On The Adjoint of Bounded Operators On A Semi-Inner Product Space

R. Respitawulan; Qori Y.  Pangestu; Elvira Kusniyanti; Fajar Yuliawan; Pudji  Astuti

doi:10.22342/jims.29.3.1598.311-321

R. Respitawulan ⁽¹⁾ , Qori Y. Pangestu ⁽²⁾ , Elvira Kusniyanti ⁽³⁾ , Fajar Yuliawan ⁽⁴⁾ , Pudji Astuti ⁽⁵⁾

(1) Algebra Research Group, Faculty of Mathematics and Natural Sciences ITB, Indonesia,

(2) Algebra Research Group, Faculty of Mathematics and Natural Sciences ITB, Indonesia,

(3) Algebra Research Group, Institut Teknologi Bandung, Indonesia,

(4) Algebra Research Group, Faculty of Mathematics and Natural Sciences ITB, Indonesia,

(5) Algebra Research Group, Faculty of Mathematics and Natural Sciences ITB, Indonesia

https://doi.org/10.22342/jims.29.3.1598.311-321

Issue
VOLUME 29 NUMBER 3 (NOVEMBER 2023)

Submitted
December 8, 2023

Published
November 30, 2023

Keywords:

semi-inner product space, adjoint, bounded operator, Riesz representation theorem

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Abstract

The notion of semi-inner product (SIP) spaces is a generalization of inner product (IP) spaces notion by reducing the positive definite property of the product to positive semi-definite. As in IP spaces, the existence of an adjoint of a linear operator on a SIP space is guaranteed when the operator is bounded. However, in contrast, a bounded linear operator on SIP space can have more than one adjoint linear operators. In this article we give an alternative proof of those results using the generalized Riesz Representation Theorem in SIP space. Further, the description of all adjoint operators of a bounded linear operator in SIP space is identified.

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References

Arias, M. L., Corach, G., and Gonzalez, M. C., ”Partial isometries in semi-Hilbertian spaces. Linear Algebra and its Applications”, 428(7) (2008), 1460-1475.

Arias, M. L., Corach, G., and Gonzalez, M. C., ”Metric properties of projections in semiHilbertian spaces”, Integral Equations and Operator Theory, 62(1) (2008), 11-28.

Ahmed, O. A. M. S., and Saddi, A., ”A-m-Isometric operators in semi-Hilbertian spaces”, Lin. Algebra and its appl., 436(10) (2012), 3930-3942.

Baklouti, H., and Namouri, S., ”Closed operator in semi-Hilbertian spaces”, Lin. and Multilinear Algebra, 70(20) (2022) 5847-5858.

Benali, A., and Mahmoud, S.A.O.A., ”(α, β)-A-Normal operators in semi-Hilbertian spaces”, Afrika Matematika, 30(5) (2019), 903-920.

Bhunia, P., Kittaneh, F., Paul, K., and Sen, A. ”Anderson’s theorem and A-spectral radius bounds for semi-Hilbertian space operators.” Lin. Alg. and its Appl., 657 (2023), 147-162.

Bovdi, V.A., Klymchuk, T., Rybalkina, T., Salima,M.A., and Sergeichuk, V.V., ”Operators on positive semi-definite inner product spaces”, Lin. Alg. and its Appl., 596 (2020),82-105.

Kittaneh, F., and Zamani,A., ”A refinement of A-Buzano inequality and applications to A-numerical radius inequalities”, Lin. Alg. and its Appl., (2023).

Krein, M. G., ”Compact linear operators on functional spaces with two norms”, Integral Equations and Operator Theory, 30(2) (1998), 140-162.

Promislow, S. D., A first course in functional analysis, Wiley-Interscience, 2008.

Saddi, A., ”A-Normal operators in Semi-Hilbertian spaces”, The Australian J. of Math. Anal. and Appl., 9(1) (2012), 1-12.

Sen, J., Sain, D., and Paul, K., ”Orthogonality and norm attainment of operators in semiHilbertian space”, Ann. of Funct. Anal., 12(1) (2021), 1-12.

Sen, J., and Paul, K., ”(ε, A)-approximate numerical radius orthogonality and numerical radius derivative”, Mathematica Slovaca 73(1) (2023), 147-158.

Sun W., ”Generalized inverse solution for general quadratic programming”, Num. Math. J. of Chinese Univ., (1991) 94-98.

Tam, T.Y and Zhang, P., Spectral decomposition of selfadjoint matrices in positive semidefinite inner product spaces and its applications, Lin. and Multilinear Algebra, 67(9)(2019),1829-1838.

Zaanen, A.C., ”Normalisable transformations in Hilbert space and systems of linear integral equations”, Acta Math., 83 (1950), 197–248.

Zamani, A., ”Birkhoff-James orthogonality of operators in semi-Hilbertian spaces and its applications”, Ann. of Funct. Anal., 10(3) (2019), 433-445

Authors

R. Respitawulan

Algebra Research Group, Faculty of Mathematics and Natural Sciences ITB

Qori Y. Pangestu

Algebra Research Group, Faculty of Mathematics and Natural Sciences ITB

Elvira Kusniyanti

Algebra Research Group, Institut Teknologi Bandung

Fajar Yuliawan

Algebra Research Group, Faculty of Mathematics and Natural Sciences ITB

yuliawan@office.itb.ac.id (Primary Contact)

Pudji Astuti

Algebra Research Group, Faculty of Mathematics and Natural Sciences ITB

Respitawulan, R., Pangestu, Q. Y. ., Kusniyanti, E., Yuliawan, F., & Astuti, P. . (2023). On The Adjoint of Bounded Operators On A Semi-Inner Product Space. Journal of the Indonesian Mathematical Society, 29(3), 311–321. https://doi.org/10.22342/jims.29.3.1598.311-321