Hankel operators and generalizations

grantor: University of TorontoIn this thesis, we investigate several properties of bounded Hankel operators in Hilbert space and study some operators which generalize Hankel operators. We prove that a bounded Toeplitz operator 'T' commutes with a Hankel operator 'Hz[variable phi]' if and only if ' T' = 'T'[variable phi] and [variable phi] is an affine function of the characteristic function of an "anti-symmetric" set of the unit circle. We also give a partial classification of algebraic Hankel operators, and some results on the reflexivity and transitivity of ultraweakly closed spaces of Hankel operators. Given a complex number [lambda], we introduce the class of [lambda]-Hankel operators as those that satisfy the operator equation 'S'*' X' - 'XS' = [lambda]'X', where ' S' is the unilateral forward shift. We investigate several properties of [lambda]-Hanker operators. In particular, we provide a sufficient condition for the compactness of a [lambda]-Hankel operator and a condition which is necessary for the boundedness of a [lambda]-Hankel operator. We also show that positivity of a [lambda]-Hankel operator is equivalent to a moment problem, whose solution gives necessary and sufficient conditions for boundedness of the operator. We also solve some other operator equations that involve the unilateral forward shift. We prove that certain spaces of non-invertible operators have the property that every compact subset of the complex plane containing zero is the spectrum of an operator in the space; the space of Hankel operators is of this kind, as is the space of [lambda]-Hankel operators for [lambda] purely imaginary. To finish, we study a generalization of Hankel operators to the Calkin algebra. We investigate some of the properties of this set, show that it is not an algebra and show that it is not the set consisting of compact perturbations of the set of Hankel operators.Ph.D