some rings are characterized and some known results are extended.     

Convex Functions with Unbounded Gradient

We show that domains, that allow for convex functions with unbounded gradient at their boundary, are convex.     

Unbounded weighted shifts and subnormality

The purpose of this paper is to characterize subnormality of operators composed of unbounded weighted shifts (or even more general, orthogonal sums of such shifts) and their adjoints.     

Unbounded asymmetry of stretch factors

A result of Handel–Mosher guarantees that the ratio of logarithms of stretch factors of any fully irreducible automorphism of the free group F_N$F_N$ and its inverse is bounded by a constant C_N$C_N$. In this short note, we show that this constant C_N$C_N$ cannot be chosen independent of the rank N.