Resumen de Integral representation of functions on sectors, functional calculus and norm estimates

Khristo N. Boyadzhiev

• We find an explicit integral representation for bounded holomorphic functions $f(z)$ on sectors $\vert Arg (z)\vert < \psi$ in terms of the kernel $z(z +\lambda)^{-2}$ and present some applications to operator theory. Namely, given a sectorial operator $A$ we define the functional calculus $A\rightarrow f(A)$ and find pointwise estimates and moment type inequalities for $\parallel f(A)x\parallel$. We show that sectorial operators have a bounded $H^\infty-$ functional calculus on a dense subspace. We also find exact estimates for the norm $\parallel e^{-\lambda A}\parallel$ of analytic semigroups.