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In this paper, we start by showing that the intertwining relationship betweentwo minimal Markov semigroups acting on Hilbert spaces implies that anyrecurrent extensions, in the sense of It\^o, of these semigroups satisfy thesame intertwining identity. Under mild additional assumptions on theintertwining operator, we prove that the converse also holds. This connection,which relies on the representation of excursion quantities as developed byFitzsimmons and Getoor, enables us to give an interesting probabilisticinterpretation of intertwining relationships between Markov semigroups viaexcursion theory: two such recurrent extensions that intertwine share, under anappropriate normalization, the same local time at the boundary point. Moreover,in the case when one of the (non-self-adjoint) semigroup intertwines with theone of a quasi-diffusion, we obtain an extension of Krein's theory of stringsbyshowing that its densely defined spectral measure is absolutely continuouswith respect to the measure appearing in the Stieltjes representation of theLaplace exponent of the inverse local time. Finally, we illustrate our resultswith the class of positive self-similar Markov semigroups and also thereflected generalized Laguerre semigroups. For the latter, we obtain theirspectral decomposition and provide, under some conditions, a perturbed spectralgap estimate for its convergence to equilibrium.

Intertwining, Excursion Theory and Krein Theory of Strings for Non-self-adjoint Markov Semigroups