Continuous g-frame and g-Riesz sequences in Hilbert spaces

A continuous g-frame is a generalization of g-frames and continuous frames, but they behave much differently from g-frames due to the underlying characteristic of measure spaces. Now, continuous g-frames have been extensively studied, while continuous g-sequences such as continuous g-frame sequence, g-Riesz sequences, and continuous g-orthonormal systems have not. This paper addresses continuous g-sequences. It is a continuation of Zhang and Li, in Numer. Func. Anal. Opt., 40 (2019), 1268-1290, where they dealt with g-sequences. In terms of synthesis and Gram operator methods, we in this paper characterize continuous g-Bessel, g-frame, and g-Riesz sequences, respectively, and obtain the Pythagorean theorem for continuous g-orthonormal systems. It is worth that our results are similar to the case of g-ones, but their proofs are nontrivial. It is because the definition of continuous g-sequences is different from that of g-sequences due to it involving general measure space.     

Fusion frames and <Emphasis Type="Italic">G</Emphasis>-frames in Banach spaces

Fusion frames and g-frames in Hilbert spaces are generalizations of frames, and frames were extended to Banach spaces. In this article we introduce fusion frames, g-frames, Banach g-frames in Banach spaces and we show that they share many useful properties with their corresponding notions in Hilbert spaces. We also show that g-frames, fusion frames and Banach g-frames are stable under small perturbations and invertible operators.     

Approximation of the inverse <Emphasis Type="Italic">G</Emphasis>-frame operator

In this paper, we introduce the concept of (strong) projection method for g-frames which works for all conditional g-Riesz frames. We also derive a method for approximation of the inverse g-frame operator which is efficient for all g-frames. We show how the inverse of g-frame operator can be approximated as close as we like using finite-dimensional linear algebra.     

Some Properties of K-Frames in Hilbert Spaces

K-frames were recently introduced by G?vru?a in Hilbert spaces to study atomic systems with respect to a bounded linear operator. From her discussions there are many differences between K-frames and ordinary frames, so in this paper we further discuss the interchangeability of two Bessel sequences with respect to a K-frame, where K is a bounded linear operator with closed range. We also give several methods to construct K-frames. In the end we discuss the stability of a more general perturbation for K-frame.     

Expansion of frames to tight frames

We show that every Bessel sequence （and therefore every frame） in a separable Hilbert space can be expanded to a tight frame by adding some elements. The proof is based on a recent generalization of the frame concept, the g-frame, which illustrates that g-frames could be useful in the study of frame theory. As an application, we prove that any Gabor frame can be expanded to a tight frame by adding one window function.     

Characterizations of Disjointness of g-Frames and Constructions of g-Frames in Hilbert Spaces

Disjointness of frames in Hilbert spaces is closely related with superframes in Hilbert spaces and it also plays an important role in construction of superframes and frames, which were introduced and studied by Han and Larson. \(G\) -frame is a generalization of frame in Hilbert spaces, which covers many recent generalizations of frame in Hilbert spaces. In this paper, we study the \(g\) -frames in Hilbert spaces. We focus on the characterizations of disjointness of \(g\) -frames and constructions of \(g\) -frames. All types of disjointness are firstly characterized in terms of disjointness of frames induced by \(g\) -frames, then are characterized in terms of certain orthogonal projections. Finally we use disjoint \(g\) -frames to construct \(g\) -frames.     

Invertible sequences of bounded linear operators

In this paper, we study the invertibility of sequences consisting of finitely many bounded linear operators from a Hilbert space to others. We show that a sequence of operators is left invertible if and only if it is a g-frame. Therefore, our result connects the invertibility of operator sequences with frame theory.     

Notes on Pointfree Disconnectivity with a Ring-theoretic Slant

We give characterizations of extremally disconnected frames, basically disconnected frames and F-frames L in terms of ring-theoretic properties of the ring RL\mathcal{R}L of continuous real-valued functions on L. Emanating from these are new (and purely ring-theoretic) proofs that a frame is extremally disconnected, basically disconnected or an F-frame iff the same holds for its Čech-Stone compactification.     

The Stone-Čech compactification of a partial frame via ideals and cozero elements

A partial frame is a meet-semilattice in which certain designated subsets are required to have joins, and finite meets distribute over these. The designated subsets are specified by means of a so-called selection function, denoted by S ; these partial frames are called S-frames.

We construct free frames over S-frames using appropriate ideals, called S-ideals. Taking S-ideals gives a functor from S-frames to frames. Coupled with the functor from frames to S-frames that takes S-Lindelöf elements, it provides a category equivalence between S-frames and a non-full subcategory of frames. In the setting of complete regularity, we provide the functor taking S-cozero elements which is right adjoint to the functor taking S-ideals. This adjunction restricts to an equivalence of the category of completely regular S-frames and a full subcategory of completely regular frames. As an application of the latter equivalence, we construct the Stone-? ech compactification of a completely regular S-frame, that is, its compact coreflection in the category of completely regular S-frames.

A distinguishing feature of the study of partial frames is that a small collection of axioms of an elementary nature allows one to do much that is traditional at the level of frames or locales and of uniform or nearness frames. The axioms are sufficiently general to include as examples of partial frames bounded distributive lattices, σ-frames, κ-frames and frames.     

The congruence frame and the Madden quotient for partial frames

Nuclei and prenuclei have proved popular for providing quotients in frame theory; moreover the collection of all nuclei is itself a frame with useful functorial properties. Another natural approach to quotients in the frame setting, much used by algebraists, uses congruences as a tool. In partial frames, nuclei no longer suffice for constructing quotients, but congruences do, and it is to these that we turn in this paper. Partial frames are meet-semilattices in which not all subsets need have joins; a selection function, \(\mathcal {S}\), specifies, for all meet-semilattices, certain subsets under consideration; an \(\mathcal {S}\)-frame then must have joins of all such subsets and binary meet must distribute over these. Examples of these are \(\sigma \)-frames, \(\kappa \)-frames and frames themselves. The first part of this paper investigates the structure and functorial properties of the congruence frame of a partial frame; the second constructs the least dense quotient, which we call the Madden quotient, in three different ways. We include some functoriality properties in the subcategory of partial frames with skeletal maps.     

Concerning P-frames, essential P-frames, and strongly zero-dimensional frames

We give characterizations of P-frames, essential P-frames and strongly zero-dimensional frames in terms of ring-theoretic properties of the ring of continuous real-valued functions on a frame. We define the m-topology on the ring RL{\mathcal{R}L} and show that if L belongs to a certain class of frames properly containing the spatial ones, then L is a P-frame iff every idealof RL{\mathcal{R}L} is m-closed. We define essential P-frames (analogously to their spatial antecedents) and show that L is a proper essential P frame iff all the nonmaximal prime ideals of RL{\mathcal{R}L} are contained in one maximal ideal. Further, we show that L is strongly zero-dimensional iff RL{\mathcal{R}L} is a clean ring, iff certain types of ideals of RL{\mathcal{R}L} are generated by idempotents.     

Frame expansions in separable Banach spaces

Banach frames are defined by straightforward generalization of (Hilbert space) frames. We characterize Banach frames (and Xd-frames) in separable Banach spaces, and relate them to series expansions in Banach spaces. In particular, our results show that we can not expect Banach frames to share all the nice properties of frames in Hilbert spaces.     

Complete nearness σ-frames

We show that complete strong nearness σ-frames are exactly the cozero parts of complete separable strong Lindelöf nearness frames. We also relate nearness σ-frames and metric σ-frames and show that every metric σ-frame admits an admissible nearness such that it is complete as a metric σ-frame if and only if it is complete in this admissible nearness.     

On some properties of continuous <Emphasis Type="Italic">g</Emphasis>-frames and Riesz-type continuous <Emphasis Type="Italic">g</Emphasis>-frames

In this paper we provide some necessary and sufficient conditions under which, a family of bounded operators is a continuous g-frame (Riesz-type continuous g-frame). Also, we study stability of duals of continuous g-frames.     

Operator Characterizations,Rigidity and Constructions of (Ω,μ)-Frames

In this paper, first, we give some operator characterizations of (Ω,μ)-frames. We obtain that normalized tight (Ω,μ)-frames are precisely the (Ω,μ)-frames which are unitary equivalent to normalized tight (Ω,μ)-frames for some closed subspace ? of L²(Ω,μ) and (Ω,μ)-frames are precisely the (Ω,μ)-frames which are similar to normalized tight (Ω,μ)-frames for some closed subspace ? of L²(Ω,μ). We also characterize the alternate dual (Ω,μ)-frames through an operator equation. Then we establish some rigidity in the pairs of dual (super) (Ω,μ)-frames related with disjointness. Finally, we consider the constructions of (Ω,μ)-frames, including the constructions of new (Ω,μ)-frames or new pair of dual (Ω,μ)-frames from known ones and the constructions of the canonical dual of a (Ω,μ)-frame under certain conditions, which generalize the corresponding results on discrete frames.     

Pair frames in Hilbert $$\varvec{C^*}$$-modules

In this paper, we introduce pair frames in Hilbert \(C^*\)-modules and show that they share many useful properties with their corresponding notions in Hilbert spaces. We also obtain the necessary and sufficient conditions for a standard Bessel sequence to construct a pair frame and get the necessary and sufficient conditions for a Hilbert \(C^*\)-module to admit a pair frame with a symbol and two standard Bessel sequences. Moreover by generalizing some of the results obtained for Bessel multipliers in Hilbert \(C^*\)-modules to pair frames and considering the stability of pair frames under invertible operators, we construct new pair frames and show that pair frames are stable under small perturbations.     

Critical Pairs of Sequences of a Mixed Frame Potential

The classical frame potential in a finite-dimensional Hilbert space has been introduced by Benedetto and Fickus, who showed that all finite unit-norm tight frames can be characterized as the minimizers of this energy functional. This was the starting point of a series of new results in frame theory, related to finding tight frames with determined lengths. The frame potential has been studied in the traditional setting as well as in the finite-dimensional fusion frame context. In this work we introduce the concept of mixed frame potential, which generalizes the notion of the Benedetto-Fickus frame potential. We study properties of this new potential, and give the structure of its critical pairs of sequences on a suitable restricted domain. For a given sequence {α _m } _{m=1,…, N} in K, where K is ? or ?, we obtain necessary and sufficient conditions in order to have a dual pair of frames {f _m } _{m=1,…, N} , {g _m } _{m=1,…, N} such that ? f _m , g _m  ? = α _m for all m = 1,…, N.     

<Emphasis Type="Italic">K</Emphasis>-fusion Frames and the Corresponding Generators for Unitary Systems

Motivated by K-frames and fusion frames, we study K-fusion frames in Hilbert spaces. By the means of operator K, frame operators and quotient operators, several necessary and sufficient conditions for a sequence of closed subspaces and weights to be a K-fusion frame are obtained, and operators preserving K-fusion frames are discussed. In particular, we are interested in the K-fusion frames with the structure of unitary systems. Given a unitary system which has a complete wandering subspace, we give a necessary and sufficient condition for a closed subspace to be a K-fusion frame generator.     

Sums of Hilbert space frames

We give simple necessary and sufficient conditions on Bessel sequences {fi} and {gi} and operators L₁, L₂ on a Hilbert space H so that {L₁fi+L₂gi} is a frame for H. This allows us to construct a large number of new Hilbert space frames from existing frames.