In this paper, we intend to define and study concepts of weight and weighted spaces of holomorphic (analytic) functions on the upper half plane. We study two special classes of these spaces of holomorphic functions on the upper half plane. Firstly, we prove these spaces of holomorphic functions on the upper half plane endowed with weighted norm supremum are Banach spaces. Then, we investigate t...

Let g and φ be holomorphic maps on D such that φ(D) ⊂ D. Define Volterra composition operators Jg,φ and Ig,φ induced by g and φ as Jg,φf(z) = Z z 0 (f ◦ φ) (ζ) (g ◦ φ)′ (ζ) dζ and Ig,φf(z) = Z z 0 (f ◦ φ)′ (ζ) (g ◦ φ) (ζ) dζ for z ∈ D and f ∈ H(D), the space of holomorphic functions on D. In this paper, we characterize boundedness and compactness of these operators acting between weighted Bergm...

In this paper, we establish bounds on the norm of multiplication operators on the Bloch space of the unit disk via weighted composition operators. In doing so, we characterize the isometric multiplication operators to be precisely those induced by constant functions of modulus 1. We then describe the spectrum of the multiplication operators in terms of the range of the symbol. Lastly, we identi...

In this paper, we establish bounds on the norm of multiplication operators on the Bloch space of the unit disk via weighted composition operators. In doing so, we characterize the isometric multiplication operators to be precisely those induced by constant functions of modulus 1. We then describe the spectrum of the multiplication operators in terms of the range of the symbol. Lastly, we identi...

Let V be a real valued potential that is smooth everywhere on R3, except at a periodic, discrete set of points, where it has singularities of the Coulomb form Z/r. We assume that the potential V is periodic with period lattice L. We study the spectrum of the Schrödinger operator H = −∆ + V acting on the space of Bloch waves with arbitrary, but fixed, wavevector k. Let T := R3/L. Let u be an eig...