On Fuzzy Soft Linear Operators in Fuzzy Soft Hilbert Spaces

Abstract

Due to the difficulty of representing problem parameters fuzziness using the soft set theory, the fuzzy soft set is regarded to be more general and flexible than using the soft set. In this paper, we define the fuzzy soft linear operator $\tilde{T}$ in the fuzzy soft Hilbert space $\tilde{H}$ based on the definition of the fuzzy soft inner product space in terms of the fuzzy soft vector ${\tilde{v}}_{f_{G\left(e\right)}}$ modified in our work. Moreover, it is shown that ${ℂ}^n\left(A\right),{ℝ}^n\left(A\right)$ and $l_2\left(A\right)$ are suitable examples of fuzzy soft Hilbert spaces and also some related examples, properties and results of fuzzy soft linear operators are introduced with proofs. In addition, we present the definition of the fuzzy soft orthogonal family and the fuzzy soft orthonormal family and introduce examples satisfying them. Furthermore, the fuzzy soft resolvent set, the fuzzy soft spectral radius, the fuzzy soft spectrum with its different types of fuzzy soft linear operators and the relations between those types are introduced. Moreover, the fuzzy soft right shift operator and the fuzzy soft left shift operator are defined with an example of each type on ${𝓁}_2\left(A\right)$.In addition, it is proved, on ${𝓁}_2\left(A\right)$, that the fuzzy soft point spectrum of fuzzy soft right shift operator has no fuzzy soft eigenvalues, the fuzzy soft residual spectrum of fuzzy soft right shift operator is equal to the fuzzy soft comparison spectrum of it and the fuzzy soft point spectrum of fuzzy soft left shift operator is the fuzzy soft open disk $\left|\widetilde{\text{λ}}\right|\widetilde{<}\tilde{1}$. Finally, it is shown that the fuzzy soft Hilbert space is fuzzy soft self-dual in this generalized setting.