Weighted Fractional Differentiation Composition Operators from Mixed-Norm Spaces to Weighted-Type Spaces

Let 𝔻 be an open unit disc in the complex plane ℂ and let φ:𝔻→𝔻 as well as u:𝔻→ℂ be analytic maps. For an analytic function f(z)=∑n=0∞anzn on 𝔻 the weighted fractional differentiation composition operator is defined as (Dφ,uβf)(z)=u(z)f[β](φ(z)), where β≥0, f[β](z)=∑n=0∞(Γ(n+1+β)/Γ(n+1))anzn, and f0z=fz. In this paper, we obtain a characterization of boundedness and compactness of weighted fractional differentiation composition operator from mixed-norm space Hp,q,ϕ to weighted-type space Hμ∞.

Weighted Composition Operator from Mixed Norm Space to Bloch-Type Space on the Unit Ball

We discuss the boundedness and compactness of the weighted composition operator from mixed norm space to Bloch-type space on the unit ball ofCn.

Multipliers on Generalized Mixed Norm Sequence Spaces

Given1≤p,q≤∞and sequences of integers(nk)kand(nk′)ksuch thatnk≤nk′≤nk+1, the generalized mixed norm spaceℓℐ(p,q)is defined as those sequences(aj)jsuch that((∑j∈Ik‍|aj|p)1/p)k∈ℓqwhereIk={j∈ℕ0 s.t. nk≤j<nk′},k∈ℕ0. The necessary and sufficient conditions for a sequenceλ=(λj)jto belong to the space of multipliers(ℓℐ(r,s),ℓ𝒥(u,v)), for different sequencesℐand𝒥of intervals inℕ0, are determined.

ESSENTIAL COMMUTATIVITY OF SOME INTEGRAL AND COMPOSITION OPERATORS

In general, multiplication of operators is not essentially commutative in an algebra generated by integral-type operators and composition operators. In this paper, we characterize the essential commutativity of the integral operators and composition operators from a mixed-norm space to a Bloch-type space, and give a complete description of the universal set of integral operators. Corresponding results for boundedness and compactness are also obtained.

On an Integral-Type Operator from Zygmund-Type Spaces to Mixed-Norm Spaces on the Unit Ball

The boundedness and compactness of an integral-type operator recently introduced by the author from Zygmund-type spaces to the mixed-norm space on the unit ball are characterized here.