Polar decompositions and spectral properties of linear operator pencils

In this paper, we introduce the spherical polar decomposition of the linear pencil of an ordered pair \(\textbf{T}= (T_{1},T_{2})\) $\mathbf{T}=(T_1,T_2)$ . Using this decomposition, we define the generalized spherical Aluthge transforms for the linear pencil of an ordered pair \(\textbf{T}= (T_{1},T_{2})\) $\mathbf{T}=(T_1,T_2)$ and give spectral properties between the linear pencil and the generalized spherical Aluthge transform of it. In detail, we study whether the spectral properties of the linear pencil of an ordered pair \(\textbf{T}= (T_{1},T_{2})\) $\mathbf{T}=(T_1,T_2)$ are invariant under the generalized spherical Aluthge transform.