In the present paper, we study frames associated with an operator ($ \mathcal{W} $-frames) in Krein spaces, and we give the definition of frames associated with an operator depending on the adjoint of the operator in the Krein space (Definition 4.1). We prove that the definition given in [A. Mohammed, K. Samir, N. Bounader, K-frames for Krein spaces, Ann. Funct. Anal., 14 (2023), 10.], which depends on the adjoint of the operator in the associated Hilbert space, is a consequence of our definition. We prove that our definition is independent of the fundamental decomposition (Theorem 4.1) and that having $ \mathcal{W} $-frames for the Krein space necessarily gives $ \mathcal{W} $-frames for the Hilbert spaces that compose the Krein space (Theorem 4.4). We also prove that orthogonal projectors generate new operators with their respective frames (Theorem 4.2). We prove an equivalence theorem for $ \mathcal{W} $-frames (Theorem 4.3), without depending on the fundamental symmetry as usually given in Hilbert spaces.
Citation: Osmin Ferrer Villar, Jesús Domínguez Acosta, Edilberto Arroyo Ortiz. Frames associated with an operator in spaces with an indefinite metric[J]. AIMS Mathematics, 2023, 8(7): 15712-15722. doi: 10.3934/math.2023802
In the present paper, we study frames associated with an operator ($ \mathcal{W} $-frames) in Krein spaces, and we give the definition of frames associated with an operator depending on the adjoint of the operator in the Krein space (Definition 4.1). We prove that the definition given in [A. Mohammed, K. Samir, N. Bounader, K-frames for Krein spaces,
| [1] | T. Azizov, I. S. Iokhvidov, Linear operators in spaces with an indefinite metric, John Wiley & Sons, 1989. |
| [2] | J. Bognár, Indefinite inner product spaces, Berlin: Springer, 1974. |
| [3] | O. Christensen, An introduction to frames and riesz bases, Boston: Birkhäuser, 2003. |
| [4] |
I. Daubechies, A. Grossmann, Y. Meyer, Painless nonorthogonal expansions, J. Math. Phys., 27 (1986), 1271–1283. https://doi.org/10.1063/1.527388 doi: 10.1063/1.527388
|
| [5] |
I. Daubechies, The wavelet transform, time-frequency localization and signal analysis, IEEE T. Inform. Theory, 36 (1990), 961–1005. https://doi.org/10.1109/18.57199 doi: 10.1109/18.57199
|
| [6] |
R. Douglas, On majorization, factorization and range inclusion of operators on Hilbert space, Proc. Amer. Math. Soc., 17 (1966), 413–415. https://doi.org/10.2307/2035178 doi: 10.2307/2035178
|
| [7] |
R. Duffin, A. Schaeffer, A class of nonharmonic Fourier series, Trans. Amer. Math. Soc., 72 (1952), 341–366. https://doi.org/10.1090/S0002-9947-1952-0047179-6 doi: 10.1090/S0002-9947-1952-0047179-6
|
| [8] |
K. Esmeral, O. Ferrer, E. Wagner, Frames in Krein spaces arising from a non-regular W-metric, Banach J. Math. Anal., 9 (2015), 1–16. http://doi.org/10.15352/bjma/09-1-1 doi: 10.15352/bjma/09-1-1
|
| [9] | L. GăvruÅ£a, Frames for operators, Appl. Comput. Harmon. A., 32 (2012), 139–144. https://doi.org/10.1016/j.acha.2011.07.006 |
| [10] | K. Gröchenig, Foundations of time-frequency analysis, Springer Science & Business Media, 2001. |
| [11] | A. Mohammed, K. Samir, N. Bounader, K-frames for Krein spaces, Ann. Funct. Anal., 14 (2023), 10. https://doi.org/10.1007/s43034-022-00223-3 |
Osmin Ferrer Villar, Jesús Domínguez Acosta, Edilberto Arroyo Ortiz. Frames associated with an operator in spaces with an indefinite metric[J]. AIMS Mathematics, 2023, 8(7): 15712-15722. doi: 10.3934/math.2023802
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