The paper summarizes, clarifies and supplements results on decompositions and a model for pairs of commuting isometries. The geometrical model is given whenever it is possible. The remaining problem of describing the model for completely non-compatible pairs of isometries is reduced to completely non-compatible pairs of unilateral shifts. Moreover, subspaces generating incompatibility are determined.

Abstract. An operator T ∈ B(H) is said to have the Putnam-Fuglede commutativity property (PF property for short) if T * X = XJ for any X ∈ B(K, H) and any isometry J ∈ B(K) such that T X = XJ * . The main purpose of this paper is to examine if paranormal operators have the PF property. We prove that k * -paranormal operators have the PF property. Furthermore, we give an example of a paranormal without the PF property.

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Abstract. Pairs (V, V ) of commuting, completely non doubly commuting isometries are studied. We show, that the space of the minimal unitary extension of V (denoted by U ) is a closed linear span of subspaces reducing U to bilateral shifts. Moreover, the restriction of V to the maximal subspace reducing V to a unitary operator is a unilateral shift. We also get a new hyperreducing decomposition of a single isometry with respect to its wandering vectors which strongly corresponds with Lebesgue decomposition. Mathematics Subject Classification (2000). Primary 47B20; Secondary 47A13.

We propose an effective regularization strategy (CW-TaLaR) for solving continual learning problems. It uses a penalizing term expressed by the Cramer-Wold distance between two probability distributions defined on a target layer of an underlying neural network that is shared by all tasks, and the simple architecture of the Cramer-Wold generator for modeling output data representation. Our strategy preserves target layer distribution while learning a new task but does not require remembering previous tasks' datasets. We perform experiments involving several common supervised frameworks, which prove the competitiveness of the CW-TaLaR method in comparison to a few existing state-of-the-art continual learning models.