Invariant subspaces of H2(T2)\mathcal{H}^2(\mathbb{T}^2) and L2(T2)L^2(\mathbb{T}^2) preserving compatibility

Operators of multiplication by independent variables on the space of square summable functions over the torus and its Hardy subspace are considered. Invariant subspaces where the operators are compatible are described.Comment: 17 pages, 3 figure

On the decomposition of families of quasinormal operators

The canonical injective decomposition of a jointly quasinormal family of operators is given. Relations between the decomposition of a quasinormal operator T and the decomposition of a partial isometry in the polar decomposition of T are described. The decomposition of pairs of commuting quasinormal partial isometries and its applications to pairs of commuting quasinormal operators is shown. Examples are given

On the decomposition of families of quasinormal operators

Tyt. z nagłówka.Bibliogr. s. 437.The canonical injective decomposition of a jointly quasinormal family of operators is given. Relations between the decomposition of a quasinormal operator T and the decomposition of a partial isometry in the polar decomposition of T are described. The decomposition of pairs of commuting quasinormal partial isometries and its applications to pairs of commuting quasinormal operators is shown. Examples are given.Dostępny również w formie drukowanej.KEYWORDS: multiple canonical decomposition, quasinormal operators, partial isometry

On dilation and commuting liftings of nn-tuples of commuting Hilbert space contractions

The $n$-tuples of commuting Hilbert space contractions are considered. We give a model of a commuting lifting of one contraction and investigate conditions under which a commuting lifting theorem holds for an $n$-tuple. A series of such liftings leads to an isometric dilation of the $n$-tuple. The method is tested on some class of triples motivated by Parrotts example. It provides also a new proof of the fact that a positive definite $n$-tuple has an isometric dilation

Generalized powers and measures

Using the winding of measures on torus in “rational directions” special classes of unitary operators and pairs of isometries are defined. This provides nontrivial examples of generalized powers. Operators related to winding Szegö-singular measures are shown to have specific properties of their invariant subspaces