On pure subnormal operators with finite rank self-commutators and related operator tuples

In this paper, we study the model of a pure subnormal operator with finite rank self-commutator and of the relatedn-tuple of commuting linear bounded operators. We also give some applications of the model to the theory ofn-tuples of commuting operators with trace class self-commutators.This work is supported in part by a NSF grant no. DMS-9400766.     

On subnormal operators

LetS be a pure subnormal operator such thatC*(S), theC*-algebra generated byS, is generated by a unilateral shiftU of multiplicity 1. We obtain conditions under which 5 is unitarily equivalent toα + βU, α andβ being scalars orS hasC*-spectral inclusion property. It is also proved that if in addition,S hasC*-spectral inclusion property, then so does its dualT andC*(T) is generated by a unilateral shift of multiplicity 1. Finally, a characterization of quasinormal operators among pure subnormal operators is obtained.     

On self-dual subnormal operators

We improve the result of C. C. Huang about self-dual subnormal operators, and consider the converse of this result.     

On unbounded subnormal operators

A minimal normal extension of unbounded subnormal operators is established and characterized and spectral inclusion theorem is proved. An inverse Cayley transform is constructed to obtain a closed unbounded subnormal operator from a bounded one. Two classes of unbounded subnormals viz analytic Toeplitz operators and Bergman operators are exhibited.     

On quasi-similarity of subnormal operators

For a compact subset K in the complex plane, let Rat(K) denote the set of the rational functions with poles off K. Given a finite positive measure with support contained in K, let R2(K,v) denote the closure of Rat(K) in L2(v) and let Sv denote the operator of multiplication by the independent variable z on R2(K, v), that is, Svf = zf for every f∈R2(K, v). SupposeΩis a bounded open subset in the complex plane whose complement has finitely many components and suppose Rat(Ω) is dense in the Hardy space H2(Ω). Letσdenote a harmonic measure forΩ. In this work, we characterize all subnormal operators quasi-similar to Sσ, the operators of the multiplication by z on R2(Ω,σ). We show that for a given v supported onΩ, Sv is quasi-similar to Sσif and only if v/■Ω■σ and log(dv/dσ)∈L1(σ). Our result extends a well-known result of Clary on the unit disk.     

On Ehresmann semigroups

We formulate an alternative approach to describing Ehresmann semigroups by means of left and right étale actions of a meet semilattice on a category. We also characterize the Ehresmann semigroups that arise as the set of all subsets of a finite category. As applications, we prove that every restriction semigroup can be nicely embedded into a restriction semigroup constructed from a category, and we describe when a restriction semigroup can be nicely embedded into an inverse semigroup.

     

On U-orthodox semigroups

As a generalization of an orthodox semigroup in the class of regular semigroups, a type W semigroup was first investigated by El-Qallali and Fountain. As an analogy of the type W semi-groups in the class of abundant semigroups, we introduce the U-orthodox semigroups. It is shown that the maximum congruence μ contained in HU on U-orthodox semigroups can be determined. As a consequence, we show that a U-orthodox semigroup S can be expressed by the spined product of a Hall semigroup WU and a V-ample semigroup ...     

On implemented semigroups

We shall be concerned with implemented semigroups of the form U(t)X := T(t)XS(t) for X ∈ L(F,E) and t ≥ 0 , where (T(t)) _{t ≥ 0} and (S(t)) _{t ≥ 0} are C ₀ -semigroups on the Banach spaces E and F , respectively. The aim of this article is to clarify the structure of this semigroup and its ``generator' with the help of inter- and extrapolation techniques. Moreover, using positivity arguments, we present a short proof of the infinite dimensional Liapunov Stability Theorem on Hilbert spaces as an application of our results. May 15, 2000     

On exclusive semigroups

The following problem was posed by the second author in ‘Semigroup Forum’, Vol. 1, No. 1, 1970, p. 91: Describe the structure of semigroups S such that for all a, b, c ε S, abc=ab, ac or bc.     

On the duals of subnormal tuples

A notion of the dual of a subnormal tuple of operators is discussed.     

On subnormal subgroups of linear groups

We describe the subnormal subgroups of 2-dimensional linear groups over local and full rings in which 2 is invertible, as well as the subnormal subgroups of symplectic groups over local rings in which 2 is invertible.