**Cauchy Dual operators, kernel condition and quasi-Brownian isometries**

Let T be a bounded linear operator on a complex Hilbert space H. For a left invertible operator T, the Cauchy dual T’ of T is given by T’=T(T*T)^{-1}. We say that T is a 2-isometry, if I-2T*T+(T*)^{2}T^{2} = 0. An operator T is called a quasi-Brownian isometry if T is a 2-isometry such that △_{T}T = (△_{T})^{1/2} T (△_{T}) ^{1/2}, where △_{T} =T*T-I. Finally, we say that an operator T satisfies the kernel condition if T*T (ker T*) ⊆ ker T*. In the talk we discuss several interesting properties of the above mentioned operators.

The talk is based on joint work with A. Anand, S. Chavan and J. Stochel.