 
Summary: ISOMETRIES AND SPECTRA OF MULTIPLICATION
OPERATORS ON THE BLOCH SPACE
ROBERT F. ALLEN AND FLAVIA COLONNA
Abstract. In this paper, we establish bounds on the norm of multiplication
operators on the Bloch space of the unit disk via weighted composition op
erators. In doing so, we characterize the isometric multiplication operators
to be precisely those induced by constant functions of modulus 1. We then
describe the spectrum of the multiplication operators in terms of the range of
the symbol. Lastly, we identify the isometries and spectra of a particular class
of weighted composition operators on the Bloch space.
1. Introduction
Let D denote the open unit disk in the complex plane. An analytic function f
on D is said to be Bloch if
f = sup
zD
(1  z
2
) f (z) < .
The mapping f f is a seminorm on the space B of Bloch functions, called the
Bloch space. Under the norm
