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\(\epsilon\)-approximations and dynamical representations of Hilbert-Schmidt frames. (English) Zbl 1502.42023

Let \(H\) and \(K\) be Hilbert spaces. A Hilbert-Schmidt (HS) frame for \(H\) with respect to \(K\) is a countable family \({\mathcal G} = \left\{{\mathcal G}_k\right\}_{k=1}^\infty\) of bounded linear operators from \(H\) into the class of Hilbert-Schmidt operators on \(K\) for which there exist constants \(A, B > 0\) such that for any \(f\in H,\) \[ A\|f\|_H \leq \left(\sum_{k=1}^\infty\|{\mathcal G}_k(f)\|_{C_2}^2\right)^{\frac{1}{2}}\leq B\|f\|_H, \] where \(\|\cdot\|_{C_2}\) denotes the Hilbert-Schmidt norm of an operator. For the more general notion of von Neumann-Schatten \(p\)-frames in separable Banach spaces see [G. Sadeghi and A. Arefijamaal, Mediterr. J. Math. 9, No. 3, 525–535 (2012; Zbl 1266.46010)]. The authors introduce and study \(\epsilon\)-approximations for HS frames and also dynamical representations, that is, HS frames having the form \(\left\{{\mathcal G}_k\right\}_{k\in {\mathbb Z}} = \left\{{\mathcal G}_0\Gamma^k\right\}_{k\in {\mathbb Z}},\) where \(\Gamma\) is an invertible operator on \(H.\) In particular they provide a sufficient condition for an \(\epsilon\)-approximation of an HS frame to be an HS frame and find an upper bound for the distance between the frame operator of an HS frame and that of its \(\epsilon\)-approximation. They also prove that if an HS frame supports a dynamical representation then the same is true for its canonical dual.

MSC:

42C15 General harmonic expansions, frames
46C50 Generalizations of inner products (semi-inner products, partial inner products, etc.)

Citations:

Zbl 1266.46010
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References:

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