It is proved that the set BpX Bessel sequences of order p in X is a Banach space;it is established that a sequence f = {fn}n∈Λ in X is a Bessel sequence of order p if and only if c>0 such that ‖∑n∈Λcnfn‖≤c‖{cn}‖q for all {cn}∈lq,where p-1+q-1=1;it is also proved that the spaces BpX and B(X*,lp) are linearly and isometrically isomorphic.

Operator perturbations of Bessel sequences in a Hilbert space

Aim To discuss some relationships between several bases for a Hilbert space H,including Hilbertian bases,Besselian bases and Riesz bases,Bessel sequences and wavelet Riesz bases in wavelet analysis.

The stability of Bessel sequences in Hilbert spaces under the perturbations with one operator is described,operator sequences or Bessel sequences are also discussed.

Characterizations of Bessel sequences and frames in a Hilbert space;

Bessel sequence of order ∞ in a Banach space;

It is shown that if {f_i}~∞_(i=1) be a Bessel sequence, frame or Riesz basis, respectively, {g_i}~∞_(i=1) is a sequence, {f_i}~∞_(i=1) and {g_i}~∞_(i=1) satisfy certain conditions, then {g_i}~∞_(i=1) be a Bessel sequence,frame or Riesz basis, respectively.