On (𝒌, 𝒎) − 𝒏 − Paranormal Operators

The new type of paranormal operators that have been defined in this study on the Hilbert space, is (𝑘, 𝑚) − 𝑛 − paranormal operators. In this paper we introduce and discuss some properties of this concept such as: the sum and product of two (𝑘, 𝑚) − 𝑛 − paranormal, the power of (𝑘, 𝑚) − 𝑛 − paranormal. Further, the relationships between the (𝐾, 𝑚) − 𝑛 − paranormal operators and other kinds of paranormal operators have been studied.


Introduction
Many authors in the current study in recent years focus own studied in special type of bounded linear operator named paranormal operators and begging give some generalized of this type of operator, such as, Arora and Thukral [1]. In 1987 also defined the class M*-paranormal and introduced topological properties with characterization of this concept. In 2012 the class K-quasi-paranormal with some basic properties and characterization has been introduced by Mecher, Salah [2]. Zuo, Fei in 2015 introduced another class which named quasi-*-nparanormal and studied spectrum properties [3]. In add, k-quasi-*-paranormal was presented with it's properties by Rashid in 2016 [4] . In this paper, we shall introduce a new class allegedly operates as a (k, m)-n-paranormal, which is generalization of paranormal operator, also giving some theorems and illustrate some examples for its properties such as: the sum and product of two (k, m)-n-paranormal, the power of (k, m)-n-paranormal , also the tensor product between (k, m)-n-paranormal and the identity operator, alongside that we explain relationships between the (K, n)-n-paranormal operators and other kinds of paranormal operators.

Main results Definition 3.1:
For any positive real number and any positive integers and , the bounded linear operator : → is called (k,m)-n-paranormal in that case: By using Definition 3.1 " ‖ ‖ 1+ ≤ ‖ ( )‖‖ ‖ for all ∈ " we may obtain, , And by the generalized arithmetic-geometric mean inequality: By the generalized arithmetic-geometric mean inequality, we get Now, we illustrate the relation among (k, m)-n-paranormal with another classes that appeared in definition 2.
is (k, m)-n-paranormal, is real number ii.
is (k, m)-n-paranormal, where is positive integer number.

Remark 3.4:
The operators and should be two (k, m)-n-paranormal operators, however, + is not always a (k,m)-n-paranormal operator, consider the example that follows  ≥ 0 Therefore, is (k, m)-n-paranormal. Now, we will point the adjoint of (k, m)-n-paranormal and explain that by example

Remark and example 3.8:
If is (k,m)-n-paranormal, then * need not likewise be (k,m)-n-paranormal. To assist illustrate that, consider the following example:  ≥ 0, so we get, ⊗ is (k, m)-n-paranormal . By same way, we can prove that ⊗ is (k, m)-n-paranormal.