Multishifts in Hilbert spaces

Abstract

We introduce and study a multishift structure in a Hilbert space. This structure is a noncommutative analog of the (simple one-sided) shift operator, well known in function theory and functional analysis. Subspaces invariant under the multishift are described. A theorem on the factorization into an inner and an outer factor is established for operators commuting with the multishift.

Applications of Affine Systems of Walsh Type to Generate Smooth Basis

“…From this, we see that the value of Further, note that all the arguments carried out remain valid for the function (1 ), 0 1. (1 )…”

“…The operator structure 01 { , } WW has an important property that lies in the fact that given two operators form a multishift in the Hilbert space 2 0 L from the viewpoint of the following definition, which was introduced and studied earlier [1].…”

Riesz Basicity for General Systems of Functions

“…Theorem 3 (see [9]). A necessary and sufficient condition for an affine system { ( )} ∞ =0 to be complete in 2 (0, 1) space is that analytic function ( ) is outer function.…”

Линейные Алгоритмы Аффинного Синтеза В Пространстве Лебега $L^1[0,1]$

“…Доказательство леммы 4. Для натурального числа n в силу равенства (11) имеем Для этого рассмотрим операторную структуру мультисдвига в пространстве L 1 , аналогичную структуре мультисдвига в гильбертовом пространстве, введенной в работе автора [20]. Для удобства обозначений мы будем считать, что каждая функция f ∈ L 1 продолжена нулем за пределы единичного отрезка, так что…”