Integr. Equ. Oper. Theory 88 (2017), 301–330
Published online July 13, 2017
Springer International Publishing AG 2017
and Operator Theory
Composition Operators on Hilbert Spaces
of Entire Functions of Several Variables
Minh Luan Doan, Le Hai Khoi and Trieu Le
Abstract. We study composition operators acting on Hilbert spaces of
entire functions in several variables. Depending on the deﬁning weight
sequence of the space, diﬀerent criteria for boundedness and compact-
ness are developed. Our work extends several known results on Fock
spaces and other spaces of entire functions.
Mathematics Subject Classiﬁcation. Primary 47B33; Secondary 32A15.
Keywords. Hilbert function spaces, Reproducing kernels,
Composition operators, Boundedness, Compactness.
Let H be a Hilbert space of functions on a certain set X. For a self-mapping
ϕ of X, the composition operator C
is deﬁned as C
f = f ◦ ϕ for f ∈H.
Researchers have been interested in the interaction between the function-
theoretic properties of ϕ and the operator-theoretic properties of C
speciﬁcally, the study of boundedness and compactness of composition oper-
ators on spaces of analytic functions has been an attractive topic in operator
theory. The books [7,16] are excellent sources for composition operators on
the Hardy spaces, Bergman spaces and other classical spaces over the unit
disk and the unit ball.
More than a decade ago, Carswell, MacCluer and Schuster  studied
composition operators on the Fock (also known as Segal–Bargmann) spaces
of entire functions over C
. They provided a complete description of the
boundedness and compactness together with a formula for the norm of C
A few years later, Guo and Izuchi  investigated composition operators
on more general Fock-type spaces over the complex plane. They obtained
Minh Luan Doan and Le Hai Khoi were supported in part by MOE’s AcRF Tier 1 Grant
M4011166.110 (RG24/13). Le Hai Khoi was also supported in part by MOE’s AcRF Tier
1 Grant M4011724.110 (RG128/16). Trieu Le was supported in part by the University of
Toledo’s Summer Research Awards and Fellowships Program.