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Register a new userGeneralized frame operator, lower semi-frames and sequences of translates
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Rosario Corso
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Given a sequence of elements $xi={xi_n}_{nin mathbb{N}}$ of a Hilbert space, an operator $T_xi$ is defined as the operator associated to a sesquilinear form determined by $xi$. This operator is in general different to the classical frame operator but possesses some remarkable properties. For instance, $T_xi$ is self-adjoint (in an specific space), unconditionally defined and, when $xi$ is a lower semi-frame, $T_xi$ gives a simple expression of a dual of $xi$. The operator $T_xi$ and lower semi-frames are studied in the context of sequences of integer translates.
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Let $Sinmathcal{M}_d(mathbb{C})^+$ be a positive semidefinite $dtimes d$ complex matrix and let $mathbf a=(a_i)_{iinmathbb{I}_k}in mathbb{R}_{>0}^k$, indexed by $mathbb{I}_k={1,ldots,k}$, be a $k$-tuple of positive numbers. Let $mathbb T_{d}(mathbf a )$ denote the set of families $mathcal G={g_i}_{iinmathbb{I}_k}in (mathbb{C}^d)^k$ such that $|g_i|^2=a_i$, for $iinmathbb{I}_k$; thus, $mathbb T_{d}(mathbf a )$ is the product of spheres in $mathbb{C}^d$ endowed with the product metric. For a strictly convex unitarily invariant norm $N$ in $mathcal{M}_d(mathbb{C})$, we consider the generalized frame operator distance function $Theta_{( N , , , S, , , mathbf a)}$ defined on $mathbb T_{d}(mathbf a )$, given by $$ Theta_{( N , , , S, , , mathbf a)}(mathcal G) =N(S-S_{mathcal G }) quad text{where} quad S_{mathcal G}=sum_{iinmathbb{I}_k} g_i,g_i^*inmathcal{M}_d(mathbb{C})^+,. $$ In this paper we determine the geometrical and spectral structure of local minimizers $mathcal G_0inmathbb T_{d}(mathbf a )$ of $Theta_{( N , , , S, , , mathbf a)}$. In particular, we show that local minimizers are global minimizers, and that these families do not depend on the particular choice of $N$.
We develop a natural generalization of vector-valued frame theory, we term operator-valued frame theory, using operator-algebraic methods. This extends work of the second author and D. Han which can be viewed as the multiplicity one case and extends to higher multiplicity (e.g., multiframes) their dilation approach. We prove several results for operator-valued frames concerning their parametrization, duality, disjointeness, complementarity, and composition and the relationship between the two types of similarity (left and right) of such frames. We then apply these notions to prove that the collection of multiframe generators for the action of a discrete group on a Hilbert space is norm pathwise-connected precisely when the von Neumann algebra generated by the right representation of the group has no minimal projections. The proof is obtained by parametrizing this collection by a class of partial isometries in a larger von Neumann algebra. In the multiplicity one case this class reduces to the unitary class which is path-connected in norm, but in the infinite multiplicity case this class is path connected only in the strong operator topology and the proof depends on properties of tensor product slice maps.
For a given finitely generated shift invariant (FSI) subspace $cWsubset L^2(R^k)$ we obtain a simple criterion for the existence of shift generated (SG) Bessel sequences $E(cF)$ induced by finite sequences of vectors $cFin cW^n$ that have a prescribed fine structure i.e., such that the norms of the vectors in $cF$ and the spectra of $S_{E(cF)}$ is prescribed in each fiber of $text{Spec}(cW)subset T^k$. We complement this result by developing an analogue of the so-called sequences of eigensteps from finite frame theory in the context of SG Bessel sequences, that allows for a detailed description of all sequences with prescribed fine structure. Then, given $0<alpha_1leq ldotsleq alpha_n$ we characterize the finite sequences $cFincW^n$ such that $|f_i|^2=alpha_i$, for $1leq ileq n$, and such that the fine spectral structure of the shift generated Bessel sequences $E(cF)$ have minimal spread (i.e. we show the existence of optimal SG Bessel sequences with prescribed norms); in this context the spread of the spectra is measured in terms of the convex potential $P^cW_varphi$ induced by $cW$ and an arbitrary convex function $varphi:R_+rightarrow R_+$.
In this paper we consider two problems in frame theory. On the one hand, given a set of vectors $mathcal F$ we describe the spectral and geometrical structure of optimal completions of $mathcal F$ by a finite family of vectors with prescribed norms, where optimality is measured with respect to majorization. In particular, these optimal completions are the minimizers of a family of convex functionals that include the mean square error and the Bendetto-Fickus frame potential. On the other hand, given a fixed frame $mathcal F$ we describe explicitly the spectral and geometrical structure of optimal frames $mathcal G$ that are in duality with $mathcal F$ and such that the Frobenius norms of their analysis operators is bounded from below by a fixed constant. In this case, optimality is measured with respect to submajorization of the frames operators. Our approach relies on the description of the spectral and geometrical structure of matrices that minimize submajorization on sets that are naturally associated with the problems above.
It is known in Hilbert space frame theory that a Bessel sequence can be expanded to a frame. Contrary to Hilbert space situation, using a result of Casazza and Christensen, we show that there are Banach spaces and approximate Bessel sequences which can not be expanded to approximate Schauder frames. We characterize Banach spaces in which one can expand approximate Bessel sequences to approximate Schauder frames.
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