Abstract/Summary

The aim of this thesis is to study the properties of multiplicative Toeplitz operators with an emphasis on boundedness and spectral points. In particular, we consider these operators acting on the sequence space ` p and the Besicovitch function space B 2 N , in which case the operator is denoted by Mf and MF respectively. First, we present conditions for Mf to be bounded when acting from ` p to ` q for 1 ≤ p ≤ q ≤ ∞. From this investigation, a surprising connection with multiplicative number theory emerges; namely, that for a particular class of f, the operator norm is attained at the multiplicative elements in ` p . Furthermore, through the Bohr lift, we consider the implication of these results in the setting of classical Toeplitz operators. Secondly, we seek to establish the spectral properties of MF : B 2 N → B2 N . For a certain class of F, we present a new result which describes the spectrum (and point spectrum). In the case of general symbols, this is much more challenging. During the investigation we illustrate how, despite their similar construction, many of the mathematical tools used to establish the spectrum of Toeplitz operators cannot be used in this multiplicative setting.