Essential norm of some extensions of the generalized composition operators between kth weighted-type spaces

Stevo Stević

doi:10.1186/s13660-017-1493-x

Abstract

We calculate the essential norm of some extensions of the generalized composition operators between kth weighted-type spaces on the unit disk in the complex plane, considerably extending some results in the literature.

Highlights

Let D be the open unit disk in the complex plane C, H(D) the class of all holomorphic functions on D, and S(D) the class of all holomorphic self-maps of D.Let μ(z) be a positive continuous function on D and k ∈ N
If k ∈ N, it is easy to see that bWμ(k) (·) is a semi-norm on Wμ(k). It is not a norm on the space since from bWμ(k) (f ) = it follows that f (k)(z) =, z ∈ D, and f (z) = pk– (z), where pk– is a polynomial of degree at most k –. It is a norm on the quotient space Wμ(k)/Pk, where Pk– is the space of all polynomials of degree less than or equal k
From ( ) and ( ) it follows that f + Pk– Wμ(k)/Pk– = bWμ(k) (f ), this fact along with the above mentioned algebraic isomorphism shows that the spaces (Wμ(k)/Pk, · ) Wμ(k)/Pk– and (Wμ(k,k) (D), bWμ(k) (·)) are isometrically isomorphic, that is, Wμ(k)/Pk– ∼= Wμ(k,k) (D)

Summary

Introduction

Let D be the open unit disk in the complex plane C, H(D) the class of all holomorphic functions on D, and S(D) the class of all holomorphic self-maps of D. From ( ) and ( ) it follows that f + Pk– Wμ(k)/Pk– = bWμ(k) (f ), this fact along with the above mentioned algebraic isomorphism shows that the spaces (Wμ(k)/Pk– , · ) Wμ(k)/Pk– and (Wμ(k,k) (D), bWμ(k) (·)) are isometrically isomorphic, that is, Wμ(k)/Pk– ∼= Wμ(k,k) (D). They can be identified, and we can regard it to be the same if we say f ∈ Wμ(k)/Pk– or f ∈ Wμ(k,k).

Let z ζk ζ

We have

Note also that

Proof First we prove

Proof First we prove that