Geodesic
On perturbations of the isometric semigroup of shifts on the semiaxis
Algebra i analiz, Tome 22 (2010) no. 4, pp. 1-20.

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Perturbations $(\widetilde\tau_t)_{t\ge0}$ of the semigroup of shifts $(\tau_t)_{t\ge 0}$ on $L^2(\mathbb R_+)$ are studied under the assumption that $\widetilde\tau_t-\tau_t$ belongs to a certain Schatten–von Neumann class $\mathfrak S_p$ with $p\ge1$. It is shown that, for the unitary component in the Wold–Kolmogorov decomposition of the cogenerator of the semigroup $(\widetilde\tau_t)_{t\ge0}$, any singular spectral type may be achieved by $\mathfrak S_1$-perturbations. An explicit construction is provided for a perturbation with a given spectral type, based on the theory of model spaces of the Hardy space $H^2$. Also, it is shown that an arbitrary prescribed spectral type may be obtained for the unitary component of the perturbed semigroup by a perturbation of class $\mathfrak S_p$ with $p>1$.
Keywords: semigroup of shifts, trace-class perturbation, Schatten–von Neumann ideals, Hardy space, inner function. 
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G. G. Amosov; A. D. Baranov; V. V. Kapustin. On perturbations of the isometric semigroup of shifts on the semiaxis. Algebra i analiz, Tome 22 (2010) no. 4, pp. 1-20. http://geodesic.mathdoc.fr/item/AA_2010_22_4_a0/

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