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Weighted Shift

Let H=l^2, (alpha_n) be a bounded sequence of complex numbers, and (xi_n) be the (usual) standard orthonormal basis of H, that is, (xi_n)(m)=delta_(nm), n,m in N, where delta_(mn) denoted the Kronecker delta, so that

zeta=sum_(n=1)^infty<zeta,xi_n>xi_n

for any zeta in H. Then the operator T in B(H) defined by Txi_n=alpha_nxi_(n+1) is called a weighted shift with the weights (alpha_n). Then ||T||=sup_(n)|alpha_n|,

r(T)=lim_(k)sup_(n)|product_(i=0)^(k-1)alpha_(n+i)|^(1/k),

and T^*xi_1=0 and T^*xi_n=alpha^__nxi_(n-1).

If alpha_n=1 for all n, then T is called unilateral shift operator.

See also

Unilateral Shift

This entry contributed by Mohammad Sal Moslehian

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References

Halmos, P. R. A Hilbert Space Problem Book. Princeton, NJ: Van Nostrand, 1967.

Referenced on Wolfram|Alpha

Weighted Shift

Cite this as:

Weisstein, Eric W., with contributions by Mohammad Sal Moslehian. "Weighted Shift." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/WeightedShift.html

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