| | | |

## Necessary Condition for Vector-Valued Model Spaces to be Invariant Under Conjugation

#### Rewayat KHAN [1] , Jamroz KHAN [2]

The $S^{*}$-invariant subspaces of the Hardy-Hilbert space $H^{2}(E)$ (where $E$ is finite dimensional Hilbert space of dimension greater than 1) on the unit disc is well known. In this study, we examine that, if $\Omega$ is a conjugation on $E$, and $\Theta$ an inner function, then there exist model spaces which are not invariant for the conjugation $C_{\Omega}:L^{2}(E)\longrightarrow L^{2}(E)$. Under what necessary condition the model spaces is mapped onto itself is under consideration.

Inner function, model spaces, conjugation
• [1] J. A. Cima, W. T. Ross, The backward shift on the Hardy space. Mathematical Surveys and Monograps, Volume 79. Amer. Math. Soc. Providence, RI, 2000. MR 1761913.
• [2] R. G. Douglas, Banach algebra techniques in operator theory , Graduate Texts in Mathematics, Volume 179, Springer-Verlag, New York, 1998.
• [3] E. Fricain, J. Mashreghi, The theory of H(b) spaces. Volume 2, New Mathematical Monographs, Volume 21, Cambridge University Press, Cambridge, 2015.
• [4] S. R. Garcia and M. Putinar, Complex symmetric operators and applications, Trans. Amer. Math. Soc. 358 (2006), 1285–1315.
• [5] S. R. Garcia, Conjugation and Clark operators. Recent advances in operator-related function theory, Contemp. Math, Volume 393, Amer. Math. Soc Providence, RI, 2006, pp. 67-111. MR 2198373 (2007b:47073).
• [6] S. R. Garcia, J. Mashreghi, W.T. Ross, Introduction to model spaces and their operators. Cambridge: Cambridge University Press. 2016.
• [7] S. R. Garcia, Conjugation, the backward shift, and Toeplitz kernel. J. Operator Theory 54/2 (2005) 239-250. MR 2186351.
Primary Language en Mathematics Research Article Author: Rewayat KHAN Author: Jamroz KHAN Publication Date : May 7, 2019
 Bibtex @research article { jnt561330, journal = {Journal of New Theory}, issn = {2149-1402}, eissn = {2149-1402}, address = {Mathematics Department, Gaziosmanpasa University 60250 Tokat-TURKEY.}, publisher = {Gaziosmanpasa University}, year = {2019}, volume = {}, pages = {1 - 4}, doi = {}, title = {Necessary Condition for Vector-Valued Model Spaces to be Invariant Under Conjugation}, key = {cite}, author = {KHAN, Rewayat and KHAN, Jamroz} } APA KHAN, R , KHAN, J . (2019). Necessary Condition for Vector-Valued Model Spaces to be Invariant Under Conjugation. Journal of New Theory , (28) , 1-4 . Retrieved from https://dergipark.org.tr/en/pub/jnt/issue/45019/561330 MLA KHAN, R , KHAN, J . "Necessary Condition for Vector-Valued Model Spaces to be Invariant Under Conjugation". Journal of New Theory (2019 ): 1-4 Chicago KHAN, R , KHAN, J . "Necessary Condition for Vector-Valued Model Spaces to be Invariant Under Conjugation". Journal of New Theory (2019 ): 1-4 RIS TY - JOUR T1 - Necessary Condition for Vector-Valued Model Spaces to be Invariant Under Conjugation AU - Rewayat KHAN , Jamroz KHAN Y1 - 2019 PY - 2019 N1 - DO - T2 - Journal of New Theory JF - Journal JO - JOR SP - 1 EP - 4 VL - IS - 28 SN - 2149-1402-2149-1402 M3 - UR - Y2 - 2020 ER - EndNote %0 Journal of New Theory Necessary Condition for Vector-Valued Model Spaces to be Invariant Under Conjugation %A Rewayat KHAN , Jamroz KHAN %T Necessary Condition for Vector-Valued Model Spaces to be Invariant Under Conjugation %D 2019 %J Journal of New Theory %P 2149-1402-2149-1402 %V %N 28 %R %U ISNAD KHAN, Rewayat , KHAN, Jamroz . "Necessary Condition for Vector-Valued Model Spaces to be Invariant Under Conjugation". Journal of New Theory / 28 (May 2019): 1-4 . AMA KHAN R , KHAN J . Necessary Condition for Vector-Valued Model Spaces to be Invariant Under Conjugation. JNT. 2019; (28): 1-4. Vancouver KHAN R , KHAN J . Necessary Condition for Vector-Valued Model Spaces to be Invariant Under Conjugation. Journal of New Theory. 2019; (28): 4-1.

Authors of the Article