Spectral theory in Hilbert space of self-similar Markov semigroups

Pierre Patie (Cornell University)

Frank Adams 2,

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.

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