Sesquilinear forms associated to sequences on Hilbert spaces

Corso, Rosario

doi:10.1007/s00605-019-01310-9

Cited by 15 publications

(29 citation statements)

References 30 publications

(30 reference statements)

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“…The motivation behind this name is that when φ is a continuous frame, then T φ = S φ . The generalized frame operator has been studied in [21,22] in the discrete case and a preliminary extension to the continuous setting has been given in [18].…”

Section: The Generalized Frame Operator T φmentioning

confidence: 99%

“…The following characterization can be proved as in [21]. (ii) Ω φ is bounded from below by m, that is:…”

Section: The Generalized Frame Operator T φmentioning

confidence: 99%

“…It turns out that most results from [21,22] can be extended to the continuous case and that is one of the aims of this present paper. The results are reported in Sect.…”

Section: Introductionmentioning

confidence: 99%

“…In recent papers, one of us (RC) [21,22] has analyzed sesquilinear forms defined by sequences in Hilbert spaces and operators associated with them by means of representation theorems. In particular, he derived results about lower semi-frames and duality.…”

Section: Introductionmentioning

confidence: 99%

“…In this paper, we do not confine ourselves to extend results of [21,22], but actually we set two more goals. From one hand, given a lower semi-frame φ : X → H with D(C φ ) dense in H, we consider general powers T −k φ φ with k ≥ 0.…”

Section: Introductionmentioning

confidence: 99%

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Lower Semi-frames, Frames, and Metric Operators

2020

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This paper deals with the possibility of transforming a weakly measurable function in a Hilbert space into a continuous frame by a metric operator, i.e., a strictly positive self-adjoint operator. A necessary condition is that the domain of the analysis operator associated with the function be dense. The study is done also with the help of the generalized frame operator associated with a weakly measurable function, which has better properties than the usual frame operator. A special attention is given to lower semi-frames: indeed, if the domain of the analysis operator is dense, then a lower semi-frame can be transformed into a Parseval frame with a (special) metric operator.

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Section: The Generalized Frame Operator T φmentioning

confidence: 99%

“…The following characterization can be proved as in [21]. (ii) Ω φ is bounded from below by m, that is:…”

Section: The Generalized Frame Operator T φmentioning

confidence: 99%

“…It turns out that most results from [21,22] can be extended to the continuous case and that is one of the aims of this present paper. The results are reported in Sect.…”

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Lower Semi-frames, Frames, and Metric Operators

2020

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Generalized frame operator, lower semiframes, and sequences of translates

Corso

2023

Mathematische Nachrichten

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Given an arbitrary sequence of elements ξ=false{ξnfalse}n∈double-struckN$\xi =\lbrace \xi _n\rbrace _{n\in \mathbb {N}}$ of a Hilbert space false(scriptH,false⟨·,·false⟩false)$(\mathcal {H},\langle \cdot ,\cdot \rangle )$, the operator Tξ$T_\xi$ is defined as the operator associated to the sesquilinear form Ωξ(f,g)=∑n∈double-struckN⟨f,ξn⟩⟨ξn,g⟩$\Omega _\xi (f,g)=\sum _{n\in \mathbb {N}} \langle f , \xi _n\rangle \langle \xi _n , g\rangle$, for f,g∈false{h∈scriptH:∑n∈double-struckNfalse|⟨h,ξn⟩|2<∞false}$f,g\in \lbrace h\in \mathcal {H}: \sum _{n\in \mathbb {N}}|\langle h , \xi _n\rangle |^2<\infty \rbrace$. This operator is in general different from the classical frame operator but possesses some remarkable properties. For instance, Tξ$T_\xi$ is always self‐adjoint with regard to a particular space, unconditionally defined, and, when ξ is a lower semiframe, Tξ$T_\xi$ gives a simple expression of a dual of ξ. The operator Tξ$T_\xi$ and lower semiframes are studied in the context of sequences of integer translates of a function of L2(R)$L^2(\mathbb {R})$. In particular, an explicit expression of Tξ$T_\xi$ is given in this context, and a characterization of sequences of integer translates, which are lower semiframes, is proved.

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Some Notes About Distribution Frame Multipliers

Corso

Tschinke

2020

Landscapes of Time-Frequency Analysis

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Sesquilinear forms associated to sequences on Hilbert spaces

Cited by 15 publications

References 30 publications

Lower Semi-frames, Frames, and Metric Operators

Lower Semi-frames, Frames, and Metric Operators

Generalized frame operator, lower semiframes, and sequences of translates

Some Notes About Distribution Frame Multipliers

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