A functional calculus for a scalar perturbation of \(\partial /\partial z\) (Q1174981)

The papers considers the problem of determining when, for \(a_ 1\), \(a_ 2\) close to 0, 1 in \(L^ \infty\)-norm, a functional calculus exists for the operator \(a_ 1(-i\partial /\partial z)+a_ 2(-i\partial/\partial \bar z)\) acting on \(L^ 2(\mathbb{R}^ 2)\). The operator \(L\) is not normal and its spectrum is the whole complex plane; thus the usual spectral theory cannot be applied. The author shows that the boundedness of some natural functions of \(L\) is equivalent to certain quantitative conditions on a surface in \(\mathbb{C}^ 2\). He studies the restrictions that the functions \(a_ 1\), \(a_ 2\) must satisfy, and exhibits a class of functions satisfying them. Then he defines \(\phi(L)\) using as a model an integral representation of \(\phi(z)\); first in the case \(\phi\in C_ 0^ \infty(\mathbb{C})\), and then for functions which are bounded and holomorphic in a conical sector in \(\mathbb{C}^ 2\).