Comment on “Continuous g-Frame in Hilbert -Modules”

Abstract and Applied Analysis ◽

10.1155/2013/243453 ◽

2013 ◽

Vol 2013 ◽

pp. 1-2

Author(s):

Zhong-Qi Xiang

Keyword(s):

Riesz Basis ◽

Hilbert Modules

The continuous g-frames in Hilbert -modules were introduced and investigated by Kouchi and Nazari (2011). They also studied the continuous g-Riesz basis and a characterization for it was presented by using the synthesis operator. However, we found that there is an error in the proof. The purpose of this paper is to improve their result by introducing the so-called modular continuous g-Riesz basis.

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Equivalency Relations between Continuous g-Frames and Stability of Alternate Duals of Continuous g-Frames in Hilbert -Modules

Journal of Applied Mathematics ◽

10.1155/2013/192732 ◽

2013 ◽

Vol 2013 ◽

pp. 1-11

Author(s):

Zhong-Qi Xiang

Keyword(s):

Hilbert Spaces ◽

Riesz Basis ◽

Sufficient Conditions ◽

Stability Result ◽

Necessary And Sufficient Conditions ◽

Hilbert Modules ◽

Necessary And Sufficient ◽

Existing Result

We introduce the modular continuous g-Riesz basis to improve one existing result for continuous g-Riesz basis in Hilbert -modules, and then we study the equivalency relations between continuous g-frames in Hilbert -modules, and, in particular, we obtain two necessary and sufficient conditions under which two continuous g-frames are similar. Finally, we generalize a stability result for alternate duals of g-frames in Hilbert spaces to alternate duals of continuous g-frames in Hilbert -modules.

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A criterion of Riesz basis property for L2 space

10.1063/1.4854776 ◽

2013 ◽

Author(s):

A. M. Sarsenbi

Keyword(s):

Riesz Basis ◽

Basis Property ◽

Riesz Basis Property

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A nonlocal problem for a differential operator of even order with involution

Journal of Applied Analysis ◽

10.1515/jaa-2020-2026 ◽

2020 ◽

Vol 26 (2) ◽

pp. 297-307

Author(s):

Petro I. Kalenyuk ◽

Yaroslav O. Baranetskij ◽

Lubov I. Kolyasa

Keyword(s):

Differential Operator ◽

Differential Equations ◽

Ordinary Differential Equations ◽

Riesz Basis ◽

Existence And Uniqueness ◽

Spectral Properties ◽

Nonlocal Problem ◽

Even Order

AbstractWe study a nonlocal problem for ordinary differential equations of {2n}-order with involution. Spectral properties of the operator of this problem are analyzed and conditions for the existence and uniqueness of its solution are established. It is also proved that the system of eigenfunctions of the analyzed problem forms a Riesz basis.

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Characterization of the potential smoothness of one-dimensional Dirac operator subject to general boundary conditions and its Riesz basis property

Journal of Mathematical Analysis and Applications ◽

10.1016/j.jmaa.2016.09.068 ◽

2017 ◽

Vol 447 (1) ◽

pp. 84-108 ◽

Cited By ~ 2

Author(s):

İlker Arslan

Keyword(s):

Dirac Operator ◽

Boundary Conditions ◽

Riesz Basis ◽

Basis Property ◽

General Boundary ◽

General Boundary Conditions ◽

One Dimensional ◽

Riesz Basis Property

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New characterizations of g-frames and g-Riesz bases

International Journal of Wavelets Multiresolution and Information Processing ◽

10.1142/s0219691318500534 ◽

2018 ◽

Vol 16 (06) ◽

pp. 1850053 ◽

Cited By ~ 2

Author(s):

Dongwei Li ◽

Jinsong Leng ◽

Tingzhu Huang

Keyword(s):

Riesz Basis ◽

Riesz Bases ◽

Gram Matrix ◽

Necessary Condition ◽

Topological Properties ◽

Bessel Sequence ◽

Bessel Sequences ◽

Sufficient And Necessary Condition

In this paper, we give some new characterizations of g-frames, g-Bessel sequences and g-Riesz bases from their topological properties. By using the Gram matrix associated with the g-Bessel sequence, we present a sufficient and necessary condition under which the sequence is a g-Bessel sequence (or g-Riesz basis). Finally, we consider the excess of a g-frame and obtain some new results.

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Exponential stability of variable coefficients Rayleigh beams under boundary feedback controls: a Riesz basis approach

Systems & Control Letters ◽

10.1016/s0167-6911(03)00205-6 ◽

2004 ◽

Vol 51 (1) ◽

pp. 33-50 ◽

Cited By ~ 15

Author(s):

Jun-min Wang ◽

Gen-qi Xu ◽

Siu-Pang Yung

Keyword(s):

Exponential Stability ◽

Riesz Basis ◽

Variable Coefficients ◽

Feedback Controls ◽

Boundary Feedback

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Hilbert \(C^{*}\)-modules in which all relatively strictly closed submodules are complemented

Glasnik Matematicki ◽

10.3336/gm.56.2.08 ◽

2021 ◽

Vol 56 (2) ◽

pp. 343-374

Author(s):

Boris Guljaš ◽

Keyword(s):

Hilbert Space ◽

Hilbert Modules ◽

Strict Topology ◽

Direct Sums ◽

Essential Ideal ◽

Closed Submodule ◽

Closed Submodules ◽

Hereditary Algebras ◽

Multiplier Algebras

We give the characterization and description of all full Hilbert modules and associated algebras having the property that each relatively strictly closed submodule is orthogonally complemented. A strict topology is determined by an essential closed two-sided ideal in the associated algebra and a related ideal submodule. It is shown that these are some modules over hereditary algebras containing the essential ideal isomorphic to the algebra of (not necessarily all) compact operators on a Hilbert space. The characterization and description of that broader class of Hilbert modules and their associated algebras is given. As auxiliary results we give properties of strict and relatively strict submodule closures, the characterization of orthogonal closedness and orthogonal complementing property for single submodules, relation of relative strict topology and projections, properties of outer direct sums with respect to the ideals in \(\ell_\infty\) and isomorphisms of Hilbert modules, and we prove some properties of hereditary algebras and associated hereditary modules with respect to the multiplier algebras, multiplier Hilbert modules, corona algebras and corona modules.

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