## Spectral theory in Hilbert space of self-similar Markov semigroups

#### Pierre Patie (Cornell University)

In this talk, relying on the concept of intertwining, we describe the spectral decomposition of the class, denoted by $$\mathcal{K}$$, of markovian self-similar $$C_0$$-semigroup on $$L^2(R+)$$. We start by showing that any two elements in $$\mathcal{K}$$,  are intertwined with a closed linear (non-necessarily positive) operator. Relying on these commutation relationships, we characterize, for each semigroup in $$\mathcal{K}$$, the spectrum including the residual part, the spectral operator, expressed in terms of (weak) Fourier kernels, and its domain. This is a joint work with M. Savov.