Pseudo-duals of continuous frames in Hilbert spaces (Q2080898)

This paper is on the study of the continuous frame \(F:\Omega\rightarrow \mathcal{H}\) defined for a Hilbert space \(\mathcal{H}\) satisfying \[ A_F\|f\|^2\le \int_\Omega |\langle f, F(\omega)\rangle|^2 d\mu(\omega)\le B_F\|f\|^2 \] for each \(f\in\mathcal{H}\) for some positive constants \(0<A_F\le B_F<\infty\), which is the generalization of the discrete (countable) frame for \(\mathcal{H}\) to the continuous setting. The definitions of tight frame (\(A_F=B_F\)), Parseval frame (\(A_F=B_F=1\)), and Bessel mapping (only right-hand side inequality holds), can be given analogously. Similarly, one can define the synthesis operator of \(F\) to be \(T_F(\varphi) = \int_\Omega\varphi(\omega)F(\omega)d\mu(\omega)\) and the analysis operator (its adjoint) of \(F\) to be \(T_F^\ast f(\omega)=\langle f, F(\omega)\rangle\). The frame operator is then given by \(S_F=T_FT_F^\ast\), which is invertible and positive when \(F\) is a continuous frame. The paper then turns to the study for a Bessel mapping \(F\), its duals (\(T_GT_F^\ast=\mathrm{Id}\)), its pseudo-duals (\(T_GT_F^\ast\) is invertible), and its approximate-duals (\(\|T_GT_F^\ast-\mathrm{Id}\|_{op}<1\)). More generally, the paper defines \(Q\)-dual (\(S_{G,Q,F}:=T_GQT_F^\ast =\mathrm{Id}\)), and similarly \(Q\)-pseudo-dual, \(Q\)-approximate-dual, for \(Q:L^2(\Omega,\mu)\rightarrow L^2(\Omega,\mu)\) of bounded linear operators. In particular, if \(Q(\varphi)=\mathcal{M}_m(\varphi)= m\varphi, \forall \varphi\in L^2(\Omega,\mu)\) for some Bessel multiplier \(m\in L^\infty(\Omega,\mu)\), then one can call it \(m\)-dual (respectively, \(m\)-pseudo-dual, \(m\)-approximate-dual). The paper gives a characterization theorem for a Bessel mapping \(F\) to be a continuous frame in terms of the \(Q\)-dual, \(Q\)-pseudo-dual, and \(Q\)-approximate-dual. In particular, if \(Q\) is self-adjoint, then \(F\) is a continuous frame if and only if \(F\) is a \(Q\)-approximate-dual of itself and if and only if there exists \(0<\varepsilon<1\) such that \((1-\varepsilon)\mathrm{Id}\le S_{F,Q,F}\le (1+\varepsilon)\mathrm{Id}\). If \(F\) is \(m\)-approximate dual of itself, the paper defines scalability (\(mF\) is Parseval) and nearly scalability (\(mF\) is nearly Parseval) of \(F\) and their relations to \(F\) when \(F\) is \(\varepsilon\)-nearly Parseval (\(A_F=1-\varepsilon, B_F=1+\varepsilon\)). Considering the relations between two Bessel mappings \(F\) and \(G\), the paper gives several characterizations and in particular, it shows that \(G\) is a \(Q\)-dual of \(F\) if and only if \(F\) is a continuous frame and a bounded linear operator \(R: L^2(\Omega,\mu)\rightarrow \mathcal{H}\) such that \(T_GQ=S_F^{-1}T_F+R(\mathrm{Id}-T_F^\ast S_F^{-1}T_F)\) exists. The next part of the paper is concerning the perturbation, closeness, and nearness of continuous frames. \(G\) is close to \(F\) if \(\|(T_F-T_G)\varphi\|\le\lambda \|T_F\varphi\|\) for all \(\varphi\in L^2(\Omega,\mu)\) for some \(\lambda\ge 0\). The infimum of such \(\lambda\)s is denoted by \(C(G,F)\) as the closeness bound. \(F\) and \(G\) are near if \(F\) is close to \(G\) and \(G\) is also close to \(F\). \(F\) is partial equivalent to \(G\) if there exists a bounded operator \(T\) such that \(TF=G\) and in addition if \(T\) is invertible, then \(F\) and \(G\) are said to be equivalent via \(T\). The paper gives the characterizations of \(G\) and \(F\) for their closeness, nearness, and equivalence, in terms of \(C(G,F) \), \(C(F,G) \), \(T \), \(\mathrm{ker} T_F \), \(\mathrm{ker} T_G\), etc. Finally, Riesz-type continuous frames, whose duals are unique (only dual is the canonical dual), are studied and characterized.