Weak $A$-frames and weak $A$-semi-frames

Yıl 2021, Cilt: 4 Sayı: 1, 104 - 118, 01.03.2021

Jean Pierre Antoıne , Giorgia Bellomonte , Camillo Trapanı

https://doi.org/10.33205/cma.835582

Cited By: 1

Öz

After reviewing the interplay between frames and lower semi-frames, we introduce the notion of lower semi-frame controlled by a densely defined operator $A$ or, for short, a weak lower $A$-semi-frame and we study its properties. In particular, we compare it with that of lower atomic systems, introduced in (GB). We discuss duality properties and we suggest several possible definitions for weak $A$-upper semi-frames. Concrete examples are presented.

Anahtar Kelimeler

$A$-frames, weak (upper and lower) $A$-semi-frames, lower atomic systems, $G$-duality

Kaynakça

• S. T. Ali, J. P. Antoine and J. P. Gazeau: Continuous Frames in Hilbert Space, Ann. Physics, 222 (1993), 1–37.
• J-P. Antoine, C. Trapani: Partial Inner Product Spaces: Theory and Applications, Lecture Notes in Mathematics, vol. 1986, Springer, Berlin (2009).
• J-P. Antoine, P. Balazs: Frames and semi-frames, J. Phys. A: Math. Theor., 44 (2011), 205201.
• J-P. Antoine, C. Trapani: Partial inner product spaces, metric operators and generalized hermiticity, J. Phys. A: Math. Theor., 46 (2013), 025204; Corrigendum, ibid. 46 (2013), 329501.
• J-P. Antoine, C. Trapani: Operator (quasi-)similarity, quasi-Hermitian operators and all that, Non-Hermitian Hamiltonians in Quantum Physics, pp. 45–65; F.Bagarello, R. Passante , C.Trapani, (eds.), Springer Proceedings in Physics, vol. 184, Springer Int. Publ. Switzerland (2016).
• J-P. Antoine, C. Trapani: Reproducing pairs of measurable functions and partial inner product spaces, Adv. Operator Th., 2 (2017), 126–146.
• J-P. Antoine, C. Trapani: Beyond frames: Semi-frames and reproducing pairs, Mathematical Structures and Applications. STEAM-H: Science, Technology, Engineering, Agriculture, Mathematics & Health, T. Diagana, B. Toni (eds), Springer, Cham (2018).
• J-P. Antoine, R. Corso and C. Trapani: Lower semi-frames and metric operators, Mediterranean J. Math., 18 (2021), 11.
• P. Balazs, J-P. Antoine and A. Gryboś: Weighted and controlled frames: Mutual relationship and first numerical properties, Int. J. Wavelets, Multires. and Inform. Proc., 8 (2010), 109–132.
• G. Bellomonte: Continuous frames for unbounded operators, arXiv:1912.13097 [math.FA], submitted.
• G. Bellomonte, R. Corso: Frames and weak frames for unbounded operators, Advances in Computational Mathematics, 46 (2020), art. n. 38, 21 pp.
• J. Bergh, J. Löfström: Interpolation Spaces. Springer, Berlin (1976).
• R. Corso: Sesquilinear forms associated to sequences on Hilbert spaces, Monatsh. Math., 189 (2019), 625–650.
• L. Gâvruţa: Frames for operators, Appl. Comput. Harmon. Anal., 32 (2012), 139–144.
• G. Kaiser:A Friendly Guide to Wavelets, Birkhäuser, Boston (1994).
• T. Kato: Perturbation Theory for Linear Operators, Springer, New York (1966).
• W. Rudin: Functional Analysis, McGraw-Hill , New York-Düsseldorf-Johannesburg (1973).
• K. Schmüdgen: Unbounded Self-adjoint Operators on Hilbert Space, Springer, Dordrecht (2012).
• A.A. Zakharova: On the properties of generalized frames, Math. Notes, 83 (2008), 190–200.