Linear maps on operator algebras that preserve elements annihilated by a polynomial

Abstract

Abstract. In this paper some purely algebraic results are given concerning linear maps on algebras which preserve elements annihilated by a polynomial of degree greater than 1 and with no repeated roots and applied to linear maps on operator algebras such as standard operator algebras, von Neumann algebras and Banach algebras. Several results are obtained that characterize such linear maps in terms of homomorphisms, anti-homomorphisms, or, at least, Jordan homomorphisms.

Linear K-Power Preservers and Trace of Power-Product Preservers

“…Watkins in [20], and by C.-K. Li and S. Pierce in [21], on BH ðÞ by P. Šemrl [22], on linear maps ψ : BH ðÞ ! BK ðÞ by Z. Bai and J. Hou in [23], and on some operator algebras by J. Hou and S. Hou in [24]. We may further explore linear f -preservers for a multivariable function fx 1 , … , x r ðÞ , that is, operator ψ satisfying ψ fA 1 , … , A r ðÞ ðÞ ¼ f ψ A 1 ðÞ , … , ψ A r ðÞ ðÞ .…”

“…V corresponds to a unique linear bijection ψ : V ! V that makes (24) hold. The next natural thing is to determine ϕ and ψ that satisfy tr ϕ A ðÞ ψ B ðÞ k ¼ tr AB k ÀÁ for a fixed k ∈ n 0, 1 fg :…”

Spectral Analysis of the Bounded Linear Operator in the Reproducing Kernel SpaceW2m(D)

“…This method has been successfully applied in Hilbert space and perfect spectral decomposition theorem [ 3 ]. In recent years, the spectral decomposition method has been developed into spectral theorems of spectral operators and decomposable operators in Banach space [ 4 , 5 ].…”

Preserving Zeros of a Polynomial

“…, x m , were studied in [5]. Linear maps that preserve operators on infinitely dimensional algebras annihilated by a polynomial in one variable, were treated in [25,32]. Bijective linear operators on the matrix algebra preserving zeros of the involutory polynomial p(x, y) = xy − yx * are classified in [14].…”