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$g$-frames and $g$-frame sequences in Hilbert spaces. (English) Zbl 1198.42032

Let $U$ and $V$ be two complex Hilbert spaces, let ${\left\{{V}_{j}\right\}}_{j\in J}$ be a sequence of closed subspaces of $V,$ where $J\subset ℤ,$ and let $L\left(U,{V}_{j}\right)$ be the collection of all bounded linear spaces from $U$ to ${V}_{j}·$ For a $g$-frame ${\left\{{{\Lambda }}_{j}:{{\Lambda }}_{j}\in L\left(U,{V}_{j}\right)\right\}}_{j\in J}$ for $V$ with respect to ${\left\{{V}_{j}\right\}}_{j\in J}$ (i.e. there exist $A,B>0$ such that

${A\parallel f\parallel }^{2}\le \sum _{j\in J}\parallel {{\Lambda }}_{j}{f\parallel }^{2}\le B{\parallel f\parallel }^{2}$

for all $f\in V\right)$ the authors find some relations between operators

$S:f\to \sum _{j\in J}{{\Lambda }}_{j}^{*}{{\Lambda }}_{j}f,\phantom{\rule{1.em}{0ex}}Q:{\left\{{g}_{j}\right\}}_{j\in J}\to \sum _{j\in J}{{\Lambda }}_{j}^{*}{g}_{j}$

and $A,B;$ moreover necessary and sufficient conditions for a $g$-frame in terms of $Q$ are given.

Further the authors define a $g$-frame sequence ${\left\{{{\Lambda }}_{j}\right\}}_{j\in J}$ for $U$ as a $g$-frame for

$W=\overline{\left\{{\sum }_{j\in {J}_{1}}{{\Lambda }}_{j}^{*}{g}_{j}\phantom{\rule{0.166667em}{0ex}}\text{for}\phantom{\rule{4.pt}{0ex}}\text{any}\phantom{\rule{4.pt}{0ex}}\text{finite}\phantom{\rule{0.166667em}{0ex}}{J}_{1}\subset J\phantom{\rule{0.166667em}{0ex}}\text{and}\phantom{\rule{4.pt}{0ex}}\text{any}\phantom{\rule{0.166667em}{0ex}}{g}_{j}\in {V}_{j},\phantom{\rule{0.222222em}{0ex}}j\in J\right\}}·$

They discuss that definition and consider the stability of a $g$-frame sequence under perturbation.

##### MSC:
 42C15 General harmonic expansions, frames 46C99 Inner product spaces, Hilbert spaces
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