Convergence of iterated Aluthge transform sequence for diagonalizable matrices II: λ-Aluthge transform
نویسندگان
چکیده
Let λ ∈ (0, 1) and let T be a r × r complex matrix with polar decomposition T = U |T |. Then, the λAluthge transform is defined by ∆λ (T ) = |T | U |T |. Let ∆nλ(T ) denote the n-times iterated Aluthge transform of T , n ∈ N. We prove that the sequence {∆nλ(T )}n∈N converges for every r × r diagonalizable matrix T . We show regularity results for the two parameter map (λ, T ) 7→ ∆∞λ (T ), and we study for which matrices the map (0, 1) ∋ λ 7→ ∆∞λ (T ) is constant. Partially supported by CONICET (PIP 4463/96), Universidad de La Plata (UNLP 11 X472) and ANPCYT (PICT03-09521). Partially supported by CNPq 1
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Convergence of iterated Aluthge transform sequence for diagonalizable matrices
Given an r × r complex matrix T , if T = U |T | is the polar decomposition of T , then, the Aluthge transform is defined by ∆ (T ) = |T |U |T |. Let ∆n(T ) denote the n-times iterated Aluthge transform of T , i.e. ∆0(T ) = T and ∆n(T ) = ∆(∆n−1(T )), n ∈ N. We prove that the sequence {∆n(T )}n∈N converges for every r× r diagonalizable matrix T . We show that the limit ∆∞(·) is a map of class C∞...
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