## 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 .

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