Abstract
We consider a class of integral operators from weighted integral transforms to Dirichlet spaces. The boundedness and compactness of these operators from weighted integral transforms to Dirichlet spaces are characterized. We also compute norm of integral operators acting between these spaces.
Highlights
Let be the open unit disk in the complex plane C, dA(z) = 1 dxdy = 1 rdrd (z = x + iy = rei ) the normalized area measure on its boundary,, H∞ the space of all bounded holomorphic functions on with the norm ‖f ‖∞ = supz∈ f (z), H( ) the class of all holomorphic functions on, and the space of all complex Borel measures on
For α > 0, the family of weighted integral transforms is the collection of functions f ∈ H( ) which admits a representation of the form f (z) = ∫ Kx (z)d (x) (z ∈ ), (1)
Sharma is an assistant professor in the School of Mathematics, Shri Mata Vaishno Devi University, Katra, Jammu and Kashmir, India. He has completed his PhD in the subject of Mathematics from Department of Mathematics, University of Jammu, Jammu, India with specialization in Operators on spaces of holomorplic functions
Summary
Several criteria for compactness and a formula for exact norm of integral-type operators between weighted integral transforms to Dirichlet-type spaces are obtained. If (X, ‖ ⋅ ‖X ) and (Y, ‖ ⋅ ‖Y ) are Banach spaces, and T is a linear operator from X to Y, T is bounded if there exists a positive constant C such that‖T(f )‖Y ≤ C‖f ‖X for all f ∈ X and the operator norm of T is defined as ‖T‖X→Y = inf{C < 0 : ‖T(f )‖Y ≤ C‖f ‖X }. We denote by (X, Y) the set of all bounded linear operators from X to Y.
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