Anti-G-hermiticity preserving linear maps that preserve strongly the invertibility of Hilbert space operators

• Main Author: Buscano, Jay D.
• Format: text
• Language: English
• Published: Animo Repository 2017
• Subjects: Linear operators; Mathematics
• Online Access: https://animorepository.dlsu.edu.ph/etd_doctoral/509
• Institution: De La Salle University

A linear map : A ! B of algebras A and B preserves strong invertibility if (x{u100000}1) = (x){u100000}1 for all x 2 A{u100000}1, where A{u100000}1 denotes the set of invertible elements of A. Let B(H) be the Banach algebra of all bounded linear operators on a separable complex Hilbert space H with dimH = 1. An operator U 2 B(H) is said to be anti-G-Hermitian if U] = {u100000}U, where U] denotes the G-adjoint of U. A linear map : B(H) ! B(H) preserves anti-G-Hermiticity if (U)] = {u100000} (U) for every anti-G- Hermitian operator U on H. In this paper, we characterize a continuous unital linear map : B(H) ! B(H) that preserves anti-G-Hermiticity and preserves strongly the invertibility of Hilbert space operators. The discussion is in the context of G- operators, that is, linear operators on H with respect to a xed but arbitrary positive de nite Hermitian operator G 2 B(H){u100000}1. From the Hilbert space H with an inner product h i, we consider a new inner product [ ] in H such that [x y] = hGx yi for all x y 2 H. We present a discussion of operators on (H [ ]) analogous to the discussion of operators on (H h i). The discussion of operators on (H [ ]) will be extended to the quotient algebra of B(H) by the ideal K(H) of compact operators on H, which is known as the Calkin algebra C(H) of operators on H. We present the anti-G-Hermiticity and invertibility preserving properties of the canonical map : B(H) ! C(H). We then introduce vii the continuous unital linear map : C(H) ! C(H) induced by the linear map : B(H) ! B(H) which preserves essentially anti-G-Hermiticity and preserves strongly the invertibility of operators on H. We also take a look at the preserving properties and the characterization of the induced map.