| Summary: | <p>The subject of this dissertation is the construction of a functional calculus for functions of two variables and pairs of commuting operators. </p>
<p>Analytic Besov functions in one variable in the context of operator theory appeared in the works of Vladimir V. Peller, Steven White, and Pascal Vitse. More recently, Charles Batty, Aleander Gomilko, and Yuri Tomilov offered a novel and unifying approach to constructing a functional calculus for the generators of bounded semigroups on Hilbert spaces and generators of bounded holomorphic semigroups on Banach spaces. Our main aim is to extend the latter construction to the setting of two commuting operators.</p>
<p>We begin by providing an overview of the basic theory of one-parameter strongly continuous semigroups, and the theory of Besov functional calculus in one variable. We establish a new result concerning the spectral features of the one-dimensional Besov calculus, and show that compositions of certain functions are in the one-dimensional Besov class, $\mathcal{B}$. </p>
<p>We define and characterise a two-dimensional analogue of the class $\mathcal B$, denoted by $\mathcal{B}^2$, and show that it shares many desirable characteristics with its one-dimensional counterpart. We prove that the class $\mathcal{B}^2$ is a Banach algebra containing the two-dimensional Hille-Phillips algebra as its proper subspace, and discuss some of its topological properties. We obtain a number of results on spectral decompositions and provide useful approximation techniques. We show that all functions in our class enjoy representations given in terms of a partial duality, and prove a convergence lemma for sequences of functions in the class.</p>
<p>We then construct a two-dimensional functional calculus for pairs of commuting semigroup generators and functions in the class $\mathcal{B}^2$. We show that our construction yields a bounded algebra homomorphism. We demonstrate that the resulting calculus extends the two-dimensional Hille-Phillips calculus, that it is compatible with the joint holomorphic and half-plane calculi, and consistent with the Besov calculus for functions in one variable. Finally, we obtain a spectral mapping theorem, and establish that our calculus is essentially unique. </p>
|
|---|