izvuzmathИзвестия высших учебных заведений. МатематикаIzvestiya Vysshikh Uchebnykh Zavedenii. Matematika0021-34462076-4626Казанский (Приволжский) федеральный университет10.26907/0021-3446-2026-7-80-94izvuzmath-323Research ArticleСтатьиНепрерывные K-g-фреймы в гильбертовых C*-модуляхContinuous K-g-frames in Hilbert C*-modulesЧешмаварДж.CheshmavarJ.

Джахангир Чешмавар

а/я 19395-4697, Тегеран

Jahangir Cheshmavar

P.O. Box: 19395-4697, Tehran

j-cheshmavar@pnu.ac.irБарадаранДж.BaradaranJ.

Джавад Барадаран

а/я 74135-111, Джахром

Javad Baradaran

P.O. Box: 74135-111, Jahrom

baradaran@jahromu.ac.irХоссейнпурА.HosseinpourA.

Асадолла Хоссейнпур

а/я 19395-4697, Тегеран

Asadollah Hosseinpour

P.O. Box: 19395-4697, Tehran

asadmath@student.pnu.ac.irУниверситет Пайаме НурИранPayame Noor UniversityIslamic Republic of IranУниверситет ДжахромаИранJahrom UniversityIslamic Republic of Iran202606082026078094Copyright © Чешмавар Д., Барадаран Д., Хоссейнпур А., 20262026Чешмавар Д., Барадаран Д., Хоссейнпур А.Cheshmavar J., Baradaran J., Hosseinpour A.Данная работа распространяется под лицензией Creative Commons Attribution 4.0.This work is licensed under a Creative Commons Attribution 4.0 License.https://izvuzmath.elpub.ru/jour/article/view/323

Целью данной статьи является объединение понятий $g$-фрейма и $K$-фрейма в гильбертовом $C^*$-модуле $U$ для оператора $K\in {\rm End}_{\mathcal{A}}^*(U)$, где ${\mathcal{A}}$ обозначает комплексную $C^{*}$-алгебру, а ${\rm End}_{\mathcal{A}}^*(U)$ — класс всех сопряжимых $\mathcal{A}$-линейных отображений на $U$. В результате вводится понятие $c$-$K$-$g$-фрейма для гильбертового $C^{*}$-модуля и приводятся его характеризации. Наконец, получены некоторые результаты о двойственных $c$-$K$-$g$-системах Бесселя в гильбертовых $C^{*}$-модулях.

This study aims at combining the concepts of $g$-frame and $K$-frame in a Hilbert $C^*$-module $U$ for an operator $K\in {\rm End}_{\mathcal{A}}^*(U)$, where ${\mathcal{A}}$ denotes a complex $C^{*}$-algebra and ${\rm End}_{\mathcal{A}}^*(U)$ denotes the class of all adjointable $\mathcal{A}$-linear maps on $U$. As a result, the concept of a $c$-$K$-$g$-frame for a Hilbert $C^{*}$-module is introduced and then characterizations on a $c$-$K$-$g$-frame are provided. Finally, some results on dual $c$-$K$-$g$-Bessel system in Hilbert $C^{*}$-modules are obtained.

гильбертов $C^*$-модуль$c$-$K$-$g$-фреймдвойственная $c$-$K$-$g$-система БесселяHilbert $C^*$-module$c$-$K$-$g$-framedual $c$-$K$-$g$-Bessel systemReferencesAli S.T., Antoine J.P., Gazeau J.P. Continuous frames in Hilbert spaces, Annals Phys. 222 (1), 1–37 (1993).Ali S.T., Antoine J.P., Gazeau J.P. Continuous frames in Hilbert spaces, Annals Phys. 222 (1), 1–37 (1993).Kaiser G. A Friendly guide to wavelets (Birkh¨auser, Boston, 1994).Kaiser G. A Friendly guide to wavelets (Birkh¨auser, Boston, 1994).Frank M., Larson D.R. Frames in Hilbert Cast -modules and Cast -algebras, J. Oper. Theory 48 (2), 273–314 (2000).Frank M., Larson D.R. Frames in Hilbert Cast -modules and Cast -algebras, J. Oper. Theory 48 (2), 273–314 (2000).Alijani A., Dehghan M.A. G-frames and their duals for Hilbert Cast -modules, Bull. Iran Math. Soc. 38 (3), 567–580 (2012).Alijani A., Dehghan M.A. G-frames and their duals for Hilbert Cast -modules, Bull. Iran Math. Soc. 38 (3), 567–580 (2012).Khosravi A., Khosravi B. Fusion frames and g-frames in Hilbert C*-modules, Int. J. Wavelets Multi. Inf. Proc. 6 (3), 433–466 (2008).Khosravi A., Khosravi B. Fusion frames and g-frames in Hilbert C*-modules, Int. J. Wavelets Multi. Inf. Proc. 6 (3), 433–466 (2008).Xiao X.C., Zeng X.M. Some properties of g-frames in Hilbert Cast -modules, J. Math. Anal. Appl. 363 (2), 399–408 (2010).Xiao X.C., Zeng X.M. Some properties of g-frames in Hilbert Cast -modules, J. Math. Anal. Appl. 363 (2), 399–408 (2010).Najati A., Saem M.M., Gavruta P. Frames and operators in Hilbert Cast -modules, Operator Matr. 10, 73–81 (2016).Najati A., Saem M.M., Gavruta P. Frames and operators in Hilbert Cast -modules, Operator Matr. 10, 73–81 (2016).Lance E.C. Hilbert Cast -modules: A Toolkit for operator algebraists, London Math. Soc. Lect. Note Ser. 210 (Cambridge Univ. Press, 1995).Lance E.C. Hilbert Cast -modules: A Toolkit for operator algebraists, London Math. Soc. Lect. Note Ser. 210 (Cambridge Univ. Press, 1995).Baradaran J., Cheshmavar J., Nikahd F., Ghorbani Z. Some properties of c-K-g-frames in Hilbert Cast -modules, Linear and MultiLinear Algebra 71 (8), 1323–1337 (2022).Baradaran J., Cheshmavar J., Nikahd F., Ghorbani Z. Some properties of c-K-g-frames in Hilbert Cast -modules, Linear and MultiLinear Algebra 71 (8), 1323–1337 (2022).Fang X.C., Yu J., Yao H.L. Solutions to operator equations on Hilbert Cast -modules, Linear Algebra Appl. 431 (11), 2142–2153 (2009).Fang X.C., Yu J., Yao H.L. Solutions to operator equations on Hilbert Cast -modules, Linear Algebra Appl. 431 (11), 2142–2153 (2009).Rashidi Kouchi M., Nazari A. Continuous g-frame in Hilbert Cast -modules (Hindawi Publishig Corporation, 2011).Rashidi Kouchi M., Nazari A. Continuous g-frame in Hilbert Cast -modules (Hindawi Publishig Corporation, 2011).G˘avruta L. Frames for operators, Appl. Comput. Harmon. Anal. 32 (1), 139–144 (2012).G˘avruta L. Frames for operators, Appl. Comput. Harmon. Anal. 32 (1), 139–144 (2012).Xiao X.C., Zhu Y.C., Gavruta L. Some properties of exact K-frames in Hilbert spaces, Result Math. 63 (3-4), 1243–1255 (2013).Xiao X.C., Zhu Y.C., Gavruta L. Some properties of exact K-frames in Hilbert spaces, Result Math. 63 (3-4), 1243–1255 (2013).

The authors declare that there are no conflicts of interest present.