Bloch functions on the unit ball of a Banach space

Abstract

The space of Bloch functions on bounded symmetric domains is extended by considering Bloch functions f on the unit ball BE of finite and infinite dimensional complex Banach spaces in two different ways: by extending the classical Bloch space considering the boundness of (1− x 2 ) f ′ (x) on BE and by preserving the invariance of the correspondiing seminorm when we compose with automorphisms ϕ of BE. We study the connection between these spaces proving that they are different in general and prove that all bounded analytic functions on BE are Bloch functions in both ways.2010 Mathematics Subject Classification. Primary 30D45, 46E50. Secondary 46G20.

Weighted composition operators between the Bloch type space and the Hardy space on the unit ball of complex Banach space

“…Before we state the main result and the proof, we state some preliminary results. A. Miralles in [26] proved the following results.…”

“…Thus there is a function f ∈ B nat (B X )\H ∞ (B X ). Combining which and the lemma 1.2, we have that the lemma 1.3 holds (see [26,Proposition 3.6]).…”

“…A class of operators that has been thoroughly studied in recent years is the weighted composition operators for the prominent role these operators play in the determination of the isometries of several well-known spaces of analytic functions. We shall be considering two classes of Banach spaces on B X as target spaces H ∞ (B X ) and B nat (B X ) defined in [26], which are defined as follows.…”

Characterizations of Integral Type for Weighted Classes of Analytic Banach Function Spaces

“…Function spaces theory is developed, extended, and generalized to spaces of several complex variables ( [8][9][10][11][12][13]29]) also using quaternion-valued functions ( [21][22][23][24][30][31][32][33]). e intention of this study is to introduce a new type of analytic function spaces, which plays an interesting and global rule of studying complex function spaces.…”