Sentence examples for weighted shift from inspiring English sources

Recall that given a bounded sequence ofpositive numbers α: α 1, α 2, α 3, … (called weights), the unilateral weighted shift W α associated with α is the operator on H = l 2 defined by W α e n : = α n e n + 1 for all n ≥ 1, where { e n } n = 1 ∞ is the canonical orthogonal basis for l 2. It is well known that W α is hyponormal if and only if α is monotonicallyincreasing.

Then T is a quasi-∗-class operator if and only if | α n | 2 ≤ | α n + 1 | | α n + 2 |. for all n ∈ Z. Lemma 2.11 Let T be a non-singular bilateral weighted shift operator with weights { α n : n ∈ Z }.

If U is the commutant of a strictly cyclic unilateral weighted shift with a monotonically decreasing weight sequence, then we show that there is a natural isomorphism of the Banach space of bounded linear maps from U into B(H) with the Banach space of bounded linear maps of the trace class operators into H, where H is a separable, infinite dimensional Hilbert space.

Given a bounded sequence of complex numbers { α n : n ∈ Z } (called weights), let T be the bilateral weighted shift on an infinite dimensional Hilbert space operator H = l 2, with the canonical orthonormal basis { e n : n ∈ Z }, defined by T e n = α n e n + 1 for all n ∈ Z. Lemma 2.10 Let T be a bilateral weighted shift operator with weights { α n : n ∈ Z }.

In the special case, we describe the commutant of balanced weighted shift only in terms of its weights.

Given a 2-variable weighted shift T whose core is of tensor form, we prove that LPCS is solvable for T if and only if LPCS is solvable for any power T m,n):="(T1m,T2n) (m,n⩾1).

We characterize joint k-hyponormality for 2-variable weighted shifts.

In addition, we prove two independent criteria for reflexivity of weighted shifts on directed trees.

N. P. Jewell and A. Lubin, "Systems of commuting weighted shifts and analytic function theory in several variables," Journal of Operator Theory, 1, 1979, 207-223.

The structure theorem shows that invertible hyperbolic composition operators are similar to cosubnormal operators built up from bilateral weighted shifts.

In particular, we show that weighted shifts which demonstrate a type of approximate self-similarity belong to ¯¯¯¯¯¯¯¯CSO\CSO.