On N-A-Metrically Equivalent and A-Metrically Equivalent Operators

Abstract

A-metrically equivalent operators may be regarded as a generalization of metrically equivalent operators. This is realized when A= I and T ]= T ∗ . Definition 1.1. Two operators S ∈ BA(H) and T ∈ BA(H) are said to be: (1). A-metrically equivalent, denoted by S ∼A-m T, provided T ]A T = S ]A S equivalently; k T ξ kA=k Sξ kA ∀ ξ ∈ H. T ]A = A †T ∗A, in which A † is the Moore-penrose inverse of A. (2). n-A-metrically equivalent, denoted by S ∼n-A-m T, provided T ]A T n = S ]A S n for a positive integer n. Definition 1.2. An operator T ∈ B(H) is (1). A-Contraction if k T ξ kA≤k ξ kA for every ξ ∈ H ⇔ T ∗AT ≤ A. (2). A-Isometry if T ∗AT = A ⇔k T ξ kA=k ξ kA for every ξ ∈ H. (3). A-Unitary if T ∗AT = T AT ∗ = A ⇔k T ∗ ξ kA=k T ξ kA=k ξ kA for every ξ ∈ H. (4). A-Normal if T ∗AT = T AT ∗ ⇔k T ξ kA=k T ∗ ξ kA for every ξ ∈ H. (5). A-Partial isometry if k T ξ kA=k ξ kA for every ξ ∈ N(AT) ⊥A .