# Weighted Composition Operators on Weighted Hardy Spaces

Weighted Composition Operators on Weighted Hardy Spaces For functions $$\mu : {\mathbb {N}}_0 \rightarrow {\mathbb {C}}$$ μ : N 0 → C and $$\theta : {\mathbb {N}}_0 \rightarrow {\mathbb {N}}_0$$ θ : N 0 → N 0 , the weighted composition operator is the operator $$\varGamma _{\mu , \theta }: {\mathcal {H}}_\beta ^p \rightarrow {\mathbb {C}}^{{\mathbb {N}}_0}$$ Γ μ , θ : H β p → C N 0 defined by $$\varGamma _{\mu , \theta }(x) = \bigl ( \sum _{m=0}^k\mu (k-m) x_{\theta (m)}\bigr )_{k =0}^\infty$$ Γ μ , θ ( x ) = ( ∑ m = 0 k μ ( k - m ) x θ ( m ) ) k = 0 ∞ for all $$x = (x_k) \in {\mathcal {H}}_\beta ^p$$ x = ( x k ) ∈ H β p . This paper deals with the properties of weighted composition operators on weighted Hardy spaces $${\mathcal {H}}_\beta ^p$$ H β p for $$1 \le p \le \infty$$ 1 ≤ p ≤ ∞ . The necessary and sufficient conditions are investigated for a weighted composition operator to be compact. A compact subset of non-negative real numbers containing the essential norm of $$\varGamma _{\mu , \theta }$$ Γ μ , θ on $${\mathcal {H}}_\beta ^p$$ H β p is also computed. The condition under which weighted composition operators commute is also explored. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Computational Methods and Function Theory Springer Journals

# Weighted Composition Operators on Weighted Hardy Spaces

, Volume 19 (3) – Jun 17, 2019
22 pages

/lp/springer-journals/weighted-composition-operators-on-weighted-hardy-spaces-kJhW7mpxiO
Publisher
Springer Journals
Subject
Mathematics; Analysis; Computational Mathematics and Numerical Analysis; Functions of a Complex Variable
ISSN
1617-9447
eISSN
2195-3724
DOI
10.1007/s40315-019-00278-9
Publisher site
See Article on Publisher Site

### Abstract

For functions $$\mu : {\mathbb {N}}_0 \rightarrow {\mathbb {C}}$$ μ : N 0 → C and $$\theta : {\mathbb {N}}_0 \rightarrow {\mathbb {N}}_0$$ θ : N 0 → N 0 , the weighted composition operator is the operator $$\varGamma _{\mu , \theta }: {\mathcal {H}}_\beta ^p \rightarrow {\mathbb {C}}^{{\mathbb {N}}_0}$$ Γ μ , θ : H β p → C N 0 defined by $$\varGamma _{\mu , \theta }(x) = \bigl ( \sum _{m=0}^k\mu (k-m) x_{\theta (m)}\bigr )_{k =0}^\infty$$ Γ μ , θ ( x ) = ( ∑ m = 0 k μ ( k - m ) x θ ( m ) ) k = 0 ∞ for all $$x = (x_k) \in {\mathcal {H}}_\beta ^p$$ x = ( x k ) ∈ H β p . This paper deals with the properties of weighted composition operators on weighted Hardy spaces $${\mathcal {H}}_\beta ^p$$ H β p for $$1 \le p \le \infty$$ 1 ≤ p ≤ ∞ . The necessary and sufficient conditions are investigated for a weighted composition operator to be compact. A compact subset of non-negative real numbers containing the essential norm of $$\varGamma _{\mu , \theta }$$ Γ μ , θ on $${\mathcal {H}}_\beta ^p$$ H β p is also computed. The condition under which weighted composition operators commute is also explored.

### Journal

Computational Methods and Function TheorySpringer Journals

Published: Jun 17, 2019

### References

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