29 research outputs found
A Helson-Szeg\"o theorem for subdiagonal subalgebras with applications to Toeplitz operators
We formulate and establish a noncommutative version of the well known
Helson-Szego theorem about the angle between past and future for subdiagonal
subalgebras. We then proceed to use this theorem to characterise the symbols of
invertible Toeplitz operators on the noncommutative Hardy spaces associated to
subdiagonal subalgebras
Functions of operators and the classes associated with them
The important classes of normally solvable, Ο΄β (Ο΄β) and strictly singular (strictly cosingular) operators have long been studied in the setting of bounded or closed operators between Banach spaces. Results by Kato, Lacey, et al (see Goldberg [16; III.1.9, III.2.1 and III.2.3] ) led to the definition of certain norm related functions of operators (Ξ, Ξ and Ξβ) which provided a powerful new way to study the classes of Ο΄β and strictly singular operators (see for example Gramsch[19], Lebow and Schechter[28] and Schechter[36]). Results by Brace and R.-Kneece[4] among others led to the definition of analogous functions (Ξ' and Ξ') which were used to study Ο΄β and strictly cosingular operators (see for example Weis, [37] and [38]). Again this problem was considered mainly for the case of bounded operators between Banach spaces. This thesis represents a contribution to knowledge in the sense that by considering the functions Ξ', Ξ' and Ξ'β, as well as the minimum modulus function in the more general setting of unbounded linear operators between normed linear spaces, we obtain the classes of Fβ and Range Open operators which turn out to be closely related to the classes of Ο΄β and normally solvable operators respectively. We also define unbounded strictly cosingular operators and find that many of the classical results on Ο΄β, normally solvable and bounded strictly cosingular operators go through for Fβ, range open and unbounded strictly cosingular operators respectively. This ties up with work done by R. W. Cross and provides a workable framework within which to study Ο΄β and Ο΄β type operators in the much more. general setting of unbounded linear operators between normed linear spaces