Composition operators between growth spaces‎ ‎on circular and strictly convex domains in complex Banach spaces‎

Document Type : Research Articles

Authors

¹ Aligudarz Branch, Islamic Azad University

² Khoy Faculty of Engineering, Urmia University, Urmia, Iran

Abstract

‎Let

Ω_{X}

be a bounded‎, ‎circular and strictly convex domain in a complex Banach space

X

‎,
‎and

H (Ω_{X})

be the space of all holomorphic functions from

Ω_{X}

C

‎.
‎The growth space

A^{ν} (Ω_{X})

consists of all

f \in H (Ω_{X})

‎
‎such that

| f (x) | ⩽ C ν (r_{Ω_{X}} (x)), x \in Ω_{X},

‎
‎for some constant

C > 0

‎, ‎whenever

r_{Ω_{X}}

is the Minkowski‎
‎functional on

Ω_{X}

and

ν ‎ : ‎ [0, 1) \to (0, \infty)

‎
‎is a nondecreasing‎, ‎continuous and unbounded function‎.
‎For complex Banach spaces

X

and

Y

‎
‎and a holomorphic map

φ : Ω_{X} \to Ω_{Y}

‎, ‎put‎
‎

C_{φ} (f) = f \circ φ, f \in H (Ω_{Y})

‎.
‎We characterize those

φ

for which the composition operator‎
‎

C_{φ} : A^{ω} (Ω_{Y}) \to A^{ν} (Ω_{X})

is a bounded or compact operator‎.

Keywords