Mathematics and Statistics Vol. 11(4), pp. 703 - 709
DOI: 10.13189/ms.2023.110412
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Self-Adjoint Operators in Bilinear Spaces

Sabarinsyah ^1,*, Hanni Garminia ², Pudji Astuti ², Zelvin Mutiara Leastari ^1
1 Department of Mathematics, Institut Teknologi Batam, Indonesia
² Algebra Research Division, Faculty of Mathematics and Natural Sciences, Institut Teknologi Bandung, Indonesia

ABSTRACT

In this research, it was agreed that a bilinear form is an extension of the inner product since a symmetry bilinear form will be equivalent to the inner product over a field of real numbers. Concepts in bilinear space, such as the concept of orthogonality of two vectors, the concept of orthogonal subspace of a subspace, the concept of adjoint operators of a linear operator and the concept of closed subspace are defined according to those prevailing in the inner product space fact assumed to be extensions of the concepts applicable in the inner product space. In the context of a cap subspace, we can identify the necessary and sufficient conditions for any linear operator in a continuous Hilbert space. These results open up opportunities to introduce the concept of pseudo-continuous linear mapping in bilinear spaces. We have obtained the result that pseudo-continuous linear mapping spaces in bilinear spaces have a relationship with linear mapping spaces that have adjoint mapping. We have also obtained the result that the structure of linear operators limited to Hilbert spaces can be extended to pseudo-continuous operator structures in bilinearal spaces. In this study, we have generalized the properties of self-adjoint operators in product spaces in infinite dimensions to bilinear, including closed properties of addition operations, and scalar multiplication, commutative properties, properties of inverse operators, properties of zero operators, properties of polynomial operators over real fields, and orthogonal properties of eigenspaces of different eigenvalues.

KEYWORDS
Self-Adjoint Operator, Non-Degenerated Bilinear Forms, Pseudo-Continuity

Cite This Paper in IEEE or APA Citation Styles
(a). IEEE Format:
[1] Sabarinsyah , Hanni Garminia , Pudji Astuti , Zelvin Mutiara Leastari , "Self-Adjoint Operators in Bilinear Spaces," Mathematics and Statistics, Vol. 11, No. 4, pp. 703 - 709, 2023. DOI: 10.13189/ms.2023.110412.

(b). APA Format:
Sabarinsyah , Hanni Garminia , Pudji Astuti , Zelvin Mutiara Leastari (2023). Self-Adjoint Operators in Bilinear Spaces. Mathematics and Statistics, 11(4), 703 - 709. DOI: 10.13189/ms.2023.110412.